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The Mathematics Companion: Mathematical Methods for Physicists and Engineers
Preview
Summary
Following the style of The Physics Companion and The Electronics Companion, this book is a revision aid and study guide for undergraduate students in physics and engineering. It consists of a series of one-page-per-topic descriptions of the key concepts covered in a typical first-year "mathematics for physics" course. The emphasis is placed on relating the mathematical principles being introduced to real-life physical problems. In common with the other companions, there is strong use of figures throughout to help in understanding of the concepts under consideration. The book will be an essential reference and revision guide, particularly for those students who do not have a strong background in mathematics when beginning their degree.
Reviews
"This is an interesting and useful little book… .it is very well done, and everything that might be expected to be there is there… .The book might also be invaluable for those undergraduate students in Mathematics, Science or Engineering, who need to undertake first and second year courses in Mathematics, and it will serve those who wish to have quick access to all those formulae that seem to be so readily forgotten." -J M Hill, Australian Physics, Vol. 43, No. 1, March/April 2006
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PUBLISHED
PRODUCT TYPE
2,116Book
2Subscription
1Training
Mathematics
Mathematics, "The Queen of Sciences" as called by Carl Friedrich Gauss, is the science of number, quantity, and space, either as abstract concepts or as applied to other disciplines (such as physics and engineering).
The distinguished authors of the top-quality books and textbooks listed under Research and Markets' Mathematics category are the world's leading researchers. These publications cover all the key areas in today's research. They are invaluable references, comprehensive and
readily accessible. When available, pre-publication titles are also included, so you can be sure not to miss the latest developments in your research field.
The readership of this category includes both graduate and undergraduate students, as well as researchers and mature mathematics.
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The second edition of A Course in Real Analysis provides a solid foundation of real analysis concepts and principles, presenting a broad range of topics in a clear and concise manner. The book is excellent...
Six Sigma methodology is a business management strategy which seeks to improve the quality of process output by identifying and removing the causes of errors and minimizing variability in manufacturing...
Practical Text Mining and Statistical Analysis for Non-structured Text Data Applications brings together all the information, tools and methods a professional will need to efficiently use text mining...
Now in its 7th edition, Mathematical Methods for Physicists continues to provide all the mathematical methods that aspiring scientists and engineers are likely to encounter as students and beginning...
This revised book provides a thorough explanation of the foundation of robust methods, incorporating the latest updates on R and S-Plus, robust ANOVA (Analysis of Variance) and regression. It guides...
The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs provides basic logic of mathematical proofs and shows how mathematical proofs work. It offers techniques for both reading and writing...
An accessible introduction to the fundamentals of calculus needed to solve current problems in engineering and the physical sciences
I ntegration is an important function of calculus, and Introduction...
February2012
EUR 104USD 127GBP 79AUD 157JPY 15,293SEK 990CHF 125CAD 148
Applied Business Statistics. Making Better Business Decisions. 7th Edition International Student Version
Black's latest outstanding pedagogy of Business Statistics includes the use of extra problems called "Demonstration Problems" to provide additional insight and explanation to working problems, and presents...
Set theory is an autonomous and sophisticated field of mathematics that is extremely successful at analyzing mathematical propositions and gauging their consistency strength. It is as a field of mathematics...
An insightful guide to understanding and visualizing multivariate statistics using SAS®, STATA®, and SPSS®
Multivariate Analysis for the Biobehavioral and Social Sciences: A Graphical Approach outlines...
Philosophy of Linguistics investigates the foundational concepts and methods of linguistics, the scientific study of human language. This groundbreaking collection, the most thorough treatment of the...
Symbolic analyzers have the potential to offer knowledge to sophomores as well as practitioners of analog circuit design. Actually, they are an essential complement to numerical simulators, since they...
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This book is indeed concise, as it manages to cover all the ideas of elementary number theory in a mere 91 pages. The proofs are also concise, leaving out many details, and this is likely to distress students. There are exercises at the end of each chapter; these are mostly straightforward applications of the ideas in the chapter.
The coverage is not well-balanced, with nearly two-thirds of the book being devoted to diophantine equations and diophantine approximation. A recent concise but less austere book with broader coverage is Underwood Dudley's 2009 A Guide to Elementary Number Theory.
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I find this to be a bit funny, because I have seen so many students who have gone through a first year traditional calculus course and can't use it. They can't write a simple integral nor a simple algebraic equation. My son got good grades in calculus, but still didn't seem to understand it. As a result he couldn't apply it. He could do the mechanics, but not much else. What is even worse I have seen students who passed calculus, but still didn't exhibit the ability to do proportional reasoning. And the type of courses they passed were all traditional using traditional texts.
They simply did not understand the concepts behind Calculus. They could sometimes recognize the name of a math principle, but recognizing that they needed to use it did not seem to happen.
John M. Clement Houston, TX
> And my point is simply that since it isn't a comprehensive > assessment of Calculus then it cannot be used to compare the > effectiveness of IE to non IE classes. The only way to make a > useful comparison between IE and non IE is to use a > comprehensive exam. There is a lot more to owning calculus > (or physics) than owning an inventory of basic principles. My > impression of IE is that it works better in first year > terminal classes than traditional methods that are geared > toward full ownership. In other words, if someone is trying > to be a nurse but has to take calculus (because the college > says so) as their last ever math class, then IE might be a > useful alternative to a real calculus class that is geared > towards students that actually need and use calculus. >
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Math Made Easy has helped over 1 million students worldwide significantly improve their math skills.
Description
Students, are you struggling with Probability and Statistics? Then Math Made Easy Probability and Statistics: Part 2 DVD courseware can help you get an A in this difficult course!
This Probability and Statistics: Part 2 tutorial is for students in college math classes and is an indispensable resource for anyone who needs a detailed review to make these complex and often confusing concepts less intimidating and easy to learn and remember. In just 2-6 weeks, you can become proficient in Probability and Statistics. Especially in the days leading up to a math exam, the easy-to-use Probability and Statistics: Part 2 DVD series will help you succeed as it keeps the math material fresh in your mind.
If you have been struggling with Probability and Statistics, then the dynamic and simple to follow live lessons along with the many interactive do-it-yourself exercises in this software will help make this complex math easy for you!
Special Bonus: If you order the Probability and Statistics: Part 2 software from Sam's Club, you will get a free 2 month membership to Tutorial Channel! Once a week for 20 minutes, you will work with a live tutor on any of your math related questions.
Specifications
Title: Probability and Statistics: Part 2
Category: Education
Publisher: Math Made Easy
Clear and comprehensive reviews of this complex math
Easy to understand step by step instruction
Colorful and illustrative graphics including charts, graphs and theorems
Helps to keep up with missed classes and the fast pace in the classroom
Used in thousands of schools and by more than one million students
Compatible with any home, portable or PC or laptop DVD player
In a 2006 survey of students using Math Made Easy and Tutorial Channel:
87% raised next math test score 10 points or more
79% raised their math grade at least one level
59% raised their math grade two or more levels
92% went from failing to passing math
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Computer and video game software must be returned in the original and unopened packaging with a receipt to receive a refund. Defective software must be returned within 90 days of the original purchase with a receipt to receive an exchange of merchandise
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Computer algebra systems are gaining more and more importance in all areas
of science and engineering. This textbook gives a thorough introduction to
the algorithmic basis of the mathematical engine in computer
algebra systems.
It is designed to accompany one- or two-semester courses for advanced
undergraduate or graduate students in computer science or mathematics.
Its comprehensiveness and authority make it also an
essential reference for professionals in the area.
Special features include: detailed study of algorithms including time
analysis; implementation reports on several topics; complete proofs of the
mathematical underpinnings; a wide variety of applications (among others, in
chemistry, coding theory, cryptography, computational logic, and the design
of calendars and musical scales). Some of this material has never appeared
before in book form. Finally, a great deal of historical information and
illustration
enlivens the text.
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...
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This Book
three-dimensional shape that is most structurally stable.
Includes one teacher resource, three posters, and one CD-ROM.
About the Series Math in a Cultural Context
This series is a supplemental math curriculum based on the traditional wisdom and practices of the Yup'ik people of southwest Alaska. The result of more than a decade of collaboration between math educators and Yup'ik elders, these modules connect cultural knowledge to school mathematics. Students are challenged to communicate and think mathematically as they solve inquiry-oriented problems, which require creative, practical and analytical thinking. Classroom-based research strongly suggests that students engaged in this curriculum can develop deeper mathematical understandings than students who engage only with a procedure-oriented, paper-and-pencil
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1991 Hardcover Good 001This very accessible guide offers a thorough introduction to the basics of differential equations and linear algebra. Expertly integrating the two topics, it explains concepts clearly and logically -without sacrificing level or rigor and supports material with a vast array of problems of varying levels for readers to choose from.
Editorial Reviews
Booknews
Introduces material on linear algebra and differential equations required in many sophomore courses for mathematics, science, and engineering majors. Coverage includes first-order differential equations, determinants, vector spaces, the Eigenvalue/Eigenvector problem, the Laplace transform, and series solutions. Appendices offer mathematical reviews. Includes worked examples, exercises, and selected solutions. A majority of exercises require some form of technology for their solution. Assumes completion of three semesters of calculus. This second edition contains a new chapter on second- order linear differential equations, and contains expanded material on slope and direction fields, vector spaces, and other
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Mathematics
What do a Daft Punk song, Call of Duty: Black Ops, and your Facebook page have in common? They're all based on mathematics.
That's right. Math. It's everywhere. And while you don't need to know how to write algorithms for high-end video games, a solid understanding of math is critical to your education and career. In fact, mathematics is one of the cornerstone general course requirements for those seeking an associate's degree.
In the Mathematics program at MCC, you'll learn important math basics, like arithmetic, geometry, and algebra. Our program is also a great for the student who is interested in transferring to a four-year math program.
Mesa Community College provides outstanding transfer and career and technical programs, workforce development, and life-long learning opportunities to residents of the East Valley area of Phoenix, Arizona.
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This course deals with the application of mathematics to problems in engineering and science. Emphasis is placed on the three phases of such an application and on the development of skills necessary to carry out each step: translation of the physical information to a mathematical model; treatment of the model by mathematical methods; and interpretation of the result in physical terms.
I cannot imagine teaching this course without Mathematica. As you can see from the topics covered, much of the material involves intense computation, and Mathematica allows us to concentrate on ideas and techniques, not algorithms. Also with Mathematica one can visualize solutions to determine their "reasonableness." I feel that Mathematica's numeric, symbolic, and graphical capabilities are "tailor made" for this course.
Students at the junior level have little difficulty with Mathematica in this course, as many have used it before, and the notebooks contain all of the necessary commands and syntax. Also, doing a few calculations by hand convinces the students very quickly of the need to learn to use technology
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Browse related Subjects
Mathematics for Economists, a new text for advanced undergraduate and beginning graduate students in economics, is a thoroughly modern treatment of the mathematics that underlies economic theory.Mathematics for Economists, a new text for advanced undergraduate and beginning graduate students in economics, is a thoroughly modern treatment of the mathematics that underlies economic theory Description: Mathematics for Economists, a new text for advanced undergraduate and beginning graduate students in economics, is a thoroughly modern treatment of the mathematics that underlies economic theory. An abundance of applications to current economic analysis, illustrative diagrams, thought-provoking exercises, careful proofs, and a flexible organization-these are the advantages that Mathematics for Economists brings to today's classroom. Contents: Introduction " One-Variable calculus: Foundations " One-Variable Calculus: Applications " One-Variable Calculus: Chain Rule " Exponents and Logarithms " PART II: Linear Algebra " Introduction to linear Algebra " Systems of Linear Equations " Matrix Algebra " Determinants: An Overview " Euclidean Spaces " Linear Independence " PART III: Calculus of several variables " Limits and open sets " Functions of several variables " Calculus of several variables " Implicil functions and their derivatives " PART IV: Optimization " Quadratic froms and definite matrices " Unconstrained optimization " Constrained optimization II " Homogeneous and homothetic functions " Concave and quasiconcave functions " Economic applications PART V: Eigenvalues and dynamics " Eigenvalues and eigenvectors " Ordinary differential equations: Scalar equations " Ordinary differential equations: systems of equations PART VI: Advanced linear algebra " Determinants: The details " Subspaces attached to a metrix " applications of linear independence " PART VII: Advanced analysis " Limits and compact sets " calculus of several variables II " PART VIII: Appendices " Index About the Author: Carl Simon is professor of mathematics at the University of Michigan. He received his Ph.D. from Northwestern University and has taught at the University of California, Berkeley, and the University of North Carolina. He is the recipient of many awards for teaching, including the University of Michigan Faculty Recognition Award and the Excellence in Education Award. Lawrence Blume is professor of economics at Cornell University. He received his Ph.D. from the University of California, Berkeley, and has taught at Harvard University's Kennedy School, the University of Michigan, and the University of Tel Aviv. Target Audience: Graduate & undergraguate student in economics Printed Pages: 956. 15 x 23 cm Hardcover. All text is legible, may contain markings, cover wear, loose/torn pages or staining and much writing. SKU: 9780393957334Good. Hardcover. May include moderately worn cover, writing, markings or slight discoloration. SKU: 9780393957334
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proven system of teaching math for classroom teachers, and for parents who are homeschooling. The easy-to-follow program requires only 20 minutes a day. Short, concise, and self-contained lessons help students to master, maintain and reinforce math skil
Customer Reviews
Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com:
78 reviews
40 of 41 people found the following review helpful
Our girls no longer complained!Feb. 24 2009
By
K. Webster
- Published on Amazon.com
Format: Paperback
Last year in our homeschool, we switched to "Mastering Essential Math Skills: Middle Grade/High School, and it was the best switch we could have made. Both girls were to the point of dreading math, and after changing to this program, they now looked forward to math class and could make sense of it because of the step-by-step instructions. Great for 7th-8th grades. Our 2nd daughter actually moved from 5th grade to 7th grade because this program was available.
67 of 78 people found the following review helpful
Not as advertisedFeb. 17 2007
By
Parent
- Published on Amazon.com
Format: Paperback
Verified Purchase
I loved the grade 6-8 edition of this book for reviewing math with my daughter. It is an excellent step by step approach to middle school math. This edition is the same book with very little added. It is not "high school" math. I wouldn't call it "Book Two". I was expecting something very different based on the title. Highly recommended for 6-7 grade math practice and review.
17 of 17 people found the following review helpful
Good math book, but has typo's.July 17 2007
By
The Way
- Published on Amazon.com
Format: Paperback
This book is helping me getting freshed up in math but it contains a few typo's. For example, on page 7 it says there are two #5 problems. There should be obviously one problem #5 and the other is problem #6. Nevertheless, it is a good math book. I just wished it gave more examples on how to actually DO some of the work.
9 of 9 people found the following review helpful
Miscategorized.Aug. 20 2011
By
Cautious Shopper
- Published on Amazon.com
Format: Paperback
Verified Purchase
This book seemed more geared toward elementary/middle school than middle school/high school. Anyone going into 9th grade might find this way too simple.
8 of 8 people found the following review helpful
Math Skills, but doesn't kill.April 4 2010
By
Jarik25
- Published on Amazon.com
Format: Paperback
Verified Purchase
I always believed that a little review, no matter your age, is good for you. This is "old enough"not to appear kid enough for really anyone to use this book. I do have one qualm though, the version I own has a few noticeable mistakes in it. One would think that they would take the time to make sure that the numbering of problems were in order. But that is the only real problem I see. I recommend giving the student/learner/you plenty of scratch paper. As there isn't really a lot of practice/workout room. My oops with this book is that I get wrapped up in doing the wheels. These wheels are multiplication/addition choose your number. I find these wheels to be lots of fun. I often find myself doing addition when I should be doing multiplication. But that's just cause I like multiplying more. I did find this book good for a person not in middle school. Practice is the same in all grades really 8-12. The more one reviews the old stuff; the easier it is to enjoy the new stuff. Unless you are like me, and hate studying the remedial math. I wish they had a workbook like this for pre-calculus through calculus on up for those trying to review on their own. This book can be a blast don't be worried about doing it perfectly. It is good review.
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and on CD-ROM features the same trusted content as the textbook in an engaging, dynamic format takes learning to a new levelProvides the structure students need to take effective notes. Includes daily vocabulary, key concepts, examples, and Check Understanding excercises for every lesson. Student notes make a great study guide for quizzes and tests.
The Teacher's Edition supports your teaching style and needs, mirroring the format of each student page, at its point of use. The Teacher's Edition includes teaching tips, additional resources, lesson planning, test preparation, classroom examples and more!
This convenient package is a real teacher time saver that provides a wealth of resources to meet individual needs and help you reach all students. For ease of use these blackline masters are organized by chapter.
Also included in the Teaching Resources package are: Cumulative Assessment,diagnostic quarterly,mid-year, and final tests,and Solution key.
Solutions for examples, problems and quizzes (this product is INCLUDED in the Teaching Resources package).
This handy workbook contains additional exercises for every lesson so students can practice their skills.
Includes technology-based classroom activities for individuals or small groups, using graphing calculators, software, spreadsheets, and more!
The Manipulatives Kit allows students to explore concepts in a hands-on way using a variety of measurement, geometry, algebra, and probability tools. The kit is designed for a class of 30 students.
Designed for use with overhead projectors, this kit helps you demonstrate concepts in a using probability tools, including algebra tiles, tangrams, a geoboard with rubber bands, a spinner, and pattern blocks.
These Diagnostic Tests pinpoint students' strengths and weaknesses on the Illinois Learning Standards for Mathematics. Tests in this book include multiple-choice items for fast, easy diagnosis. Also included are individualized Diagnostic Reports for prescribing reteaching resources.
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2 years ago
Calculus ground work. Any suggestions were to start except from the beginning?
Start with a solid review of algebra. Law of powers and things like that. A review of basic trig relations (what sin, cos and tan are) is very helpful. Reviewing the natural log (especially when combined with powers) is good. Things like solving for a variable and simplifying an equation should be automatic for you.
And just as a personal opinion, I found getting used to using WolframAlpha and Mathematica to plot equations very helpful in visualizing what is going on. When you plot an equation and its derivative, you can visually see the relationship of the curve to the slope.
Hope this helps!
As far as topics are concerned....
limits:
existence of a limit - graph analysis
evaluating a limit by use of a table of values.
evaluating a limit at a specific value (denominator not equal to zero)
evaluating a limit at a specific value (denominator equal to zero)
evaluating a limit to infinity
derivatives:
definition of a derivative using limit as h approaches zero.
graph analysis of secant and tangent lines
power rule
constant rule
product rule
quotient rule
chain rule
trig rules
a^x
e^x
ln(x)
inverse trig rules
implicit differentiation
graph analysis of f,f',f""
integrals:
integration of basic functions
integration of trig functions
applications:
related rates
particle motion
area between two curves
volumes of revolution about a line or axis
differential equations with an initial condition
slope fields
trapezoidal approximations
Riemann Sums
These are just a few of the common topics discussed in an introductory Calculus course....I'm sure I left out a few....they will come to me later I'm sure....
Gerbrand - not sure if you are in the US, but any bookstore/Amazon has calculus problem books, usually with solved questions. Pick up one of these, as calculus at this level is very similar across textbooks. You can also try some places like Google Books as well - "The Calculus" by Davis and Brenke (1912) really isn't all that different from modern Calculus books!
Honestly, you can get a lot from taking these older books and using WolframAlpha to solve the problems - use the 'Show Steps' feature of WolframAlpha to demonstrate how the answer was arrived at.
Good luck!
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Maple For Math Majors
Here are links to a collection of Maple worksheets that make up an introductory textbook on using and programming Maple. More information about these worksheets is given in the Introduction. This is a work in progress. Not all of these worksheets are complete. These worksheets are currently being converted from Maple V Release 5 to Maple 8, so most of them are Maple 8 worksheets but some of them are still Maple V Release 5 worksheets. If you have need for them, there is also a complete set of Maple V Release 5 worksheets available.
You need a copy of Maple to work with these worksheets. Maple 8 will work best and Maple 6, 7, 9, or 9.5 should work with most of the worksheets. You can use a web browser to download a worksheet and then you can use Maple to open and work with the worksheet. (You can also use a web browser to view a PDF version of each worksheet.)
If after you click on one of these links you get a page full of nonsense, then return to this page and "right click" (i.e., use the right mouse button) on the link that you want. Most browsers will pop up a "context menu." Depending on your browser, you should click on the "Save Target As ..." menu item or the "Save Link As ..." menu item. Then the browser will ask you where you would like to save the worksheet.
An introduction to the concepts of data types and data structures, some of Maple's built in data types, some commands for making and manipulating Maple's data structures, and why Maple expressions are data structures.
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Find a Mount Berry Algebra 1I'm familiar with both the newer versions as well as the older versions where most functions had to be memorized as to reference the cell, table, and complex calculations involving multi-step process, from defining names of source data, creations of queries. Without going into every possible fun
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students that extra boost they need to acquire important concepts in specific areas of math. The goal of these how to books is to provide the information and practice necessary to master the math skills established by the National Council of Teachers of Mathematics. Each book is divided into units containing concepts, rules, terms, and formulas, followed by corresponding practice pages
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books.google.co.uk - Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems. Drawing... Optimization
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Synopses & Reviews
Publisher Comments:
This textbook for graduate students introduces integrable systems through the study of Riemann surfaces, loop groups, and twistors. The introduction by Nigel Hitchin addresses the meaning of integrability, discussing in particular how to recognize an integrable system. He then develops connections between integrable systems and algebraic geometry and introduces Riemann surfaces, sheaves, and line bundles. In the next part, Graeme Segal takes the Korteweg-de Vries and nonlinear Schrödinger equations as central examples and discusses the mathematical structures underlying the inverse scattering transform. He also explains loop groups, the Grassmannian, and algebraic curves. In the final part of the book, Richard Ward explores the connection between integrability and self-dual Yang-Mills equations and then describes the correspondence between solutions to integrable equations and holomorphic vector bundles over twistor space.
Synopsis:
This textbook for graduate students introduces integrable systems through the study of Riemann surfaces, loop groups, and twistors. Topics include Korteweg-de Vries and nonlinear Schrodinger equations, the inverse scattering transform, algebraic curves, and self-dual Yang-Mills equations
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Integration as a summation resources
The second major component of the Calculus is called integration. This
may be introduced as a means of finding areas using summation and limits. We
shall adopt this approach in the present Unit. In later units, we shall also
see how integration may be related to differentiation. (Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.
| 677.169 | 1 |
More About
This Textbook
Overview
These authors understand what it takes to be successful in mathematics, the skills that students bring to this course, and the way that technology can be used to enhance learning without sacrificing math skills. As a result, they have created a textbook with an overall learning system involving preparation, practice, and review to help students get the most out of the time they put into studying. In sum, Sullivan and Sullivan's Precalculus: Enhanced with Graphing Utilities gives students a model for success in mathematics.
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Meet the Author
Mike Sullivan is a Professor of Mathematics at Chicago State University and received a Ph.D. in mathematics from Illinois Institute of Technology. Mike has taught at Chicago State for over 30 years and has authored or co-authored over fifty books. Mike has four children, all of whom are involved with mathematics or publishing: Kathleen, who teaches college mathematics; Mike III, who co-authors this series and teaches college mathematics; Dan, who is a Pearson Education sales representative; and Colleen, who teaches middle-school mathematics. When he's not writing, Mike enjoys gardening or spending time with his family, including nine grandchildren.
Mike Sullivan III is a professor of mathematics at Joliet Junior College. He holds graduate degrees from DePaul University in both mathematics and economics. Mike is an author or co-author on more than 20 books, including a statistics book and a developmental mathematics series. Mike is the father of three children and an avid golfer who tries to spend as much of his limited free time as possible on the golf
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Reinforces student understanding and aids in test preparation with detailed explanations, worked-out examples, and practice problems. Lists key ideas to master and builds problem-solving skills. Includes worked solutions to the odd-numbered problems in the text.
| 677.169 | 1 |
three gifted—and funny—teachers, How to Ace Calculus provides humorous and readable explanations of the key topics of calculus without the technical details and fine print that would be found in a more formal text. Capturing the tone of students exchanging ideas among themselves, this unique guide also explains how calculus is taught, how to get the best teachers, what to study, and what is likely to be on exams—all the tricks of the trade that will make learning the material of first-semester calculus a piece of cake. Funny, irreverent, and flexible, How to Ace Calculus shows why learning calculus can be not only a mind-expanding experience but also fantastic fun.
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"This is a marvelous, user-friendly introduction to the basic ideas of calculus. It is effective, humorous and eminently practical. The book that 100,000 calculus students have been searching for is finally here."—Ron Graham, Chief Scientist, AT&T Labs, former President of the American Mathematical Society, and author of Concrete Mathematics: A Foundation of Computer Science
"Can a calculus book be lighthearted and engaging? Surprisingly, yes, and here is one that does the job."—Thomas Banchoff, Professor of Mathematics, Brown University, President-Elect of the Mathematics Association of America, and author of Beyond the Third Dimension
"This book is dangerously clear, direct, and funny. It should be suppressed before it jeopardizes the time-tested function of the calculus sequence to befuddle and filter surplus students."—William Thurston, Professor of Mathematics, University of California at Davis, Fields Medalist, and former Director of the Mathematical Sciences Research Institute
"Comic opera meets college math in this amusing and edifying roller coaster of an introduction to calculus."—Ivars Peterson, author of The Mathematical Tourist
About the Author
Colin Adams is Professor of Mathematics at Williams College. He is the author of The Knot Book and winner of the Mathematical Association of America Distinguished Teaching Award for 1998. Joel Hass is Professor of Mathematics at the University of California at Davis, and Abigail Thompson is also Professor of Mathematics at the University of California at Davis. Both have held fellowships from the Sloan Foundation and the National Science Foundation.
Most Helpful Customer Reviews
I am a math professor at a large state university. Like math departments everywhere, we depend on calculus students for our very existence. Students who fail Calculus, or repeat customers as we like to call them, are of course the most highly appreciated of all. An otherwise good student, given the right combination of obscure lectures and unreadable texts, may pay for the same Calculus course 2, 3 or even 4 times. The Streetwise Guide is too thin and too cheap. Moreover, the writing style gives ordinary students what they need to master Calculus the first time they take it. In other words, this book is a threat to mathematics departments everywhere.
My dad bought me this book the summer before my senior year in high school (this year) - I looked at it and thought, "yeah, right." I'm a humanities person, I got a C in my last term of Pre-calculus, and I didn't think some book was going to help. However, once my AP Calc class started, I thought "hey, might as well give it a try." That was the best idea I've had in a long time. Mr. Adams and friends explain the concepts so clearly that I have to wonder what the authors of standard calculus texts were thinking when they wrote their books. Not only do I have an A- in the class, I have a much better understanding of what we're actually learning than the students who used to do better than me by just following all the formulas. In short, I am acing calculus, and (this is very hard for me to admit) enjoying it as well. I would highly recommend this book to anyone who panics at the thought of calculus.
Unless you're a mathematician, it is probably a rare thing when you find a math book that you would enjoy reading in your free time. It is utterly inconceivable that you would find a CALCULUS book that you wouldn't mind for some light night reading. Believe it or not, inconceivable does not imply impossible, and here is the proof in hard copy. I HATE calculus and I found this book both easy to understand and actually amusing! It was an excellent refresher for my 2nd (and surprisingly successful) attempt to pass calculus II. I really wish I'd had this thing earlier, when I was slaving away through first-quarter calculus. For anyone who is taking calculus or wants to review, I give this book my highest recommendation. Now, if only these folks would write a full-length textbook!
I am horrible at math. I used this book in my Calculus I class last semester (Fall 2000) and part of Calculus II and it really helped me out. With this book and a lot of determination I was able to get an 'A' in Cal I and, depending on how my final exam turns out, it looks like I might get an 'A' in Cal II. I am completely convinced I would not have done as well with just my text book alone. I was impressed that all though the book was 1/5 the size of my Calculus text, everytime I stumbled in class I was able to flip right to the section I was having problems with and it explained almost everything in a clear and relevent manner. Some of the humor in the book is a bit of stretch and there were a few topics on which I wish the book contained a bit more information. Particularly the chapter "Fancy Pants Techniques of Integration." I really could have used a little more help on Trigonometric Substitutions, Misc Substitutions, and Partial Fractions. Other than that, this book is easily one of the best educational supplements I've ever purchased. I have "How to Ace the Rest of Calculus" on order now. I hope it gets to me before the end of the semester.
I am heading back to school to finish up a Mathematics degree. Been away from school for a while, and even though I've completed Calc I, II and III (over 8 years ago), Calc is the prereq for my remaining advanced math courses. What to do?!?
Found this little gem three weeks ago, ordered it, and finished it tonight (an hour or two each night, and still living my life)! Given the textbook racket that seems to have become more costly than courses - not to mention filled with sometimes _too_ much information - I'm happy to say that this book is cleverly written, to the point, yet thorough with the principles and explanations of the general principles of Calculus.
I recommend this book to anyone who is:
a) about to take a beginning Calculus course b) in need of an excellent refresher on the topic c) would like to learn Calculus for fun, in a fun way!
Thanks goes out to the previous reviewers, and to the authors for taking the time to care about the student by setting the egos aside (next week it's on to How to Ace: the Rest of Calculus)!
This book is aptly subtitled, as so much of the advice could have been written by knowledgeable students rather than three mathematics professors. The second chapter gives you sound advice regarding how to choose a calculus instructor and what the different academic ranks often mean. While students learn this very quickly through informal channels, it is surprising to read three professors publicly stating that the instructor is the single most determining factor as to whether you enjoy or hate calculus. Of course it is often true and we all know it. Many professors view teaching undergraduates as some form of penance and it shows in the quality of their teaching. Furthermore, although things are changing a bit, working hard to instruct well is often negatively reflected in a tenure decision. The remainder of the book is a combination of a relaxed, joking style and sound advice that is part of the standard exam speech given by many teachers.
However, presented in the form of jokes, it is possible that the message will penetrate a little deeper. The authors also do one other very admirable thing. Rather than try to boost sales by deleting or grossly simplifying mathematical expressions, all of the major formulas of first year calculus are here, and in the same form as they appear in standard calculus texts. However, the approach is much more relaxed, which makes it more understandable than a formal text. In the days when I was teaching calculus, so many of the problems that students had was a consequence of being intimidated by the formal structure of the text and the rigor of the proofs.Read more ›
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Differential Games [NOOK Book] ...
Customers Who Bought This Also Bought
More About
This Book delayed information) and N-person games.
Geared toward graduate students, Differential Games will be of particular interest to professionals in the fields of electrical engineering, industrial engineering, economics, and mathematics. Although intended primarily for self-study, it can be used as a core or ancillary text in courses in differential games, game theory, and control theory.
Related Subjects
Table of Contents
Introduction
1. Definition of a Differential Game
2. Games of Fixed Duration
3. Games of Pursuit and Evasion
4. Computation of Saddle Points
5. Games of Survival
6. Games with Restricted Phase Coordinates
7. Selected Topics
8. N-Person Games
Bibliographical Remarks
Bibliography
Index
Index of Conditions
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Description of Key to Algebra Book 8: Graphs by Key Curriculum Press
Too many students end their study of mathematics before ever taking an algebra course. Others attempt to study algebra, but are unprepared and cannot keep up. "Key to Algebra" was developed with the belief that anyone can learn basic algebra if the subject is presented in a friendly, non-threatening manner and someone is available to help when needed.
Some teachers find that their students benefit by working through these books before enrolling in a regular algebra course--thus greatly enhancing their chances of success. Others use "Key to Algebra" as the basic text for an individualized algebra course, while still others use it as a supplement to their regular hardbound text. Allow students to work at their own pace.
The "Key to Algebra" books are informal and self-directing. The authors suggest that you allow the student to proceed at his or her own pace. Book 8 covers Graphs.
Product:
Key to Algebra Book 8: Graphs
Vendor:
Key Curriculum Press
Binding Type:
Paperback
Media Type:
Book
Minimum Grade:
5th Grade
Maximum Grade:
12th Grade
Number of Pages:
36
Weight:
0.25 pounds
Length:
8.25 inches
Width:
0.25 inches
Height:
10.5 inches
Vendor Part Number:
53008
Subject:
Algebra, Calculus & Trig, Math
Curriculum Name:
Key Curriculum Press
Learning Style:
Kinesthetic, Visual
Teaching Method:
Traditional, Unit Study
There are currently no reviews for Key to Algebra Book 8: Graphs.
Items Related to Key to Algebra Book 8: Graphs by Key Curriculum Press
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I took geometry in 9th grade, in a class that only did Euclidean proofs. This book covers alot of measuring, and gets into the very basics of analyticI took geometry in 9th grade, in a class that only did Euclidean proofs. This book covers alot of measuring, and gets into the very basics of analytic geometry and trigonometry. Those were both full classes that I had in high school. So I'm wondering what makes this for college students? Anyway, the presentation was clear.
In the Meno, Plato argues that knowledge is a form of memory, and he uses math to illustrate the point. I was reminded of the Meno here because everything in this book was calling up distant memories. The funny thing is that at one point, I had to fill out my address on a form, and I caught myself giving my childhood street address. So, returning to this stuff has had a different kind of effect on the memory, subconsciously transporting me back to my youth....more
Mostly really simple stuff, with the addition of brief introductions to linear algebra, probability and series, and conics. The approach was easy to fMostly really simple stuff, with the addition of brief introductions to linear algebra, probability and series, and conics. The approach was easy to follow, but at the expense of not including proofs for almost anything that has any complexity to it. It's hard for me to believe that this is really college level stuff....more
Even for a textbook I thought this was bad. I haven't ever taken statistics before, and almost everything I know is stuff I've picked up either from sEven for a textbook I thought this was bad. I haven't ever taken statistics before, and almost everything I know is stuff I've picked up either from studying trading, or from trying to figure out the meaning of scientific studies. I had hoped that a textbook would clear up some of the things I've always found confusing. No such luck.
I would have liked the book to explain the formulas, give some insight into how they were developed, and discuss why they were needed. Instead, the book typically presents a fairly complicated formula, without anything in the way of prior explanation. Sometimes it will present the formula before it even defines the terms that the formula contains, a practice that makes me want to shove needles into my eyes.
Then, the book will go straight into examples of how to use the formula. Most of the time, this means a few pages of examples where the only engagement by the reader is going to a table and looking up a number from rows and columns. Sometimes you go to Table 6 in Appendix B. Other times you go to Table 8. And that's about the depth of the insight you get. I really did not need to have extensive instruction in how to look something up in a stupid table. And frankly, the table look-up is pretty pointless nowadays, because anyone with access to a computer and the internet can probably do better than the tables here.
Then there are the ideas that I always found mystifying when reading statistics, like "degrees of freedom." There are many formulas presented here that contain an entry where you apply the appropriate degrees of freedom. But nowhere is there even a word about what a degree of freedom is, why it might be needed in a formula, or anything else to give a person an inkling of what is really going on with any formula involving a degree of freedom. (In my frustration I did some looking on google to get some idea. It involves n-dimensional vector spaces and constraints along some dimensions within that space. This is a pretty advanced mathematical idea, and I understand why the book doesn't go into any detail. But it wouldn't be too hard to at least mention the idea. It wouldn't be that hard even to explain what happens if you take an object in 3 dimensional space and limit it to motion on a flat plane. What you have done is removed one of the "degrees of freedom" of that object. There! Was that really so hard that it wouldn't even deserve a mention.
Then, even with ideas that are pretty simple, the book offers no explanation for how it gets to the formulas a person is to apply. For example, late in the book we get the Sign Test for a population median. Here's how the book presents it without any other explanation. Someone claims that X is the median for some population. Take a sample and assign + signs when the sample datum is greater than X and a - sign when its less than X. Count the number of + and - signs. If the sample size is 25 or less, then choose the lower count. If greater than 25, then calculate (x+.5) - .5n/(square root n/2). Then do a table lookup depending on the level of significance important to you, but you go to different tables depending on whether the sample size is less than or greater than 25 (without explaining why). Then you compare your sample calculation with the number from the table. The examples then walk you through the procedures of the algorithm. Even on its own, this method depends almost entirely on rote, and doesn't give any understanding about what's going on.
But in the case above, its even worse because it would be very simple to explain what's going on with the sign test. First, remind people of what a median is. It's the number in the population where half the population lies above the number and half lies below it. This means that if you take a random sample of one from the population, there's a 50/50 chance of him falling above or below the median. For any random sample, it's a coin toss. Thus, the derivation of the formula should be really simple. Thus, this problem can be reduced to a simple probability problem involving throwing a coin. If the results deviate too much from 50/50, that suggests that the stated median is probably incorrect. This book doesn't even attempt this, or any other, sort of explanation.
If anyone knows of a good book on probability and statistics, I would welcome the suggestion. This ain't it....more
Very nice and very clear for the most part. This is pretty much the opposite of the dreadful statistics textbook I recently read. Here, you have to doVery nice and very clear for the most part. This is pretty much the opposite of the dreadful statistics textbook I recently read. Here, you have to do quite a bit of the work for yourself. This means that Clark will fairly often present a theorem and then immediately leave the proof as an exercise. This procedure basically requires an understanding of the material -- otherwise, within a few pages the reader will become hopelessly lost. Thus, while the book is very short, it's way more dense than the other math texts I've recently read. All in all, I prefer this approach, but its definitely not for the lazy reader.
As far as I know, this book is available only as a download from the author's website. He has made it available for free, and its easily worth the price. ...more
This was much better than the last book on statistics I read. At least it would either prove something, or state that the math involved was beyond theThis was much better than the last book on statistics I read. At least it would either prove something, or state that the math involved was beyond the scope of the book and leave a proposition unproven. It also talked some about the limitations of the ideas and approximations that were being used. It even made an attempt to explain "degrees of freedom."
I like this all the way through the basic hypothesis testing chapters. I liked the approach taken here to non-parametric testing. But that's about the point where things started to get fuzzy. And it was bit fuzzier with Chi squared distributions. And by the time we got to F Distributions, it happened again: I felt like bees were living in my head.
What I take away from this is that I pretty much dislike statistics. It all seems like a huge house of cards to me: alot of very elegant math built on what may or may not be some pretty shady assumptions. That said, I did take away some very useful ideas that might have some application to testing trading systems....more
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BOT 108 - Using Calculators to Solve Business Problems
Prerequisite:
None
Corequisite:
None
Recommended
Preparation:
Assessment
recommendation for ESL 106
Hours:
Assignments to be completed on campus
in room E120 during open
lab hours
Introduces the 10-key, digital display electronic
calculator. Students will build skill in performing fundamental
arithmetic operations using a calculator. Topics include use of
decimals, fractions, constants, discounts, percentages and memory
keys.
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Shipping prices may be approximate. Please verify cost before checkout.
About the book:
This advanced undergraduate-level text was recommended for teacher education by The American Mathematical Monthly and praised as a "most readable book." An ideal introduction to groups and Galois theory, it provides students with an appreciation of abstraction and arbitrary postulational systems, ideas that are central to automation. The authors take the algebraic equation and the discovery of the insolubility of the quintic as their theme. Starting with treatments of groups, rings, fields, and polynomials, they advance to Galois theory, radicals and roots of unity, and solution by radicals. Thirteen appendixes supplement this volume, along with numerous examples, illustrations, commentaries, and exercises. Students who have completed a first-year college course in algebra or calculus will find it an accessible and well-written treatment.
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Better World Books via United States Used - Good. Shows some signs of wear, and may have some markings on the inside. Shipped to over one million happy customers. Your purchase benefits world literacy!
Hardcover, ISBN 0721661874 Publisher: W.B. Saunders Company, 1971 Good. US Edition. Shows some signs of wear, and may have some markings on the inside. 100% Money Back Guarantee. Shipped to over one million happy customers. Your purchase benefits world literacy!.
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Abstract Algebra A Geometric Approach
9780133198317
ISBN:
0133198316
Pub Date: 1995 Publisher: Prentice Hall
Summary: Appropriate for a 1 or 2 term course in Abstract Algebra at the Junior level. This book explores the essential theories and techniques of modern algebra, including its problem-solving skills, basic proof techniques, many unusual applications, and the interplay between algebra and geometry. It takes a concrete, example-oriented approach to the subject matter.
Shifrin, Theodore is the author of Abstract Algebr...a A Geometric Approach, published 1995 under ISBN 9780133198317 and 0133198316. Two hundred eighty nine Abstract Algebra A Geometric Approach textbooks are available for sale on ValoreBooks.com, fifty four used from the cheapest price of $61.06, or buy new starting at $97.71
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More About
This Textbook
Overview
Differential and integral equations involve important mathematical techniques, and as such will be encountered by mathematicians, and physical and social scientists, in their undergraduate courses. This text provides a clear, comprehensive guide to first- and second-order ordinary and partial differential equations, whilst introducing important and useful basic material on integral equations. Readers will encounter detailed discussion of the wave, heat and Laplace equations, of Green's functions and their application to the Sturm-Liouville equation, and how to use series solutions, transform methods and phase-plane analysis. The calculus of variations will take them further into the world of applied analysis.
Providing a wealth of techniques, but yet satisfying the needs of the pure mathematician, and with numerous carefully worked examples and exercises, the text is ideal for any undergraduate with basic calculus to gain a thorough grounding in 'analysis for applications
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An open access, self-paced Prealgebra Module on Integers with a pretest, 4 sections, and a posttest.PreAlgebra Module 2:...
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An open access, self-paced Prealgebra Module on Integers with a pretest, 4 sections, and a posttest.PreAlgebra Module 2: IntegersLO: Student will use properties of integers to solve simple equations and application problems• PA Module 2 Section 1: Integers and the Number LineStudent will compare integer values using properties of inequalities, opposites, and absolute values.• PA Module 2 Section 2: Operations with IntegersStudent will apply the rules of addition, subtraction, multiplication and division to simplify expressions involving positive and negative integers.• PA Module 2 Section 3: Order of Operations with IntegersStudent will use the properties and methods for the order of operations when evaluating and simplifying mathematical expressions.• PA Module 2 Section 4: Solving Linear One Variable EquationsStudent will use properties and methods to solve linear equations with one variable.
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2014 Books Gift Guide for Children & Teens
Browse our featured books to find gift ideas for the boys or girls on your holiday shopping list this year!
This introduction to computational geometry focuses on algorithms. Motivation is provided from the application areas as all techniques are related to particular applications in robotics, graphics, CAD/CAM, and geographic information systems. Modern insights in computational geometry are used to provide solutions that are both efficient and easy to understand and implement.
Product Description
Review
"An excellent introduction to the field is given here, including a general motivation and usage cases beyond simple graphics rendering or interaction." from the ACM Reviews by William Fahle, University of Texas at Dallas, USA
Most helpful customer reviews
Easy reading, excellent text on the topic. I'm coming from a Geomatics (CompSci/Geog) based background and in my 4th year of University.
Every chapter starts with an overview of the problem with real world examples, simple solutions to this that are not optimized nor consider degenerate cases, and then goes into a 'how can we make this better' style of discussion with excellent justifications along the way.
There is an expectation that you are familiar with basic algorithm design, performance analysis, and data structures.
Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com:
13 reviews
3 of 3 people found the following review helpful
Good overview but lacking detailMarch 17 2011
By
David
- Published on Amazon.com
Format: Hardcover
This is a very well written book and covers a wide range of computational geometry problems. It is a very good introduction/overview to computational geometry. I would have preferred more detail, specifically code examples. This is a good book to gain an understanding of the topic but not so good if you are actually trying to implement the concepts in code or make the best use existing code libraries.
A textbook for undergraduatesNov. 4 2014
By
Clive McCarthy
- Published on Amazon.com
Format: Hardcover
Verified Purchase
This is a computer science textbook for undergraduates. Lots of "real world" uses for computational geometry to egg-on the unmotivated. Not so good for those that ARE already motivated.
The text glosses over basic tasks such as "whether a point lies to the left or right of a directed line" (page 4) with the expectation that some unnamed library function will do this. For me, not so. Moreover I'd would have liked to read a geometric proof of such things. The foundations are left out, yet elsewhere they waste space to give us Pythagoras' Theorem.
Good introduction to the topics but is not good at explaningApril 8 2014
By
mackster
- Published on Amazon.com
Format: Hardcover
Verified Purchase
This is the standard text book for CG, and it nicely introduces us to a lot of concepts. But, unfortunately it is not the best book I have read. Most of the examples albeit few, does not make much sense. The algorithms discussed sometimes cannot be grasped. I often went online to read more about the subject to understand the topics. This also could mean that my grasping of the subject is low :), but that should not matter as long as the material is explained clearly.
Loved it.Oct. 30 2013
By
J. Netzley
- Published on Amazon.com
Format: Hardcover
Verified Purchase
This book is a great all-purpose volume to own. Covers lots of topics in a very intuitive way. I highly recommend it.
Need this for classOct. 6 2013
By
Yulun Wu
- Published on Amazon.com
Format: Hardcover
Verified Purchase
This book is pretty damn good. It explains the content well. I like it. My professor likes it. The hardcover is nice.
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Optional Mathematics Extensions: Each of the four core lessons are designed to be taught in 80 minutes. There is also a final, optional mathematics extension that explains how to use the Octave computer programming language as a numerical simulator.
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This course examines how numerical methods are used by engineers to translate the language of mathematics and physics into...
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This course examines how numerical methods are used by engineers to translate the language of mathematics and physics into information that may be used to make engineering decisions. Often, this translation is implemented so that calculations may be done by machines (computers). This free course may be completed online at any time. See course site for detailed overview and learning outcomes. (Mechanical Engineering 205)
This is a free, online textbook "is written primarily for engineering and science undergraduates taking a course in Numerical...
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This is a free, online textbook "is written primarily for engineering and science undergraduates taking a course in Numerical Methods. The textbook offers a unique treatise to numerical methods which is based on a holistic approach and short chapters.״ In addition to the text, one can also access Video Lectures, PowerPoint Presentations,Worksheets, Multiple-Choice Tests, and Anecdotes.
'This book entitled Numerical Methods with Applications is written primarily for engineering undergraduates taking a course...
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'This book entitled Numerical Methods with Applications is written primarily for engineering undergraduates taking a course in Numerical Methods. The textbook offers a unique treatise to numerical methods which is based on a holistic approach and short chapters. This book is a product of many years of work on educational projects funded since 2002 by the NSF. Features: 1) Examples of real-life applications are available from seven different engineering majors. 2) Each chapter is followed by multiple-choice questions. 3) Supplemental material such as primers on differential and integral calculus, and ordinary differential equations are available on the web.'
For systems of two linear equations in two variables, this Formula Solver program walks you through the steps for finding the...
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For systems of two linear equations in two variables, this Formula Solver program walks you through the steps for finding the solution using the elimination method. You can even work with your own values!
Solving a system of two linear equations in two variables just got a little easier. This Formula Solver program will walk...
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Solving a system of two linear equations in two variables just got a little easier. This Formula Solver program will walk you through the steps for finding the solution using the substitution method, and you can even work with your own values!
'This textbook was born of a desire to contribute a viable, free, introductory Numerical Analysis textbook for instructors...
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'This textbook was born of a desire to contribute a viable, free, introductory Numerical Analysis textbook for instructors and students of mathematics. The ultimate goal of Tea Time Numerical Analysis is to be a complete, one-semester, single-pdf, downloadable textbook designed for mathematics classes.'
A collection of webMathematica scripts on a variety of topics in mathematics including calculus, number theory and abstract...
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A collection of webMathematica scripts on a variety of topics in mathematics including calculus, number theory and abstract algebra. In each script, html form input is processed by a remote version of Mathematica and output is returned. No Mathematica expertise is required other than familiarity with some notation (e. g. Pi for pi ).
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QuickMath is an automated service for answering common math problems over the internet. ...
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QuickMath is an automated service for answering common math problems over the internet. Think of it as an online calculator that solves equations and does all sorts of algebra and calculus problems - instantly and automatically! When you submit a question to QuickMath, it is processed by Mathematica, the largest and most powerful computer algebra package available today. The answer is then sent back to you and displayed right there on your browser, usually within a couple of seconds. Best of all, QuickMath is 100% free!
This Macromedia ShockWave applet illustrates the principle of accommodation in the eye. The user adjusts the distance between...
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This Macromedia ShockWave applet illustrates the principle of accommodation in the eye. The user adjusts the distance between eye and object by clicking and dragging. The applets updates the ray tracing diagram, the image size, and the lens shape in real time.
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More About
This Textbook
Overview
The Multivariable portion of the Soo Tan Calculus textbook tackles complex concepts with a strong visual approach. Utilizing a clear, concise writing style, and use of relevant, real world examples, Soo Tan introduces abstract mathematical concepts with his intuitive style that brings abstract multivariable concepts to life. The Multivariable text provides a great deal of visual help by introducing unique videos that assist students in drawing complex calculus artwork by hand. In keeping with this emphasis on conceptual understanding, each exercise set begins with concept questions and each end-of-chapter review section includes fill-in-the-blank questions which are useful for mastering the definitions and theorems in each chapter. Additionally, many questions asking for the interpretation of graphical, numerical, and algebraic results are included among both the examples and the exercise sets.
Related Subjects
Meet the Author
Soo T. Tan
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Elementary Algebra: MyMathLab Edition: Concepts and Applications
Browse related Subjects
...Read More applications and exercises to help them apply and retain their knowledge. New features such as Translating for Success and Visualizing for Success unlock the way students think, making math accessible to them. KEY TOPICS: Introduction to Algebraic Expressions; Equations, Inequalities, and Problem Solving; Introduction to Graphing; Polynomials; Polynomials and Factoring; Rational Expressions and Equations; Systems and More Graphing; Radical Expressions and Equations; Quadratic Equations MARKET: For all readers interested in algebra 0321641388 Cover has moderate wear. Binding good. No markings to text. Small stain to bottom corner of a few early pages. NOTE: Does NOT come with access code for math lab Ebook-exercise sets are in the E
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Course Description:(Non-credit
for mathematics major or minor.) Special emphasis for teachers of grades P-8.
Broadens understanding of the fundamental concepts of probability and
statistics with particular attention to specific methods and materials of
instruction.
an
understanding of standard vocabulary and symbols associated with
probability and statistics;
a
better understanding of fundamental concepts in probability and
statistics, including the organization of data, numerical descriptive
measures, discrete random variables and their probability distributions,
and continuous random variables and the normal distribution;
a
better understanding of appropriate strategies for teaching mathematics
concepts at the P-8 level; and
a
better understanding of the uses of a variety of manipulatives and other
materials for the P-8 level.
Grading Methods: Tests, Quizzes, Final
Exam, Homework; Percentages to be determined by the instructor.
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This is a sub-page of the large and comprehensive Eric Weisstein's World of Mathematics site which is separately reviewed...
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This is a sub-page of the large and comprehensive Eric Weisstein's World of Mathematics site which is separately reviewed elsewhere on MERLOT. At the time of review this probability and statistics area listed 19 subtopics including Bayesian analysis, descriptive statistics, probability, random numbers, random walks, and statistical tests. Approximately 350 separate items on probability and statistics were includedThis site provides about 35 graphical applets on topics relative to Algebra, Precalculus, Calculus, and Statistics. These...
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This site provides about 35 graphical applets on topics relative to Algebra, Precalculus, Calculus, and Statistics. These are designed for classroom demonstrations of various mathematical/statistical concepts.
This applet is a web based lab that explores the properties of linear functions. It is one in a series of other precalculus...
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This applet is a web based lab that explores the properties of linear functions. It is one in a series of other precalculus labs by the same author. The directions for using Graph Explorer are contained in the Cartesian Coordinates appletComment:
I'm the publisher of the book. It should be classified asMathematics and Statistics/Mathematics/AnalysisnotMathematics and...
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Comment:
I'm the publisher of the book. It should be classified asMathematics and Statistics/Mathematics/AnalysisnotMathematics and Statistics/Mathematics/Numerical AnalysisThe primary audience is definitely not Lower Division College, it's Upper Division College and probably mainly Graduate Courses in Mathematics
Comment:
i liked this site because iam really into the probabilites of things because of statistics. i play baseball so probabilities...
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Comment:
i liked this site because iam really into the probabilites of things because of statistics. i play baseball so probabilities are very interesting when it comes to statistics. unfortunatly this is a part of math that not too many people are interested in. i think one of the main reasons i like probabilites so much is because everything else is so confusing to me.
Comment:
A power write up that simplifies statistics, particularly to those with a mathematical background. The content is very...
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Comment:
A power write up that simplifies statistics, particularly to those with a mathematical background. The content is very accurately. The concepts and the principles of statistics are clearly articulated. students were pleased with contentTechnical Remarks:Basic ICT skills
Comment:
The MacTutor History of Math site which is from Scottland has the largest amount of general math concepts on any site I have...
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Comment:
The MacTutor History of Math site which is from Scottland has the largest amount of general math concepts on any site I have ever seen. It's basically the mathematical equivlent of Google. After viewing certain topics such as the Fibonachi Sequence and Statistics I wouldnt go anywhere else for math related stuff!
Comment:
Java based software package that allows user to solve some statistical problems only using browser only. A good tool for a...
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Comment:
Java based software package that allows user to solve some statistical problems only using browser only. A good tool for a student who does not have an access to comercial statistics software packages.Technical Remarks: A very friendly interface.
Comment:
The Applet gives good examples that connect the idea to real world applications, such as the movement of atoms and molecules....
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Comment:
The Applet gives good examples that connect the idea to real world applications, such as the movement of atoms and molecules. This would be a good site to recomend to kids in a math class because it gives an activity that helps demonstrate the idea, and it also poses some questions at the end. It is straightforward and easy to understand.
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Designed to incorporate the power of a graphing calculator into the classroom, Discovering Algebra presents step-by-step keystroke instruction in a convenient, easy to follow format. This manual is intended as a supplement to any standard algebra text and provides beginners with the tools necessary to succeed in algebra without anxiety.
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Overview
This popular and successful text was originally written for a one-semester course in linear algebra at the sophomore undergraduate level. Consequently, the book deals almost exclusively with real finite dimensional vector spaces, but in a setting and formulation that permits easy generalisation to abstract vector spaces. A wide selection of examples of vector spaces and linear transformation is presented to serve as a testing ground for the theory. In the second edition, a new chapter on Jordan normal form was added which reappears here in expanded form as the second goal of this new edition, after the principal axis theorem. To achieve these goals in one semester it is necessary to follow a straight path, but this is compensated by a wide selection of examples and exercises. In addition, the author includes an introduction to invariant theory to show that linear algebra alone is incapable of solving these canonical forms problems. A compact, but mathematically clean introduction to linear algebra with particular emphasis on topics in abstract algebra, the theory of differential equations, and group representation theory.
Larry Smith
SMITH Magazine founding editor Larry Smith has worked as an editor at Men's Journal, ESPN: The Magazine, and Might. His writing has appeared in the New York Times, the Los Angeles Times, Popular Science, on Salon.com, and many other places. Larry lives
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Algebra refers to your ability to manipulate variables and unknowns based on rules and properties. Matrix algebra is extremely similar to the algebra you already know for numbers with a few important differences. What are these differences?
Guidance
Two matrices of the same order can be added by summing the entries in the corresponding positions.
Two matrices of the same order can be subtracted by subtracting the entries in the corresponding positions.
You can find the product of matrix
and matrix
if the number of columns in matrix
matches the number of rows in matrix
. Another way to remember this is when you write the orders of matrix
and matrix
next to each other they must be connected by the same number. The resulting matrix has the number of rows from the first matrix and the number of columns from the second matrix.
To compute the first entry of the resulting
matrix you should match the first row from the first matrix and the first column of the second matrix. The arithmetic operation to combine these numbers is identical to taking the dot product between two vectors.
The entry in the first row first column of the new matrix is computed as
.
The entry in the second row first column of the new matrix is computed as
.
The rest of the entries of this product are left to Example A.
Properties of Matrix Algebra
Commutatively holds for matrix addition. This means that when matrices
and
can be added (when they have matching orders), then:
Complete the entries of the matrix multiplication introduced in the guidance section.
Solution:
Two of the arithmetic operations are shown.
Example B
Show the commutative property does not hold by demonstrating
Solution:
Example C
Compute the following matrix arithmetic:
.
Solution:
When a matrix is multiplied by a scalar (such as with
), multiply each entry in the matrix by the scalar.
Since the associative property holds, you can either distribute the ten or multiply by matrix
next.
Concept Problem Revisited
The main difference between matrix algebra and regular algebra with numbers is that matrices do not have the commutative property for multiplication. There are other complexities that matrices have, but many of them stem from the fact that for most matrices
.
Vocabulary
Matrix operations
are addition, subtraction and multiplication. Division involves a multiplicative inverse that is not discussed at this point.
Guided Practice
1. Show that a
identity matrix works as the multiplicative identity.
2. Use your calculator to input and compute the following matrix operations.
3. Matrix multiplication can be used as a transformation in the coordinate system. Consider the triangle with coordinates (0, 0) (1, 2) and (1, 0) the following matrix:
What does the new picture look like?
Answers:
1. A
matrix multiplied by the identity should yield the original matrix.
2. Most graphing calculators like the TI-84 can do operations on matrices. Find where you can enter matrices and enter the two matrices.
Then type in the appropriate operation and see the result. The TI-84 has a built in Transpose button.
The actual numbers on this guided practice are less important than the knowledge that your calculator can perform all of the matrix algebra demonstrated in this concept. It is useful to fully know the capabilities of the tools at your disposal, but it should not replace knowing why the calculator does what it does.
3. The matrix simplifies to become:
When applied to each point as a transformation, a new point is produced. Note that
is a matrix representing each original point and
is the new point. The
is read as "
prime" and is a common way to refer to a result after a transformation.
Notice how the matrix transformation rotates graphs in a counterclockwise direction
.
The matrix transformation applied in the following order will rotate a graph clockwise
.
Practice
Do #1-#11 without your calculator.
1. Find
. If not possible, explain.
2. Find
. If not possible, explain.
3. Find
. If not possible, explain.
4. Find
. If not possible, explain.
5. Find
. If not possible, explain.
6. Find
. If not possible, explain.
7. Find
. If not possible, explain.
8. Find
. If not possible, explain.
9. Find
. If not possible, explain.
10. Show that
.
11. Show that
.
Practice using your calculator for #12-#15.
12. Find
.
13. Find
.
14. Find
.
15. Find
.
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Learning Objectives
Here you will add, subtract and multiply matrices. As a result you will discover the algebraic properties of matrices.
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Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for).
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Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for).
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04951084050495108405Ships from: Alpharetta, GAAlgebra is accessible and engaging with this popular text from Charles "Pat" McKeague! INTERMEDIATE ALGEBRA is infused with McKeague's passion for teaching mathematics. With years of classroom experience, he knows how to write in a way that you will understand and appreciate. McKeague's attention to detail and exceptionally clear writing style help you to move through each new concept with ease. Real-world applications in every chapter of this user-friendly book highlight the relevance of what you are learning. And studying is easier than ever with the book's multimedia learning resources, including CengageNOW for INTERMEDIATE ALGEBRA, a personalized online learning companion.
Editorial Reviews
Booknews
A standard textbook intended for a lecture-format class, with each section suited for discussion in a 45- to 50-minute class. New to this edition (fourth was 1990), each chapter begins with an opening that includes an introduction, general overview, and (in the first six chapter openings) study skills, i.e., skills intended to help students become organized and efficient with their time. Annotation c. Book News, Inc., Portland, OR (booknews.com)
Related Subjects
Meet the Author
Charles P. "Pat" McKeague earned his B.A. in Mathematics from California State University, Northridge, and his M.S. in Mathematics from Brigham Young University. A well-known author and respected educator, he is a full-time writer and a part-time instructor at Cuesta College. He has published twelve textbooks in mathematics covering a range of topics from basic mathematics to trigonometry. An active member of the mathematics community, Professor McKeague is a popular speaker at regional conferences, including the California Mathematics Council for Community Colleges, the American Mathematical Association of Two-Year Colleges, the National Council of Teachers of Mathematics, the Texas Mathematics Association of Two-Year Colleges, the New Mexico Mathematics Association of Two-Year Colleges, and the National Association for Developmental Education. He is a member of the American Mathematics Association for Two-Year Colleges, the Mathematics Association of America, the National Council of Teachers of Mathematics, and the California Mathematics Council for Community Colleges
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...
PAPERBACK Good 157685650X Ex library book with stickers and stampings. Overall good condition with clean text and good binding unless otherwise noted. Most items ship within 24 ...hours.Read moreShow Less
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apply algebra, fractions, decimals, percents, ratios and proportions, statistics and probability, formulas, and geometry Overcoming word problem pitfalls that can take down even a seasoned math aficionado With Express Review Guides: Math Word Problems, even math-phobic students can quickly gain the understanding to solve even the most challenging problems that involve money, distance, work, time, and more. In addition, this book includes: Fun brainstorming ideas, notes, and trivia Hundreds of practice questions and puzzlers Clear and complete explanations of key mathematics concepts Glossary of terms and appendices Samples of every type of math word problem found on standardized tests A pretest that pinpoints where to focus further study A posttest that provides proof of improvement-and the feeling of a job well done!
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Meet the Author
The team at LearningExpress is dedicated to publishing the best and most up-to-date skill-building, academic test-preparation, and career/vocational test-preparation titles available. We're constantly watching market trends, which allows us to be extraordinarily creative and authoritative as we stay ahead of the curve in our editorial approach. LearningExpress makes a point of being cutting edge in bringing you the best and most sought-after titles for all your skill-building
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97806181313 Study Skills Workbook And College Survival Insert
This best-selling workbook provides accessible, clearly written guidance to help students effectively study and learn mathematics. Examining students' strengths, weaknesses and learning styles, this text offers proven study tips, a homework system, and concrete techniques for such skills as note-taking, reducing math anxiety, improving test-taking, and much more. Students are encouraged to personalize the workbook's strategies and tools to help them succeed in their specific math course. Intended for use in any math class, math lab, study skills class, or developmental studies course, the workbook can be used for independent study or as a supplement to class lectures.
Test-taking coverage includes advice on managing and reducing negative self-talk, understanding the stages of memory and using memory techniques, and understanding and preventing specific test-taking errors.
Students learn to assess their own learning styles and to use the Math Autobiography Appendix to understand their personal math history.
Chapter Openers clearly outline a list of skills and strategies students will learn within the chapter.
Boxed features and examples throughout the text illustrate study skills and learning styles in a context that is relevant to students; this method of learning to recognize their strengths and weakness helps them shape their own set of study
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MAT5010T: Trig (2014-2015)
Trigonometry is often considered a "gateway" course because its content is necessary for further study in upper level mathematics and the sciences. This course introduces students to the fundamental concepts and application of trigonometry. Students will learn about the basic trigonometric functions and how to graph these functions. Topics covered in Trigonometry include: nonlinear functions, how to solve right triangle properties, how to use Law of Sines and Law of Cosines, trigonometric functions, the unit circle, radian measure, trigonometric identities, trigonometric graphs, and advanced algebra
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Key Message: Fundamentals of Differential Equations Fundamentals of Differential Equations presents the basic theory of differential equations and offers a variety of modern applications in science and engineering. Available in two versions, these flexible texts offer the instructor many choices in syllabus design, course emphasis (theory, methodology, applications, and numerical methods), and in using commercially available computer software
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Short Course In Discrete Mathematics
9780486439464
ISBN:
0486439461
Pub Date: 2004 Publisher: Dover Pubns
Summary: What sort of mathematics do I need for computer science? In response, a pair of professors at the University of California at San Diego created this text. Explores Boolean functions and computer arithmetic; logic; number theory and cryptography; sets and functions; equivalence and order; and induction, sequences, and series. Assumes some familiarity with calculus. Original 2005 edition.
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The two-line display scientific calculator combines statistics and advanced scientific functions and is a durable and affordable calculator for the classroom. The two-line display helps students explore math and science concepts in the classroom.
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About the Book
About the Contents:
Pretest
Helps you pinpoint where you need the most help and directs you to the corresponding sections of the book
Topic Area Reviews
Math Basics
Numbers (Signed Numbers and Fractions)
Linear Equations and Algebraic Fractions
Polynomials and Factoring
Inequalities, Absolute Value Inequalities, and Radicals
Introducing Quadratic Equations--Testing Solutions
Graphing and Systems of Equations
Functions
Story Problems
Customized Full-Length Exam
Covers all subject areas
Pretest that pinpoints what you need to study most
Clear, concise reviews of every topic
Targeted example problems in every chapter with solutions and explanations
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time has now come when graph theory should be part of the education of every serious student of mathematics and computer science, both for its own sake and to enhance the appreciation of mathematics as a whole. This book is an in-depth account of graph theory, written with such a student in mind; it reflects the current state of the subject and emphasizes connections with other branches of pure mathematics. The volume grew out of the author's earlier book, "Graph Theory - An Introductory Course", but its length is well over twice that of its predecessor, allowing it to reveal many exciting new developments in the subject. Recognizing that graph theory is one of several courses competing for the attention of a student, the book contains extensive descriptive passages designed to convey the flavor of the subject and to arouse interest. In addition to a modern treatment of the classical areas of graph theory such as coloring, matching, extremal theory, and algebraic graph theory, the book presents a detailed account of newer topics, including Szemer'edi's Regularity Lemma and its use, Shelah's extension of the Hales-Jewett Theorem, the precise nature of the phase transition in a random graph process, the connection between electrical networks and random walks on graphs, and the Tutte polynomial and its cousins in knot theory. In no other branch of mathematics is it as vital to tackle and solve challenging exercises in order to master the subject. To this end, the book contains an unusually large number of well thought-out exercises: over 600 in total. Although some are straightforward, most of them are substantial, and others will stretch even the most able reader. (source: Nielsen Book Data)
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Mathematics, Course 3 - 07 edition
Summary: The Student Edition develops skills that stretch beyond the classroom, such as higher-order thinking and the ability to read and write about math. All this is in a framework that begins to prepare students for algebra as soon as they open the bookAcceptable
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2006 Hardcover Fair 2006. HEAVY COVER WEAR. A used copy with heavy cover wear. Cardboard showing at the corners. Usual school stamps and labels which may include a book number or scribbling on the...show more
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Online Christian Education
Academics
Algebra 1 & 2
Algebra 1
This course provides the foundation for secondary mathematics study. Topics covered include algebraic expressions including polynomials and rational expressions, solving linear and quadratic equations, inequalities and systems of equations, and radicals and exponents. Emphasis is placed on problem solving and graphing.
Algebra 2
The course focuses on the study of linear, quadratic, logarithmic, exponential, rational and irrational algebraic functions as well as systems of linear equations and inequalities. Emphasis will be placed on graphing and mathematical modeling.
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A First Course in Numerical Methods
Written for undergraduate and beginning graduate students, this book provides comprehensive coverage of modern techniques in scientific computing. The book provides an in-depth treatment of fundamental issues and methods, as well as the reasons behind the success and failure of numerical software. Topics include numerical algorithms, round-off errors, nonlinear equations in one variable, and linear algebra.
MATLAB is used to solve numerous examples in the book. In addition, a supplemental set of MATLAB code files is available for download.
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Bucknell offers two essentially different introductory calculus courses, both of which presuppose a high school course in pre-calculus. MATH 201 is the first course in the core calculus sequence for mathematics, science, and engineering students; MATH 192 is a one-semester course designed for students in the social sciences.
You think math is beautiful and elegant. And you want to learn the basic language of the sciences and engineering and also be a great writer, speaker and thinker.
You'll learn from and work alongside faculty who are active researchers. Our mathematics department maintains a vibrant schedule of events and speakers and can connect you with summer research and study abroad opportunities. You'll connect with classmates in student organizations and have a shot at making a splash at the Putnam Competition.
Courses You Could Take
280.
Logic, Sets, and Proofs.
Skills and tools for independent reading, problem solving and exploration.
Associate Professor
The ability to decode patterns is a key that opens many locks. Whether you're talking about data security, business, medicine, chemistry or even art, seeing patterns means seeing solutions to real world problems
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. Consider a set of scalar quantities arranged in arectangular array containing
m
rows and
n
columns:This array will be called a
rectangular matrix
of order
m
by
n
, or, briefly, an
m
×
n
matrix. Notevery rectangular array is a matrix; to qualify as such it must obey the operational rules discussed below.The quantities
a
i j
are called the
entries
or
components
of the matrix. Preference will be given tothe latter unless one is talking about the computer implementation. As in the case of vectors, theterm "matrix element" will be avoided to lessen the chance of confusion with finite elements. Thetwo subscripts identify the row and column, respectively.Matrices are conventionally identified by
bold uppercase
letters such as
A
,
B
, etc. The entries of matrix
A
may be denoted as
A
i j
or
a
i j
, according to the intended use. Occassionally we shall usethe short-hand component notation
A
=
[
a
i j
]
.
Where Do Matrices Come From?
Although we speak of "matrix algebra" as embodying vectors as special cases of matrices, in practice the quantities of primary interest to the structural engineer are vectors rather than matrices.For example, an engineer may be interested in displacement vectors, force vectors, vibrationeigenvectors, buckling eigenvectors. In finite element analysis even stresses and strains are oftenarrangedas vectors although they are really tensors.On the other hand, matrices are rarely the quantities of primary interest: they work silently in the background where they are normally engaged in operating on vectors.
Why use Matrices?
We use matrices in mathematics and engineering because often we need to deal with severalvariables at once—e.g. the coordinates of a point in the plane are written (x, y) or in space as(x, y, z) and these are often written as column matrices in the form:It turns out that many operations that are needed to be performed on coordinates of points arelinear operations and so can be organized in terms of rectangular arrays of numbers, matrices.Then we find that matrices themselves can under certain conditions be added, subtracted andmultiplied so that there arises a whole new set of algebraic rules for their manipulation.In general, an (n × m)–matrix A looks like:Here, the entries are denoted aij.This branch of mathematics is used by engineers and applied scientists to design and analyzecomplex systems. Civil engineers use this to design and analyze load-bearing structures such as bridges. Mechanical engineers use it to design and analyze suspension systems, and electricalengineers use it to design and analyze electrical circuits. Electrical, biomedical, and aerospaceengineers use it to enhance X rays, tomographs, and images from space.
Matrices used in science and engineering
Cabibbo-Kobayashi-Maskawa matrix — a unitary matrix used in particle physics to describe thestrength of flavour-changing weak decays.Density matrix — a matrix describing the statistical state of a quantum system. Hermitian non-negative and with trace 1.Fundamental matrix (computer vision) — a 3 × 3 matrix in computer vision that relatescorresponding points in stereo images.Fuzzy associative matrix — a matrix in artificial intelligence, used in machine learning processes.Gamma matrices — 4 × 4 matrices in quantum field theory.Gell-Mann matrices — a generalization of the Pauli matrices, these matrices are one notablerepresentation of the infinitesimal generators of the special unitary group, SU(3).
Hamiltonian matrix — a matrix used in a variety of fields, including quantum mechanics andlinear quadratic regulator (LQR) systems.Irregular matrix — a matrix used in computer science which has a varying number of elementsin each row.Overlap matrix — a type of Gramian matrix, used in quantum chemistry to describe the inter-relationship of a set of basis vectors of a quantum system.S matrix — a matrix in quantum mechanics that connects asymptotic (infinite past and future) particle states.State transition matrix — Exponent of state matrix in control systems.Substitution matrix — a matrix from bioinformatics, which describes mutation rates of aminoacid or DNA sequences.Z-matrix — a matrix in chemistry, representing a molecule in terms of its relative atomicgeometry.
An example of use of matrix in civil engineering
In this example, we are trying to solve for the forces located in the beams.Since we do not haveany initial conditions, we must solve for variables, which is good. With variables we can changethem at will with very minimal hassle to observe the effects. We will apply the loads only on the joints. You can see that we have applied compression forces at all members, and labeled them for ease. Applying the method of joints, we get these equations, labeled 1-6 for the joints they aretaken from:
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Excellent condition. Interior is tight, bright and clean. Comes with library quality, richly printed hardcovers. Minor scuffing on the covers due to shelf wear. 100% Satisfaction ...Guaranteed. All items are carefully enclosed with bubble wrap. We ship promptly and worldwide via US Post and will email you a tracking number. Read moreShow Less
Ships from: Emigrant, MTIntended to follow the usual introductory physics courses, this book has the unique feature of addressing the mathematical needs of sophomores and juniors in physics, engineering and other related fields. Many original, lucid, and relevant examples from the physical sciences, problems at the ends of chapters, and boxes to emphasize important concepts help guide the student through the material.
Beginning with reviews of vector algebra and differential and integral calculus, the book continues with infinite series, vector analysis, complex algebra and analysis, ordinary and partial differential equations. Discussions of numerical analysis, nonlinear dynamics and chaos, and the Dirac delta function provide an introduction to modern topics in mathematical physics.
This new edition has been made more user-friendly through organization into convenient, shorter chapters. Also, it includes an entirely new section on Probability and plenty of new material on tensors and integral transforms.
Some praise for the previous
—
Editorial Reviews
From the Publisher
From the reviews of the second — —Zentralblatt MATH
"This new edition … of Mathematical Methods is designed to be used in an upper-division undergraduate course for physics and engineering majors. … The order of presentation is particularly good. … An additional strength of the book is the inclusion of chapters on nonlinear dynamics and probability. These chapters give a good introduction to chaos theory and the basic concepts of probability for beginning statistical mechanics students. … Overall, the volume is well written … . Summing Up: Highly recommended. Upper-division undergraduates." (E. Kincanon, Choice, Vol. 46 (8), April, 2009)
"The purpose of this book is to provide a comprehensive survey of the mathematics underlying theoretical physics at the level of graduate students entering research … . It is also intended to serve the research scientist or engineer who needs a quick refresher course in the subject … . This volume is intended to help bridge the wide gap separating the level of mathematical sophistication expected of students of introductory physics from that expected of students of advanced courses of undergraduate physics and engineering." (Teodora-Liliana Radulescu, Zentrablatt MATH, Vol. 1153
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Summary: This manual is organized to follow the sequence of topics in the text, and provides an easy-to-follow, step-by-step guide with worked-out examples to help students fully understand and get the most out of their graphing calculator. Compatible models include the popular TI-83/84 Plus and MathPrint. This manual will be packaged with every text.
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Beginning Algebra - With CD - 4th edition
Summary: For college-level courses in beginning or elementary algebra.
Elayn Martin-Gay's success as a developmental math author and teacher starts with a strong focus on mastering the basics through well-written explanations, innovative pedagogy and a meaningful, integrated program of learning resources. The revisions provide new pedagogy and resources to build student confidence, help students develop basic skills and understand concepts, and provide the highest ...show morelevel of instructor and adjunct support.
Martin-Gay's series is well known and widely praised for an unparalleled ability to:
Relate to students through real-life applications that are interesting, relevant, and practical.
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Test better: The new Chapter Test Prep Video shows Martin-Gay working step-by-step video solutions to every problem in each Chapter Test to enhance mastery of key chapter content.
Study better: New, integrated Study Skills Reminders reinforce the skills introduced in section 1.1, "Tips for Success in Mathematics" to promote an increased focus on the development of all-important study skills.
Learn better: The enhanced exercise sets and new pedagogy, like the Concept Checks, mean that students have the tools they need to learn successfully.
Martin-Gay believes that every student can succeed, and with each successive edition enhances her pedagogy and learning resources to provide evermore relevant and useful tools to help students and instructors achieve success09 +$3.99 s/h
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1 of 6
Algebra Academy Curriculum Components
Individualized Prescriptive Lessons™ (IPLs)
During Phase I of Algebra Readiness and Algebra I, students work their way individually through a series of Individualized Prescriptive Lessons™ (IPLs). Students begin this process by completing a diagnostic assessment designed to measure a student's knowledge and level of understanding of a wide range of basic math concepts. Based on the outcome of the assessment, each student is prescribed lessons in math concepts for which they need remediation. Each lesson begins with a practice test, and students then proceed with the lesson in preparation for successfully completing a mastery test at the conclusion of the lesson. Students must pass each prescribed mastery test before moving on to the next lesson. IPLs are computer based and used extensively in Phase I and also for targeted remediation in Phases II and III. Students are required to achieve mastery of basic math concepts in Phase I before they can begin applying their skills during the project-based curriculum delivered in Phase II and Phase III.
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0131107348Math for Merchandising: A Step-by-Step Approach (3rd Edition)
This book takes users step by step through the concepts of merchandising math. It is organized so that the chapters parallel a career path in the merchandising industry. The book begins with coverage of fundamental math concepts used in merchandising and progresses through the forms and math skills needed to buy, price, and re-price merchandise. Next readers learn the basics of creating and analyzing six-month plans. The final section of the book introduces math and merchandising concepts that are typically used at the corporate level. For individuals pursuing a career in merchandising
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Chapter 3: Graphs, Linear Equations, and Functions 3.1 The Rectangular Coordinate System 3.2 The Slope of a Line 3.3 Linear Equations in Two Variables Summary Exercises on Slopes and Equations of Lines 3.4 Linear Inequalities in Two Variables 3.5 Introduction to FunctionsBlue Cloud Books Phoenix, AZ
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Simple Equations App is for students from 4th to 8th grades. About180 problems on the above topics are included. Tutorial on each topic is included.
Simple Equations App Features - Simple Equations is for students from 4th to 8th grades. - There are about 180 equations at different levels to solve. - If you are familiar with simple math like addition/ subtraction/ multiplication/ division then this app is for you. - A Tutorial with examples on solving the equation is provided. - This app will make student ready for algebra and helps in improving the critical thinking. - This is a must app if you are preparing for any magnet/ gifted and talented exams. • This app teaches about - Variables, Constants, algebraic expressions - Writing algebraic expressions - Evaluating algebraic expressions - Like Terms, Combining Like Terms - Solving single step equations - Solving two step equations.
2) General Features - These are common features on all apps from sarmacs.com: The screens are designed for simplicity and easy navigation. No display of ads on any screen. No location tracking. No offline notifications or messages.
3) Our Privacy Policy: We do not collect any information from the user. We do not ask the user for any personal details like name, age etc. We do not transmit any information on user. We want to provide a very safe environment for kids to use our apps.
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What is the MathBridge Program?
The MathBridge program is a Pre-Calculus math course designed to allow students to navigate learning paths based on their level of readiness, creating a customized Pre Calculus course that meets each student's particular learning needs. The course includes an extensive algebra and geometry review and will prepare students to be successful in a high school or college level Calculus I course.
When is the MathBridge Program?
The Math Bridge will last one week during the Summer 2014 semesters at UCCS. Students will will spend 3 hours each day on MathBridge Online Coursework in a UCCS math lab with professional assistance.
The 2014 session runs during the afternoon (1-4 PM), Monday through Friday, June 2-6 on the UCCS campus.
What will I get from the MathBridge Program?
MathBridge students will be able to address any gaps in their own math understanding to gain a solid foundation for success in Calculus I. Coursework will cover Algebra II, Geometry, Trigonometry, and Pre-Calculus concepts. The course is designed to allow students to enter a high school or college level Calculus I course without any need for remediation.
What is the format of the MathBridge course?
MathBridge students will do the majority of the coursework online using Blackboard, ALEKS, and other internet resources in a UCCS math lab. The instructor will be there during class times to support student understanding and success.
Who can take the MathBridge Pre-Calculus Course?
Any student who has taken Algebra II can enroll.
What is the cost?
The cost of the MathBridge Pre-Calculus course is $150for the 1-week course.
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Thinkfinity Lesson Plans
Title: Mathematical Proofs
Description:
In this lesson, from Science NetLinks, students explore the nature of logic, evidence, and proofs in the context of mathematics. This lesson would be appropriate after students are familiar with the Pythagorean Theorem.
Standard(s): [MA2013] MI1 (9-12) 12: Summarize the history of probability, including the works of Blaise Pascal; Pierre de Fermat; Abraham de Moivre; and Pierre-Simon, marquis de Laplace. (Alabama)
Subject: Mathematics Title: Mathematical Proofs Description: In this lesson, from Science NetLinks, students explore the nature of logic, evidence, and proofs in the context of mathematics. This lesson would be appropriate after students are familiar with the Pythagorean Theorem. Thinkfinity Partner: Science NetLinks Grade Span: 9,10,11,12
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king how one does mathematical research is like asking how a composer creates a masterpiece. No one really knows. However, it is a recognized fact that problem solving plays an important role in training the mind of a researcher. It would not be an exaggeration to say that the ability to do mathematical research lies essentially asking 'well-posed' questions. The approach taken by the authors in "Problems in Algebraic Number Theory" is based on the principle that questions focus and orient the mind. The book is a collection of about 500 problems in algebraic number theory, systematically arranged to reveal ideas and concepts in the evolution of the subject. While some problems are easy and straightforward, others are more difficult. For this new edition the authors added a chapter and revised several sections. The text is suitable for a first course in algebraic number theory with minimal supervision by the instructor.The exposition facilitates independent study, and students having taken a basic course in calculus, linear algebra, and abstract algebra will find these problems interesting and challenging. For the same reasons, it is ideal for non-specialists in acquiring a quick introduction to the subject. (source: Nielsen Book Data)
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The two books under review are introductory combinatorics texts, also suitable for individual study. I shall refer to them by the combination of the first letters of the authors' last names: AS and WG. Both book are thoughtfully written, contain plenty of material and exercises. Understandably, there is a considerable overlap. Common to the books are the essential topics, such as permutations and combinations, the Inclusion-Exclusion principle, generating functions and recurrence relations, graphs and graph algorithms, groups of permutations, counting patterns, Pólya counting, the pigeonhole principle and Ramsey theory, and Catalan and Sterling numbers.
AS features chapters on integer partitions, barely touched in WG, and, in general, covers more expansively the common topics, especially, group and Ramsey theories, matching and marriages. WG, on the other hand, devotes separate chapters to coding theory, Latin squares, balanced incomplete block designs, and applications of linear algebra to combinatorics. AS (p. 228) observes that "Latin squares are interesting and important combinatorial objects, but because of shortage of space we are not able to discuss them in this book." There are briefer, say section-level, differences: rook polynomials, Euclidean Ramsey theory (AS), sum-free sets, exponential generating functions (WG).
In both books exercises come in two categories: A and B. The A exercises are supplied with solutions, the B exercises (which in WG are collected in separate sections) are not. WG also offers lists of problems — generally more sophisticated questions — that are accompanied by hints and solutions at the end of the book.
Both books begin with extensive Introductions that outline the contents by posing questions and problems that the books deal with subsequently. AS offers short biographical information in footnotes; WG in a separate Appendix. In addition, WG covers some background information (proof techniques, matrices and vectors) in two separate appendices.
WG provides numerous fragments of Mathematica code and this is a nice touch. For example, on pp. 259–260 the authors discuss the problem of determining the number of ways to cover a 2×6 board with nonoverlapping dominos. They form a digraph and its adjacency matrix A and then rely on Mathematica's
B = Inverse[IdentityMatrix[6] - x*A];
B[[2,2]]
to obtain the generating function
(1 – 2x2 + x4)/(1- x – 6x2 – x3 + 6x4 + 2x5 – x6)
which they then expand into a power series with
Series[%,{x,0,10}]
and subsequently elucidate the meanings of every coefficient.
For a balance check, AS goes to a considerably greater length introducing elements of group theory as a tool towards Frobenius and Pólya counting theorems.
At this point I would like to confess that the two books have been sitting on my desk for a while now. Somehow my first impression was so distressing that I could not see the point of reviewing them. As luck would have it, I opened the books the first time on pages with misprints. For AS, it was a confusing swap of ">" for "<" on page 296. For WG, it was a section titled "Applications of P(n, k) and C(n, k)" that had no mention of P(n, k).
Truth be told, I actually came across only a few more typos in AS and eventually found the text very readable and useful. I can't say the same of WG. The number of typos there is bewildering. For example, on pages 196–197 there are three:
B1A4B3B4B5B6B7 instead of B1B2B3B4B5B6B7
ainC instead of a∈C, and
N instead of n.
These were actually trifles. On page 73, Example 4.3 refers to Example 2.2 (p. 27), while the reference should be to Example 2.6 (p. 31) The diagram in the latter example is exactly the same as that accompanying Example 4.3, even though the diagram is not quite suitable for the earlier stage.
I truly came to grief with the WG's index. None of the terms I wanted to check could be found on the pages the Index pointed me to. Only after I decided to record the discrepancies, I realized that a major part of the index was off by 2 pages. It's not that, bad one may say, but not every student will be as savvy or as determined to discover the truth. Further, the Index contains the word "Arrangement", but it is not defined (far as I can see) anywhere in the book; what the index poinst to is the definition of "Derangement".
To give one more example, Theorem 13.3 reads, "The balanced bipartite graph B has exactly Per(B(G)) perfect matchings." As a matter of fact the term "balanced bipartite graph" is not found in the index and, as far as I can judge, is not defined anywhere in the book, at least not where I would expect it to be. The same holds for the "Cancellation law" entry.
He brings out the best in his editors and asks as much of them as he does of himself. As such, working with John is a joyous challenge and one of the highlights of my career.
This note gave me a point of reference and an impulse to pull myself together. Whether or not this is due to the authors' failure to engage the editors at CRC, the fact is that the latter did a lousy job editing the books, crucially so for WG.
Partitions and Generating Functions The Generating Function for the Partition Numbers A Quick(ish) Way of Finding p(n) An Upper Bound for the Partition Numbers The Hardy–Ramanujan Formula The Story of Hardy and Ramanujan
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Algebra and Trigonometry: A Graphing Approach
Browse related Subjects
...Read More-edge design, and innovative resources, make teaching easier and help students succeed in mathematics. This edition, intended for algebra and trigonometry courses that require the use of a graphing calculator, includes a moderate review of algebra to help students entering the course with weak algebra skills
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College Algebra - With CD - 4th edition
ISBN13:978-0534405991 ISBN10: 0534405991 This edition has also been released as: ISBN13: 978-0534406196 ISBN10: 053440619X
Summary: James Stewart, author of the worldwide, best-selling Calculus texts, along with two of his former Ph.D. students, Lothar Redlin and Saleem Watson, collaborated in writing this text to address a problem they frequently saw in their calculus courses: many students were not prepared to think mathematically but attempted instead to memorize facts and mimic examples. College Algebra was written specifically to help students learn to think mathematically and to develop tru...show moree problem-solving skills. This comprehensive, evenly paced book highlights the authors' commitment to encouraging conceptual understanding. To implement this goal, Stewart, Redlin, and Watson incorporate technology, the rule of four, real-world applications, and extended projects and writing exercises to enhance a central core of fundamental skills.
Benefits:
NEW!-Each chapter now has a new Chapter Overview that gives the overarching concept of the chapter and reveals how the ideas of the chapter are relevant in modeling real-life situations.
NEW!-A new Instructor's Guide contains points to stress, suggested time to allot, text discussion topics, core materials for lecture, workshop/discussion suggestions, group work exercises in a form suitable for handout, and suggested homework problems.
NEW!-Available with the text, BCA Testing is a revolutionary, Internet-ready, text-specific testing suite that allows instructors to customize exams and track student progress in an accessible, browser-based format. BCA offers full algorithmic generation of problems and free response mathematics.
NEW!-Each new copy of this text includes FREE access to BCA Tutorial, a text-specific, interactive, Web-based tutorial system. Like BCA Testing, it is browser-based, making it an intuitive mathematical guide even for students with little technological proficiency. So sophisticated, it's simple, BCA Tutorial allows students to work with real math notation in real time, providing instant analysis and feedback. The entire textbook is available in PDF format through BCA Tutorial, as are section-specific video tutorials, unlimited practice problems, and additional student resources such as a glossary, Web links, and more. The tracking program built into the instructor version of the software enables instructors to carefully monitor student progress. Results flow automatically to your gradebook.
An extensive Focus on Problem Solving Section concludes the "Preliminaries" chapter, to provide students with the training they need before moving into new material.
Mathematical Modeling sections, which follow the ends of every chapter, show how algebra can be applied to model real-life situations.
Discovery/Discussion exercises and Projects encourage group learning, extended thinking about a problem, and learning by writing about mathematics.
The material on graphing calculators is incorporated where it is most appropriate in the regular sections. However, subsections, examples, and exercises that deal with graphing devices are still labeled with an icon so that those who prefer not to use the graphing calculator can skip this material.
Short vignettes called "Mathematics in the Modern World," show that mathematics is a living science crucial to the scientific and technological progress of recent times, as well as to the social, behavioral, and life sciences.
NEW!-Packaged FREE with every text! Accessed seamlessly through BCA Tutorial, vMentor provides tutorial help that can substantially improve student performance, increase test scores, and enhance technical aptitude. Your students will have access, via the Web, to highly qualified tutors with thorough knowledge of our textbooks. When students get stuck on a particular problem or concept, they need only log on to vMentor, where they can talk (using their own computer microphones) to vMentor tutors who will skillfully guide them through the problem using the interactive whiteboard for illustration.
Real-World Applications: applications from engineering, physics, chemistry, business, biology, environmental studies, and other fields show how mathematics is used to model real-life situations.
NEW!-A new icon in the text's Exercise sets distinguishes and highlights the application exercises. Each application exercise also has a brief title so the topic of the exercise can be quickly determined.
Mathematical Vignettes: short biographies of interesting mathematicians as well as applications of algebra to the real world.
Review Sections and Chapter Tests: Each chapter ends with an extensive review section, including a Chapter Test designed to help students gauge their progress. Brief answers to the odd-numbered exercises in each section, and to all questions in the Chapter Tests, are given at the back of the book.
Graphing Calculators and Computers: optional technology integrated throughout the text is marked by a special logo.
NEW!-The former Chapter 1 is a review chapter and has been renamed Chapter P, "Prerequisites."
NEW!-With more Focus on Modeling sections, this edition provides an even greater emphasis on modeling. These now appear at the end of every chapter, with the exception of the "Preliminary" chapter, which concludes with a Focus on Problem Solving section.
NEW!-In response to reviewers, the authors have moved the "Equations" chapter before the "Graphing" chapter. What used to be Chapter 3 is now Chapter 1 with added material on linear equations in Section 1.1.
NEW!-The chapter dealing with Systems of Linear Equations and Inequalities has been divided into two chapters (Chapters 6 and 7) to accomodate users whose course syllabus does not include matrices.
NEW!-Packaged with each new copy of the book, the Interactive Video Skillbuilder CD-ROM contains one video lesson for each section of the book. Problems worked are listed next to the video, so that students can work them ahead of time, if they choose. To help students evaluate their progress, each section contains a 10-question Web quiz, and each chapter contains a chapter test.
NEW!-Icons found throughout the text point students to appropriate material on the CD-ROM and the videos.
NEW!-A Resource Integration Guide in the Annotated Instructor's Edition demonstrates how the wide variety of text-specific and stand-alone supplements integrate with and complement the main text.
Chapter Overview. What is a Function?. Graphs of Functions. Increasing and Decreasing Functions: Average Rate of Change. Transformations of Functions. Quadratic Functions; Maxima and Minima. Combining Functions. Discovery Project: Iteration and Chaos. One-to-One Functions and Their Inverses. Focus on Modeling: Modeling with Functions.
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Please Read This Description- Used - Acceptable. Sorry, CD missing. Disc NOT included.Binding in good condition, light wear around edges. Light scuffs and scratches on front and back cover.No apparent...show more writing/highlighting.front Sorry, CD missing. Disc NOT included. Binding in good condition, light wear around edges. Light scuffs and scratches on front and back cover....show more No apparent writing/highlighting. front of OC Santa Ana, CA
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Calculus is a series of videos that introduce the fundamental concepts of calculus to both high school and college students. Renowned mathematics professor, Gilbert Strang, will guide students through a number of calculus topics to help them understand why calculus is relevant and important to understand.
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Course Resources
Mathematics Department
Welcome to the Mathematics department of Los Angeles Mission College. Our department offers a full spectrum of courses from basic mathematics through Calculus, Linear Algebra, and Differential Equations.
To best accommodate the needs of both our full-time and evening students, courses are scheduled throughout the day form 8:00 a.m. to 9:30 p.m. weekdays, as well as some classes offered on a Saturday schedule.
In addition to traditional lecture courses, several courses are taught through the Title V Math Center. The center has a fully equipped computer lab including state of the art lecture/presentation facilities. A separate room serves as the tutoring center and includes computers supporting interactive tutorial software. The center is also staffed by qualified tutors during operating hours and is supervised by full time staff assistants. Several small rooms are also available for group tutoring or consultations with instructors. Textbook-specific video tapes are available for checkout and viewing by students. Since its inception the center has become our students' first choice for supplemental instruction.
Tutoring and videos are also provided through the Library/Learning Center.
A Beautiful Formula: eiπ+ 1 = 0
By relating five of the most basic quantities in mathematics this formula becomes one of its most beautiful. Even the most elemental algebraic structures require the concept of zero and one. Pi, the ratio of the circumference of a circle to its diameter, is invariant throughout the universe. The properties of Euler's constant e have inspired whole mathematical systems. And while Euler offered a "proof" that negative numbers were not numbers, the imaginary unit i was readily accepted and its applications in physics and electrical engineering thoroughly developed.
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ISBN13:978-0618248575 ISBN10: 0618248579 This edition has also been released as: ISBN13: 978-0030256820 ISBN10: 0030256828
Summary: Ostebee and Zorn's approach applies reform principles to a rigorous calculus text. Conceptual understanding is the main goal of the text, and looking at mathematics from many representations (graphical, symbolic, numerical) is the main strategy for achieving this type of understanding. The key strengths of the text include combining symbolic manipulation with graphical and numerical representation, exercises of a varied nature and difficulty, and explanations written...show more to be understandable to student readers.
A student-friendly and approachable tone, numerous examples, critical-thinking questions, and supportive details and commentary help students successfully read and use the text.
Representation of mathematical concepts through a variety of viewpoints supports different learning styles. Students see the math worked out through multiple representations--graphically, numerically, and symbolically--to enhance conceptual understanding.
Proofs presented at point of use contribute significantly to helping students understand rigorous calculus concepts and develop analytic skills.
Varied exercise sets offer instructors more options for creating homework assignments. Basic Exercises, which are straightforward and focus on a single idea, help students build basic skills.
Further Exercises are a little more ambitious and may require the synthesis of several ideas, a deeper or more sophisticated understanding of basic concepts, or the use of a computer algebra system such as Maple or Mathematica. These are available for professors to assign when they would like to challenge their students and incorporate technology into their course.
Answers to Select Exercises can be found in the back of the text, enabling students to get immediate feedback and assess their understanding of the material.
Interludes are brief project-oriented expositions, with related exercises, that extend the concepts presented in the chapter. Professors have the opportunity to include these topics found at the end of the chapter as independent work, group work, or as a classroom activity. The Interludes include theoretical problems and proofs intended to enhance student understanding of the key calculus concepts.
11.1 Sequences and Their Limits 11.2 Infinite Series, Convergence, and Divergence 11.3 Testing for Convergence; Estimating Limits 11.4 Absolute Convergence; Alternating Series 11.5 Power Series 11.6 Power Series as Functions 11.7 Taylor Series Interlude: Fourier series
16.1 Line Integrals 16.2 More on Line Integrals; A Fundamental Theorem 16.3 Green's Theorem: Relating Line and Area Integrals 16.4 Surfaces and Their Parameterizations 16.5 Surface Integrals 16.6 Derivatives and Integrals of Vector Fields 16.7 Back to Fundamentals: Stokes's Theorem and the Divergence Theorem
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PrefaceMany people think there is only one "right" way to teach geometry. Fortwo millennia, the "right" way was Euclid's way, and it is still good inmany respects. But in the 1950s the cry "Down with triangles!" was heardin France and new geometry books appeared, packed with linear algebrabut with no diagrams. Was this the new "right" way, or was the "right" waysomething else again, perhaps transformation groups? In this book, I wish to show that geometry can be developed in fourfundamentally different ways, and that all should be used if the subject is tobe shown in all its splendor. Euclid-style construction and axiomatics seemthe best way to start, but linear algebra smooths the later stages by replacingsome tortuous arguments by simple calculations. And how can one avoidprojective geometry? It not only explains why objects look the way theydo; it also explains why geometry is entangled with algebra. Finally, oneneeds to know that there is not one geometry, but many, and transformationgroups are the best way to distinguish between them. Two chapters are devoted to each approach: The first is concrete andintroductory, whereas the second is more abstract. Thus, the first chapteron Euclid is about straightedge and compass constructions; the second isabout axioms and theorems. The first chapter on linear algebra is aboutcoordinates; the second is about vector spaces and the inner product. Thefirst chapter on projective geometry is about perspective drawing; the sec-ond is about axioms for projective planes. The first chapter on transforma-tion groups gives examples of transformations; the second constructs thehyperbolic plane from the transformations of the real projective line. I believe that students are shortchanged if they miss any of these fourapproaches to the subject. Geometry, of all subjects, should be about tak-ing different viewpoints, and geometry is unique among the mathematicaldisciplines in its ability to look different from different angles. Some prefer vii
7.
viii Prefaceto approach it visually, others algebraically, but the miracle is that they areall looking at the same thing. (It is as if one discovered that number theoryneed not use addition and multiplication, but could be based on, say, theexponential function.) The many faces of geometry are not only a source of amazement anddelight. They are also a great help to the learner and teacher. We all knowthat some students prefer to visualize, whereas others prefer to reason orto calculate. Geometry has something for everybody, and all students willfind themselves building on their strengths at some times, and workingto overcome weaknesses at other times. We also know that Euclid hassome beautiful proofs, whereas other theorems are more beautifully provedby algebra. In the multifaceted approach, every theorem can be given anelegant proof, and theorems with radically different proofs can be viewedfrom different sides. This book is based on the course Foundations of Geometry that I taughtat the University of San Francisco in the spring of 2004. It should bepossible to cover it all in a one-semester course, but if time is short, somesections or chapters can be omitted according to the taste of the instructor.For example, one could omit Chapter 6 or Chapter 8. (But with regret, Iam sure!)AcknowledgementsMy thanks go to the students in the course, for feedback on my raw lecturenotes, and especially to Gina Campagna and Aaron Keel, who contributedseveral improvements. Thanks also go to my wife Elaine, who proofread the first version of thebook, and to Robin Hartshorne, John Howe, Marc Ryser, Abe Shenitzer,and Michael Stillwell, who carefully read the revised version and saved mefrom many mathematical and stylistic errors. Finally, I am grateful to the M. C. Escher Company – Baarn – Hollandfor permission to reproduce the Escher work Circle Limit I shown in Figure8.19, and the explicit mathematical transformation of it shown in Figure8.10. This work is copyright (2005) The M. C. Escher Company. J OHN S TILLWELL San Francisco, November 2004 South Melbourne, April 2005
11.
1Straightedge and compassP REVIEW For over 2000 years, mathematics was almost synonymous with the geometry of Euclid's Elements, a book written around 300 BCE and used in school mathematics instruction until the 20th century. Eu- clidean geometry, as it is now called, was thought to be the founda- tion of all exact science. Euclidean geometry plays a different role today, because it is no longer expected to support everything else. "Non-Euclidean geome- tries" were discovered in the early 19th century, and they were found to be more useful than Euclid's in certain situations. Nevertheless, non-Euclidean geometries arose as deviations from the Euclidean, so one first needs to know what they deviate from. A naive way to describe Euclidean geometry is to say it concerns the geometric figures that can be drawn (or constructed as we say) by straightedge and compass. Euclid assumes that it is possible to draw a straight line between any two given points, and to draw a circle with given center and radius. All of the propositions he proves are about figures built from straight lines and circles. Thus, to understand Euclidean geometry, one needs some idea of the scope of straightedge and compass constructions. This chapter reviews some basic constructions, to give a quick impression of the extent of Euclidean geometry, and to suggest why right angles and parallel lines play a special role in it. Constructions also help to expose the role of length, area, and angle in geometry. The deeper meaning of these concepts, and the related role of numbers in geometry, is a thread we will pursue throughout the book. 1
12.
2 1 Straightedge and compass1.1 Euclid's construction axiomsEuclid assumes that certain constructions can be done and he states theseassumptions in a list called his axioms (traditionally called postulates). Heassumes that it is possible to: 1. Draw a straight line segment between any two points. 2. Extend a straight line segment indefinitely. 3. Draw a circle with given center and radius.Axioms 1 and 2 say we have a straightedge, an instrument for drawingarbitrarily long line segments. Euclid and his contemporaries tried to avoidinfinity, so they worked with line segments rather than with whole lines.This is no real restriction, but it involves the annoyance of having to extendline segments (or "produce" them, as they say in old geometry books).Today we replace Axioms 1 and 2 by the single axiom that a line can bedrawn through any two points. The straightedge (unlike a ruler) has no scale marked on it and hencecan be used only for drawing lines—not for measurement. Euclid separatesthe function of measurement from the function of drawing straight linesby giving measurement functionality only to the compass—the instrumentassumed in Axiom 3. The compass is used to draw the circle through agiven point B, with a given point A as center (Figure 1.1). A B Figure 1.1: Drawing a circle
13.
1.1 Euclid's construction axioms 3 To do this job, the compass must rotate rigidly about A after beinginitially set on the two points A and B. Thus, it "stores" the length of theradius AB and allows this length to be transferred elsewhere. Figure 1.2is a classic view of the compass as an instrument of measurement. It isWilliam Blake's painting of Isaac Newton as the measurer of the universe. Figure 1.2: Blake's painting of Newton the measurer The compass also enables us to add and subtract the length |AB| ofAB from the length |CD| of another line segment CD by picking up thecompass with radius set to |AB| and describing a circle with center D(Figure 1.3, also Elements, Propositions 2 and 3 of Book I). By addinga fixed length repeatedly, one can construct a "scale" on a given line, effec-tively creating a ruler. This process illustrates how the power of measuringlengths resides in the compass. Exactly which lengths can be measured inthis way is a deep question, which belongs to algebra and analysis. Thefull story is beyond the scope of this book, but we say more about it below. Separating the concepts of "straightness" and "length," as the straight-edge and the compass do, turns out to be important for understanding thefoundations of geometry. The same separation of concepts reappears indifferent approaches to geometry developed in Chapters 3 and 5.
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4 1 Straightedge and compass |A B| |CD| − |AB|C D |CD| + |AB| Figure 1.3: Adding and subtracting lengths1.2 Euclid's construction of the equilateral triangleConstructing an equilateral triangle on a given side AB is the first proposi-tion of the Elements, and it takes three steps: 1. Draw the circle with center A and radius AB. 2. Draw the circle with center B and radius AB. 3. Draw the line segments from A and B to the intersection C of the two circles just constructed.The result is the triangle ABC with sides AB, BC, and CA in Figure 1.4. C A B Figure 1.4: Constructing an equilateral triangle Sides AB and CA have equal length because they are both radii of thefirst circle. Sides AB and BC have equal length because they are both radiiof the second circle. Hence, all three sides of triangle ABC are equal.
15.
1.2 Euclid's construction of the equilateral triangle 5 This example nicely shows the interplay among • construction axioms, which guarantee the existence of the construc- tion lines and circles (initially the two circles on radius AB and later the line segments BC and CA), • geometric axioms, which guarantee the existence of points required for later steps in the construction (the intersection C of the two cir- cles), • and logic, which guarantees that certain conclusions follow. In this case, we are using a principle of logic that says that things equal to the same thing (both |BC| and |CA| equal |AB|) are equal to each other (so |BC| = |CA|). We have not yet discussed Euclid's geometric axioms or logic. Weuse the same logic for all branches of mathematics, so it can be assumed"known," but geometric axioms are less clear. Euclid drew attention toone and used others unconsciously (or, at any rate, without stating them).History has shown that Euclid correctly identified the most significant ge-ometric axiom, namely the parallel axiom. We will see some reasons forits significance in the next section. The ultimate reason is that there areimportant geometries in which the parallel axiom is false. The other axioms are not significant in this sense, but they should alsobe identified for completeness, and we will do so in Chapter 2. In particu-lar, it should be mentioned that Euclid states no axiom about the intersec-tion of circles, so he has not justified the existence of the point C used inhis very first proposition!A question arising from Euclid's constructionThe equilateral triangle is an example of a regular polygon: a geometricfigure bounded by equal line segments that meet at equal angles. Anotherexample is the regular hexagon in Exercise 1.2.1. If the polygon has nsides, we call it an n-gon, so the regular 3-gon and the regular 6-gon areconstructible. For which n is the regular n-gon constructible? We will not completely answer this question, although we will showthat the regular 4-gon and 5-gon are constructible. The question for generaln turns out to belong to algebra and number theory, and a complete answerdepends on a problem about prime numbers that has not yet been solved: mFor which m is 22 + 1 a prime number?
16.
6 1 Straightedge and compassExercisesBy extending Euclid's construction of the equilateral triangle, construct:1.2.1 A regular hexagon.1.2.2 A tiling of the plane by equilateral triangles (solid lines in Figure 1.5).1.2.3 A tiling of the plane by regular hexagons (dashed lines in Figure 1.5). Figure 1.5: Triangle and hexagon tilings of the plane1.3 Some basic constructionsThe equilateral triangle construction comes first in the Elements becauseseveral other constructions follow from it. Among them are constructionsfor bisecting a line segment and bisecting an angle. ("Bisect" is from theLatin for "cut in two.")Bisecting a line segmentTo bisect a given line segment AB, draw the two circles with radius AB asabove, but now consider both of their intersection points, C and D. Theline CD connecting these points bisects the line segment AB (Figure 1.6).
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1.3 Some basic constructions 7 C A B D Figure 1.6: Bisecting a line segment AB Notice also that BC is perpendicular to AB, so this construction can beadapted to construct perpendiculars. • To construct the perpendicular to a line L at a point E on the line, first draw a circle with center E, cutting L at A and B. Then the line CD constructed in Figure 1.6 is the perpendicular through E. • To construct the perpendicular to a line L through a point E not on L , do the same; only make sure that the circle with center E is large enough to cut the line L at two different points.Bisecting an angleTo bisect an angle POQ (Figure 1.7), first draw a circle with center O cut-ting OP at A and OQ at B. Then the perpendicular CD that bisects the linesegment AB also bisects the angle POQ. P A C D O Q B Figure 1.7: Bisecting an angle POQ
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8 1 Straightedge and compass It seems from these two constructions that bisecting a line segment andbisecting an angle are virtually the same problem. Euclid bisects the anglebefore the line segment, but he uses two similar constructions (Elements,Propositions 9 and 10 of Book I). However, a distinction between line seg-ments and angles emerges when we attempt division into three or moreparts. There is a simple tool for dividing a line segment in any number ofequal parts—parallel lines—but no corresponding tool for dividing angles.Constructing the parallel to a line through a given pointWe use the two constructions of perpendiculars noted above—for a pointoff the line and a point on the line. Given a line L and a point P outside L ,first construct the perpendicular line M to L through P. Then constructthe perpendicular to M through P, which is the parallel to L through P.Dividing a line segment into n equal partsGiven a line segment AB, draw any other line L through A and markn successive, equally spaced points A1 , A2 , A3 , . . . , An along L using thecompass set to any fixed radius. Figure 1.8 shows the case n = 5. Thenconnect An to B, and draw the parallels to BAn through A1 , A2 , . . . , An−1 .These parallels divide AB into n equal parts. L A5 A4 A3 A2 A1 A B Figure 1.8: Dividing a line segment into equal parts This construction depends on a property of parallel lines sometimes at-tributed to Thales (Greek mathematician from around 600 BCE): parallelscut any lines they cross in proportional segments. The most commonlyused instance of this theorem is shown in Figure 1.9, where a parallel toone side of a triangle cuts the other two sides proportionally.
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1.3 Some basic constructions 9 The line L parallel to the side BC cuts side AB into the segments APand PB, side AC into AQ and QC, and |AP|/|PB| = |AQ|/|QC|. A L P Q B C Figure 1.9: The Thales theorem in a triangle This theorem of Thales is the key to using algebra in geometry. Inthe next section we see how it may be used to multiply and divide linesegments, and in Chapter 2 we investigate how it may be derived fromfundamental geometric principles.Exercises1.3.1 Check for yourself the constructions of perpendiculars and parallels de- scribed in words above.1.3.2 Can you find a more direct construction of parallels? Perpendiculars give another important polygon—the square.1.3.3 Give a construction of the square on a given line segment.1.3.4 Give a construction of the square tiling of the plane. One might try to use division of a line segment into n equal parts to dividean angle into n equal parts as shown in Figure 1.10. We mark A on OP and B atequal distance on OQ as before, and then try to divide angle POQ by dividing linesegment AB. However, this method is faulty even for division into three parts. P Q A B O Figure 1.10: Faulty trisection of an angle1.3.5 Explain why division of AB into three equal parts (trisection) does not al- ways divide angle POQ into three equal parts. (Hint: Consider the case in which POQ is nearly a straight line.)
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10 1 Straightedge and compass The version of the Thales theorem given above (referring to Figure 1.9) hasan equivalent form that is often useful.1.3.6 If A, B,C, P, Q are as in Figure 1.9, so that |AP|/|PB| = |AQ|/|QC|, show that this equation is equivalent to |AP|/|AB| = |AQ|/|AC|.1.4 Multiplication and divisionNot only can one add and subtract line segments (Section 1.1); one can alsomultiply and divide them. The product ab and quotient a/b of line seg-ments a and b are obtained by the straightedge and compass constructionsbelow. The key ingredients are parallels, and the key geometric propertyinvolved is the Thales theorem on the proportionality of line segments cutoff by parallel lines. To get started, it is necessary to choose a line segment as the unit oflength, 1, which has the property that 1a = a for any length a.Product of line segmentsTo multiply line segment b by line segment a, we first construct any triangleUOA with |OU| = 1 and |OA| = a. We then extend OU by length b to B 1and construct the parallel to UA through B1 . Suppose this parallel meetsthe extension of OA at C (Figure 1.11). By the Thales theorem, |AC| = ab. B1 b Multiply by a U 1 a ab O C A Figure 1.11: The product of line segments
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1.4 Multiplication and division 11Quotient of line segmentsTo divide line segment b by line segment a, we begin with the same triangleUOA with |OU| = 1 and |OA| = a. Then we extend OA by distance b toB2 and construct the parallel to UA through B2 . Suppose that this parallelmeets the extension of OU at D (Figure 1.12). By the Thales theorem, |UD| = b/a. D b/a Divide by a U 1 a b O B2 A Figure 1.12: The quotient of line segments The sum operation from Section 1.1 allows us to construct a segmentn units in length, for any natural number n, simply by adding the segment1 to itself n times. The quotient operation then allows us to construct asegment of length m/n, for any natural numbers m and n = 0. These arewhat we call the rational lengths. A great discovery of the Pythagoreanswas that some lengths are not rational, and that some of these "irrational"lengths can be constructed by straightedge and compass. It is not knownhow the Pythagoreans made this discovery, but it has a connection with theThales theorem, as we will see in the next section.ExercisesExercise 1.3.6 showed that if PQ is parallel to BC in Figure 1.9, then |AP|/|AB| =|AQ|/|AC|. That is, a parallel implies proportional (left and right) sides. Thefollowing exercise shows the converse: proportional sides imply a parallel, or(equivalently), a nonparallel implies nonproportional sides.1.4.1 Using Figure 1.13, or otherwise, show that if PR is not parallel to BC, then |AP|/|AB| = |AR|/|AC|.
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12 1 Straightedge and compass A L P Q R B C Figure 1.13: Converse of the Thales theorem1.4.2 Conclude from Exercise 1.4.1 that if P is any point on AB and Q is any point on AC, then PQ is parallel to BC if and only if |AP|/|AB| = |AQ|/|AC|. The "only if" direction of Exercise 1.4.2 leads to two famous theorems—thePappus and Desargues theorems—that play an important role in the foundationsof geometry. We will meet them in more general form later. In their simplestform, they are the following theorems about parallels.1.4.3 (Pappus of Alexandria, around 300 CE) Suppose that A, B,C, D, E, F lie al- ternately on lines L and M as shown in Figure 1.14. M C E A L O F B D Figure 1.14: The parallel Pappus configuration Use the Thales theorem to show that if AB is parallel to ED and FE is parallel to BC then |OA| |OC| = . |OF| |OD| Deduce from Exercise 1.4.2 that AF is parallel to CD.
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1.5 Similar triangles 131.4.4 (Girard Desargues, 1648) Suppose that points A, B,C, A , B ,C lie on con- current lines L , M , N as shown in Figure 1.15. (The triangles ABC and A B C are said to be "in perspective from O.") L A A B B M O C C N Figure 1.15: The parallel Desargues configuration Use the Thales theorem to show that if AB is parallel to A B and BC is parallel to B C , then |OA| |OA | = . |OC| |OC | Deduce from Exercise 1.4.2 that AC is parallel to A C .1.5 Similar trianglesTriangles ABC and A B C are called similar if their corresponding anglesare equal, that is, if angle at A = angle at A (= α say), angle at B = angle at B (= β say), angle at C = angle at C (= γ say).It turns out that equal angles imply that all sides are proportional, so wemay say that one triangle is a magnification of the other, or that they havethe same "shape." This important result extends the Thales theorem, andactually follows from it.
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1.5 Similar triangles 15 We got this result by making the angles α in the two triangles coincide.If we make the angles β coincide instead, we similarly find that the sidesopposite to α and γ are proportional. Thus, in fact, all corresponding sidesof similar triangles are proportional. This consequence of the Thales theorem has many implications. Ineveryday life, it underlies the existence of scale maps, house plans, engi-neering drawings, and so on. In pure geometry, its implications are evenmore varied. Here is just one, which shows why square roots and irrationalnumbers turn up in geometry. √The diagonal of the unit square is 2The diagonals of the unit square cut it into four quarters, each of which isa triangle similar to the half square cut off by a diagonal (Figure 1.17). d/ 2 d/ 2 d/ 2 d/ 2 1 Figure 1.17: Quarters and halves of the square Each of the triangles in question has one right angle and two half rightangles, so it follows from the theorem above that corresponding sides ofany two of these triangles are proportional. In particular, if we take the halfsquare, with short side 1 and long side d, and compare it with the quartersquare, with short side d/2 and long side 1, we get short 1 d/2 = = . long d 1 √Multiplying both sides of the equation by 2d gives 2 = d 2 , so d = 2.
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16 1 Straightedge and compass √ The great, but disturbing, discovery of the Pythagoreans is that 2 is √irrational. That is, there are no natural numbers m and n such 2 = m/n. If there are such m and n we can assume that they have no common √divisor, and then the assumption 2 = m/n implies 2 = m2 /n2 squaring both sides 2 2hence m = 2n multiplying both sides by n2hence m2 is evenhence m is even since the square of an odd number is oddhence m = 2l for some natural number lhence m2 = 4l 2 = 2n2hence n2 = 2l 2hence n2 is evenhence n is even since the square of an odd number is odd.Thus, m and n have the common divisor 2, contrary to assumption. Ouroriginal assumption is therefore false, so there are no natural numbers m √and n such that 2 = m/n.Lengths, products, and areaGeometry obviously has to include the diagonal of the unit square, hencegeometry includes the study of irrational lengths. This discovery trou-bled the ancient Greeks, because they did not believe that irrational lengthscould be treated like numbers. In particular, the idea of interpreting theproduct of line segments as another line segment is not in Euclid. It firstappears in Descartes' G´ om´ trie of 1637, where algebra is used systemat- e eically in geometry for the first time. The Greeks viewed the product of line segments a and b as the rectan-gle with perpendicular sides a and b. If lengths are not necessarily num-bers, then the product of two lengths is best interpreted as an area, and theproduct of three lengths as a volume—but then the product of four lengthsseems to have no meaning at all. This difficulty perhaps explains why al-gebra appeared comparatively late in the development of geometry. On theother hand, interpreting the product of lengths as an area gives some re-markable insights, as we will see in Chapter 2. So it is also possible thatalgebra had to wait until the Greek concept of product had exhausted itsusefulness.
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1.6 Discussion 17ExercisesIn general, two geometric figures are called similar if one is a magnification of the long sideother. Thus, two rectangles are similar if the ratio short side is the same for both. √ √ 2+1 2−1 1 1 Figure 1.18: A pair of similar rectangles √ 2+1 √11.5.1 Show that 1 = 2−1 and hence that the two rectangles in Figure 1.18 are similar.1.5.2 Deduce that if a rectangle with long side a and short side b has the same shape as the two above, then so has the rectangle with long side b and short side a − 2b. √ This simple observation gives another proof that 2 is irrational: √1.5.3 Suppose that 2 + 1 = m/n, where m and n are natural numbers √ m as with small as possible. Deduce from Exercise 1.5.2 that we also have 2 + 1 = n/(m − 2n). This is a contradiction. Why? √1.5.4 It follows from Exercise 1.5.3 that 2+1 is irrational. Why does this imply √ that 2 is irrational?1.6 DiscussionEuclid's Elements is the most influential book in the history of mathemat-ics, and anyone interested in geometry should own a copy. It is not easyreading, but you will find yourself returning to it year after year and notic-ing something new. The standard edition in English is Heath's translation,which is now available as a Dover reprint of the 1925 Cambridge Univer-sity Press edition. This reprint is carried by many bookstores; I have evenseen it for sale at Los Angeles airport! Its main drawback is its size—threebulky volumes—due to the fact that more than half the content consists of
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18 1 Straightedge and compassHeath's commentary. You can find the Heath translation without the com-mentary in the Britannica Great Books of the Western World, Volume 11.These books can often be found in used bookstores. Another, more recent,one-volume edition of the Heath translation is Euclid's Elements, edited byDana Densmore and published by Green Lion Press in 2003. A second (slight) drawback of the Heath edition is that it is about 80years old and beginning to sound a little antiquated. Heath's English issometimes quaint, and his commentary does not draw on modern researchin geometry. He does not even mention some important advances that wereknown to experts in 1925. For this reason, a modern version of the El-ements is desirable. A perfect version for the 21st century does not yetexist, but there is a nice concise web version by David Joyce at This Elements has a small amount of commentary, but I mainly rec-ommend it for proofs in simple modern English and nice diagrams. Thediagrams are "variable" by dragging points on the screen, so each diagramrepresents all possible situations covered by a theorem. For modern commentary on Euclid, I recommend two books: Euclid:the Creation of Mathematics by Benno Artmann and Geometry: Euclid andBeyond by Robin Hartshorne, published by Springer-Verlag in 1999 and2000, respectively. Both books take Euclid as their starting point. Artmannmainly fills in the Greek background, although he also takes care to makeit understandable to modern readers. Hartshorne is more concerned withwhat came after Euclid, and he gives a very thorough analysis of the gapsin Euclid and the ways they were filled by modern mathematicians. Youwill find Hartshorne useful supplementary reading for Chapters 2 and 3,where we examine the logical structure of the Elements and some of itsgaps. The climax of the Elements is the theory of regular polyhedra in BookXIII. Only five regular polyhedra exist, and they are shown in Figure 1.19.Notice that three of them are built from equilateral triangles, one fromsquares, and one from regular pentagons. This remarkable phenomenonunderlines the importance of equilateral triangles and squares, and drawsattention to the regular pentagon. In Chapter 2, we show how to constructit. Some geometers believe that the material in the Elements was chosenvery much with the theory of regular polyhedra in mind. For example,Euclid wants to construct the equilateral triangle, square, and pentagon inorder to construct the regular polyhedra.
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1.6 Discussion 19 Cube Dodecahedron Tetrahedron Octahedron Icosahedron Figure 1.19: The regular polyhedra It is fortunate that Euclid did not need regular polygons more complexthan the pentagon, because none were constructed until modern times. Theregular 17-gon was constructed by the 19-year-old Carl Friedrich Gaussin 1796, and his discovery was the key to the "question arising" from theconstruction of the equilateral triangle in Section 1.2: for which n is theregular n-gon constructible? Gauss showed (with some steps filled in byPierre Wantzel in 1837) that a regular polygon with a prime number p of msides is constructible just in case p is of the form 22 + 1. This result givesthree constructible p-gons not known to the Greeks, because 24 + 1 = 17, 28 + 1 = 257, 216 + 1 = 65537 mare all prime numbers. But no larger prime numbers of the form 2 2 + 1 areknown! Thus we do not know whether a larger constructible p-gon exists. These results show that the Elements is not all of geometry, even ifone accepts the same subject matter as Euclid. To see where Euclid fitsin the general panorama of geometry, I recommend the books Geometryand the Imagination by D. Hilbert and S. Cohn-Vossen, and Introductionto Geometry by H. S. M. Coxeter (Wiley, 1969).
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2Euclid's approach to geometryP REVIEW Length is the fundamental concept of Euclid's geometry, but several important theorems seem to be "really" about angle or area—for example, the theorem on the sum of angles in a triangle and the Pythagorean theorem on the sum of squares. Also, Euclid often uses area to prove theorems about length, such as the Thales theorem. In this chapter, we retrace some of Euclid's steps in the theory of angle and area to show how they lead to the Pythagorean theorem and the Thales theorem. We begin with his theory of angle, which shows most clearly the influence of his parallel axiom, the defining axiom of what is now called Euclidean geometry. Angle is linked with length from the beginning by the so-called SAS ("side angle side") criterion for equal triangles (or "congruent trian- gles," as we now call them). We observe the implications of SAS for isosceles triangles and the properties of angles in a circle, and we note the related criterion, ASA ("angle side angle"). The theory of area depends on ASA, and it leads directly to a proof of the Pythagorean theorem. It leads more subtly to the Thales the- orem and its consequences that we saw in Chapter 1. The theory of angle then combines nicely with the Thales theorem to give a second proof of the Pythagorean theorem. In following these deductive threads, we learn more about the scope of straightedge and compass constructions, partly in the exercises. Interesting spinoffs from these investigations include a process for cutting any polygon into pieces that form a square, a construction for the square root of any length, and a construction of the regular pentagon. 20
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2.1 The parallel axiom 212.1 The parallel axiomIn Chapter 1, we saw how useful it is to have rectangles: four-sided poly-gons whose angles are all right angles. Rectangles owe their existence toparallel lines—lines that do not meet—and fundamentally to the parallelaxiom that Euclid stated as follows.Euclid's parallel axiom. If a straight line crossing two straight linesmakes the interior angles on one side together less than two right angles,then the two straight lines will meet on that side. Figure 2.1 illustrates the situation described by Euclid's parallel axiom,which is what happens when the two lines are not parallel. If α + β is lessthan two right angles, then L and M meet somewhere on the right. N M β α L Figure 2.1: When lines are not parallel It follows that if L and M do not meet on either side, then α + β = π .In other words, if L and M are parallel, then α and β together make astraight angle and the angles made by L , M , and N are as shown inFigure 2.2. N M α π −α π −α α L Figure 2.2: When lines are parallel
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22 2 Euclid's approach to geometry It also follows that any line through the intersection of N and M , notmeeting L , makes the angle π − α with N . Hence, this line equals M .That is, if a parallel to L through a given point exists, it is unique. It is a little more subtle to show the existence of a parallel to L througha given point P, but one way is to appeal to a principle called ASA ("angleside angle"), which will be discussed in Section 2.2. Suppose that the lines L , M , and N make angles as shown in Fig-ure 2.2, and that L and M are not parallel. Then, on at least one sideof N , there is a triangle whose sides are the segment of N between Land M and the segments of L and M between N and the point wherethey meet. According to ASA, this triangle is completely determined bythe angles α , π − α and the segment of N between them. But then anidentical triangle is determined on the other side of N , and hence L andM also meet on the other side. This result contradicts Euclid's assumption(implicit in the construction axioms discussed in Section 1.1) that there isa unique line through any two points. Hence, the lines L and M are infact parallel when the angles are as shown in Figure 2.2. Thus, both the existence and the uniqueness of parallels follow fromEuclid's parallel axiom (existence "follows trivially," because Euclid's par-allel axiom is not required). It turns out that they also imply it, so theparallel axiom can be stated equivalently as follows.Modern parallel axiom. For any line L and point P outside L , there isexactly one line through P that does not meet L . This form of the parallel axiom is often called "Playfair's axiom," af-ter the Scottish mathematician John Playfair who used it in a textbook in1795. Playfair's axiom is simpler in form than Euclid's, because it doesnot involve angles, and this is often convenient. However, we often needparallel lines and the equal angles they create, the so-called alternate in-terior angles (for example, the angles marked α in Figure 2.2). In suchsituations, we prefer to use Euclid's parallel axiom.Angles in a triangleThe existence of parallels and the equality of alternate interior angles implya beautiful property of triangles.Angle sum of a triangle. If α , β , and γ are the angles of any triangle,then α + β + γ = π .
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2.1 The parallel axiom 23 To prove this property, draw a line L through one vertex of the trian-gle, parallel to the opposite side, as shown in Figure 2.3. L α γ β α γ Figure 2.3: The angle sum of a triangle Then the angle on the left beneath L is alternate to the angle α in thetriangle, so it is equal to α . Similarly, the angle on the right beneath L isequal to γ . But then the straight angle π beneath L equals α + β + γ , theangle sum of the triangle.ExercisesThe triangle is the most important polygon, because any polygon can be builtfrom triangles. For example, the angle sum of any quadrilateral (polygon withfour sides) can be worked out by cutting the quadrilateral into two triangles.2.1.1 Show that the angle sum of any quadrilateral is 2π . A polygon P is called convex if the line segment between any two points inP lies entirely in P. For these polygons, it is also easy to find the angle sum.2.1.2 Explain why a convex n-gon can be cut into n − 2 triangles.2.1.3 Use the dissection of the n-gon into triangles to show that the angle sum of a convex n-gon is (n − 2)π .2.1.4 Use Exercise 2.1.3 to find the angle at each vertex of a regular n-gon (an n-gon with equal sides and equal angles).2.1.5 Deduce from Exercise 2.1.4 that copies of a regular n-gon can tile the plane only for n = 3, 4, 6.
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24 2 Euclid's approach to geometry2.2 Congruence axiomsEuclid says that two geometric figures coincide when one of them can bemoved to fit exactly on the other. He uses the idea of moving one figureto coincide with another in the proof of Proposition 4 of Book I: If two tri-angles have two corresponding sides equal, and the angles between thesesides equal, then their third sides and the corresponding two angles arealso equal. His proof consists of moving one triangle so that the equal angles ofthe two triangles coincide, and the equal sides as well. But then the thirdsides necessarily coincide, because their endpoints do, and hence, so do theother two angles. Today we say that two triangles are congruent when their correspond-ing angles and side lengths are equal, and we no longer attempt to provethe proposition above. Instead, we take it as an axiom (that is, an unprovedassumption), because it seems simpler to assume it than to introduce theconcept of motion into geometry. The axiom is often called SAS (for "sideangle side").SAS axiom. If triangles ABC and A B C are such that |AB| = |A B |, angle ABC = angle A B C , |BC| = |B C |then also|AC| = |A C |, angle BCA = angle B C A , angle CAB = angle C A B . For brevity, one often expresses SAS by saying that two triangles arecongruent if two sides and the included angle are equal. There are similarconditions, ASA and SSS, which also imply congruence (but SSA doesnot—can you see why?). They can be deduced from SAS, so it is notnecessary to take them as axioms. However, we will assume ASA here tosave time, because it seems just as natural as SAS. One of the most important consequences of SAS is Euclid's Proposi-tion 5 of Book I. It says that a triangle with two equal sides has two equalangles. Such a triangle is called isosceles, from the Greek for "equal sides."The spectacular proof below is not from Euclid, but from the Greek math-ematician Pappus, who lived around 300 CE.Isosceles triangle theorem. If a triangle has two equal sides, then theangles opposite to these sides are also equal.
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2.2 Congruence axioms 25 Suppose that triangle ABC has |AB| = |AC|. Then triangles ABC andACB, which of course are the same triangle, are congruent by SAS (Figure2.4). Their left sides are equal, their right sides are equal, and so are theangles between their left and right sides, because they are the same angle(the angle at A). A A B C C B Figure 2.4: Two views of an isosceles triangle But then it follows from SAS that all corresponding angles of thesetriangles are equal: for example, the bottom left angles. In other words, theangle at B equals the angle at C, so the angles opposite to the equal sidesare equal. A useful consequence of ASA is the following theorem about parallel-ograms, which enables us to determine the area of triangles. (Remember,a parallelogram is defined as a figure bounded by two pairs of parallellines—the definition does not say anything about the lengths of its sides.)Parallelogram side theorem. Opposite sides of a parallelogram are equal. To prove this theorem we divide the parallelogram into triangles by adiagonal (Figure 2.5), and try to prove that these triangles are congruent.They are, because • they have the common side AC, • their corresponding angles α are equal, being alternate interior an- gles for the parallels AD and BC, • their corresponding angles β are equal, being alternate interior an- gles for the parallels AB and DC.
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26 2 Euclid's approach to geometry D C β α α β A B Figure 2.5: Dividing a parallelogram into triangles Therefore, the triangles are congruent by ASA, and in particular wehave the equalities |AB| = |DC| and |AD| = |BC| between correspondingsides. But these are also the opposite sides of the parallelogram.Exercises2.2.1 Using the parallelogram side theorem and ASA, find congruent triangles in Figure 2.6. Hence, show that the diagonals of a parallelogram bisect each other. Figure 2.6: A parallelogram and its diagonals2.2.2 Deduce that the diagonals of a rhombus—a parallelogram whose sides are all equal—meet at right angles. (Hint: You may find it convenient to use SSS, which says that triangles are congruent when their correspond- ing sides are equal.)2.2.3 Prove the isosceles triangle theorem differently by bisecting the angle at A.2.3 Area and equalityThe principle of logic used in Section 1.2—that things equal to the samething are equal to each other—is one of five principles that Euclid callscommon notions. The common notions he states are particularly importantfor his theory of area, and they are as follows:
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2.3 Area and equality 27 1. Things equal to the same thing are also equal to one another. 2. If equals are added to equals, the wholes are equal. 3. If equals are subtracted from equals, the remainders are equal. 4. Things that coincide with one another are equal to one another. 5. The whole is greater than the part.The word "equal" here means "equal in some specific respect." In mostcases, it means "equal in length" or "equal in area," although Euclid's ideaof "equal in area" is not exactly the same as ours, as I will explain below.Likewise, "addition" can mean addition of lengths or addition of areas, butEuclid never adds a length to an area because this has no meaning in hissystem. A simple but important example that illustrates the use of "equals" isEuclid's Proposition 15 of Book I: Vertically opposite angles are equal.Vertically opposite angles are the angles α shown in Figure 2.7. α α β Figure 2.7: Vertically opposite angles They are equal because each of them equals a straight angle minus β .The square of a sumProposition 4 of Book II is another interesting example. It states a propertyof squares and rectangles that we express by the algebraic formula (a + b)2 = a2 + 2ab + b2 .Euclid does not have algebraic notation, so he has to state this equation inwords: If a line is cut at random, the square on the whole is equal to thesquares on the segments and twice the rectangle contained by the segments.Whichever way you say it, Figure 2.8 explains why it is true. The line is a + b because it is cut into the two segments a and b, andhence
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28 2 Euclid's approach to geometry b ab b2 a a2 ab a b Figure 2.8: The square of a sum of line segments • The square on the line is what we write as (a + b)2 . • The squares on the two segments a and b are a2 and b2 , respectively. • The rectangle "contained" by the segments a and b is ab. • The square (a+b)2 equals (in area) the sum of a2 , b2 , and two copies of ab. It should be emphasized that, in Greek mathematics, the only inter-pretation of ab, the "product" of line segments a and b, is the rectanglewith perpendicular sides a and b (or "contained in" a and b, as Euclid usedto say). This rectangle could be shown "equal" to certain other regions,but only by cutting the regions into identical pieces by straight lines. TheGreeks did not realize that this "equality of regions" was the same as equal-ity of numbers—the numbers we call the areas of the regions—partly be-cause they did not regard irrational lengths as numbers, and partly becausethey did not think the product of lengths should be a length. As mentioned in Section 1.5, this belief was not necessarily an obstacleto the development of geometry. To find the area of nonrectangular regions,such as triangles or parallelograms, one has to think about cutting regionsinto pieces in any case. For such simple regions, there is no particularadvantage in thinking of the area as a number, as we will see in Section2.4. But first we need to investigate the concept mentioned in Euclid'sCommon Notion number 4. What does it mean for one figure to "coincide"with another?
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2.4 Area of parallelograms and triangles 29ExercisesIn Figure 2.8, the large square is subdivided by two lines: one of them perpendic-ular to the bottom side of the square and the other perpendicular to the left side ofthe square.2.3.1 Use the parallel axiom to explain why all other angles in the figure are necessarily right angles. Figure 2.8 presents the algebraic identity (a + b)2 = a2 + 2ab + b2 in geo-metric form. Other well-known algebraic identities can also be given a geometricpresentation.2.3.2 Give a diagram for the identity a(b + c) = ab + ac.2.3.3 Give a diagram for the identity a2 − b2 = (a + b)(a − b).Euclid does not give a geometric theorem that explains the identity (a + b)3 =a3 + 3a2 b + 3ab2 + b3 . But it is not hard to do so by interpreting (a + b)3 as a cubewith edge length a + b, a3 as a cube with edge a, a2 b as a box with perpendicularedges a, a, and b, and so on.2.3.4 Draw a picture of a cube with edges a+b, and show it cut by planes (parallel to its faces) that divide each edge into a segment of length a and a segment of length b.2.3.5 Explain why these planes cut the original cube into eight pieces: • a cube with edges a, • a cube with edges b, • three boxes with edges a, a, b, • three boxes with edges a, b, b.2.4 Area of parallelograms and trianglesThe first nonrectangular region that can be shown "equal" to a rectanglein Euclid's sense is a parallelogram. Figure 2.9 shows how to use straightlines to cut a parallelogram into pieces that can be reassembled to form arectangle. = =Figure 2.9: Assembling parallelogram and rectangle from the same pieces
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30 2 Euclid's approach to geometry Only one cut is needed in the example of Figure 2.9, but more cuts areneeded if the parallelogram is more sheared, as in Figure 2.10. 3 3 2 = 2 1 1 Figure 2.10: A case in which more cuts are required In Figure 2.10 we need two cuts, which produce the pieces labeled 1, 2,3. The number of cuts can become arbitrarily large as the parallelogram issheared further. We can avoid large numbers of cuts by allowing subtrac-tion of pieces as well as addition. Figure 2.11 shows how to convert anyrectangle to any parallelogram with the same base OR and the same heightOP. We need only add a triangle, and then subtract an equal triangle. P Q S T O RFigure 2.11: Rectangle and parallelogram with the same base and height To be precise, if we start with rectangle OPQR and add triangle RQT ,then subtract triangle OPS (which equals triangle RQT by the parallelo-gram side theorem of Section 2.2), the result is parallelogram OST R. Thus,the parallelogram is equal (in area) to a rectangle with the same base andheight. We write this fact as area of parallelogram = base × height.To find the area of a triangle ABC, we notice that it can be viewed as "half"of a parallelogram by adding to it the congruent triangle ACD as shown inFigure 2.5, and again in Figure 2.12. D C A B Figure 2.12: A triangle as half a parallelogram
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2.4 Area of parallelograms and triangles 31 Clearly,area of triangle ABC +area of triangle ACD = area of parallelogram ABCD,and the two triangles "coincide" (because they are congruent) and so theyhave equal area by Euclid's Common Notion 4. Thus, 1 area of triangle = base × height. 2 This formula is important in two ways: • As a statement about area. From a modern viewpoint, the formula gives the area of the triangle as a product of numbers. From the an- cient viewpoint, it gives a rectangle "equal" to the triangle, namely, the rectangle with the same base and half the height of the triangle. • As a statement about proportionality. For triangles with the same height, the formula shows that their areas are proportional to their bases. This statement turns out to be crucial for the proof of the Thales theorem (Section 2.6). The proportionality statement follows from the assumption that eachline segment has a real number length, which depends on the acceptanceof irrational numbers. As mentioned in the previous section, the Greeksdid not accept this assumption. Euclid got the proportionality statement bya lengthy and subtle "theory of proportion" in Book V of the Elements.ExercisesTo back up the claim that the formula 1 base × height gives a way to find the area 2of the triangle, we should explain how to find the height.2.4.1 Given a triangle with a particular side specified as the "base," show how to find the height by straightedge and compass construction.The equality of triangles OPS and RQT follows from the parallelogram side the-orem, as claimed above, but a careful proof would explain what other axioms areinvolved.2.4.2 By what Common Notion does |PQ| = |ST |?2.4.3 By what Common Notion does |PS| = |QT |?2.4.4 By what congruence axiom is triangle OPS congruent to triangle RQT ?
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32 2 Euclid's approach to geometry2.5 The Pythagorean theoremThe Pythagorean theorem is about areas, and indeed Euclid proves it im-mediately after he has developed the theory of area for parallelograms andtriangles in Book I of the Elements. First let us recall the statement of thetheorem.Pythagorean theorem. For any right-angled triangle, the sum of thesquares on the two shorter sides equals the square on the hypotenuse. We follow Euclid's proof, in which he divides the square on the hy-potenuse into the two rectangles shown in Figure 2.13. He then shows thatthe light gray square equals the light gray rectangle and that the dark graysquare equals the dark gray rectangle, so the sum of the light and darksquares is the square on the hypotenuse, as required. Figure 2.13: Dividing the square for Euclid's proof First we show equality for the light gray regions in Figure 2.13, and infact we show that half of the light gray square equals half of the light grayrectangle. We start with a light gray triangle that is obviously half of thelight gray square, and we successively replace it with triangles of the samebase or height, ending with a triangle that is obviously half of the light grayrectangle (Figure 2.14).
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2.5 The Pythagorean theorem 33 Start with half of the light gray square Same base (side of light gray square) and height Congruent triangle, by SAS (the included angle is the sum of the same parts) Same base (side of square on hypotenuse) and height; new triangle is half the light gray rectangle Figure 2.14: Changing the triangle without changing its area The same argument applies to the dark gray regions, and thus, thePythagorean theorem is proved. Figure 2.13 suggests a natural way to construct a square equal in areato a given rectangle. Given the light gray rectangle, say, the problem is toreconstruct the rest of Figure 2.13.
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34 2 Euclid's approach to geometry We can certainly extend a given rectangle to a square and hence recon-struct the square on the hypotenuse. The main problem is to reconstructthe right-angled triangle, from the hypotenuse, so that the other vertex lieson the dashed line. See whether you can think of a way to do this; a reallyelegant solution is given in Section 2.7. Once we have the right-angledtriangle, we can certainly construct the squares on its other two sides—inparticular, the gray square equal in area to the gray rectangle.ExercisesIt follows from the Pythagorean theorem that a right-angled triangle with sides 3 √ √and 4 has hypotenuse 32 + 42 = 25 = 5. But there is only one triangle withsides 3, 4, and 5 (by the SSS criterion mentioned in Exercise 2.2.2), so puttingtogether lengths 3, 4, and 5 always makes a right-angled triangle. This triangle isknown as the (3, 4, 5) triangle.2.5.1 Verify that the (5, 12, 13), (8, 15, 17), and (7, 24, 25) triangles are right- angled.2.5.2 Prove the converse Pythagorean theorem: If a, b, c > 0 and a2 + b2 = c2 , then the triangle with sides a, b, c is right-angled.2.5.3 How can we be sure that lengths a, b, c > 0 with a2 + b2 = c2 actually fit together to make a triangle? (Hint: Show that a + b > c.) Right-angled triangles can be used to construct certain irrational lengths. Forexample, we saw in Section 1.5 that the right-angled triangle with sides 1, 1 has √hypotenuse 2. √2.5.4 Starting from the triangle√ with sides 1, 1, and 2, find a straightedge and compass construction of 3. √2.5.5 Hence, obtain constructions of n for n = 2, 3, 4, 5, 6, . . ..2.6 Proof of the Thales theoremWe mentioned this theorem in Chapter 1 as a fact with many interestingconsequences, such as the proportionality of similar triangles. We are nowin a position to prove the theorem as Euclid did in his Proposition 2 ofBook VI. Here again is a statement of the theorem.The Thales theorem. A line drawn parallel to one side of a triangle cutsthe other two sides proportionally.
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2.6 Proof of the Thales theorem 35 The proof begins by considering triangle ABC, with its sides AB and ACcut by the parallel PQ to side BC (Figure 2.15). Because PQ is parallel toBC, the triangles PQB and PQC on base PQ have the same height, namelythe distance between the parallels. They therefore have the same area. A P Q B C Figure 2.15: Triangle sides cut by a parallel If we add triangle APQ to each of the equal-area triangles PQB andPQC, we get the triangles AQB and APC, respectively. Hence, the lattertriangles are also equal in area. Now consider the two triangles—APQ and PQB—that make up trian-gle AQB as triangles with bases on the line AB. They have the same heightrelative to this base (namely, the perpendicular distance of Q from AB).Hence, their bases are in the ratio of their areas: |AP| area APQ = . |PB| area PQBSimilarly, considering the triangles APQ and PQC that make up the triangleAPC, we find that |AQ| area APQ = . |QC| area PQCBecause area PQB equals area PQC, the right sides of these two equationsare equal, and so are their left sides. That is, |AP| |AQ| = . |PB| |QC|In other words, the line PQ cuts the sides AB and AC proportionally.
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36 2 Euclid's approach to geometryExercisesAs seen in Exercise 1.3.6, |AP|/|PB| = |AQ|/|QC| is equivalent to |AP|/|AB| =|AQ|/|AC|. This equation is a more convenient formulation of the Thales theoremif you want to prove the following generalization:2.6.1 Suppose that there are several parallels P1 Q1 , P2 Q2 , P3 Q3 . . . to the side BC of triangle ABC. Show that |AP1 | |AP2 | |AP3 | |AB| = = = ··· = . |AQ1 | |AQ2 | |AQ3 | |AC| We can also drop the assumption that the parallels P1 Q1 , P2 Q2 , P3 Q3 . . . fallacross a triangle ABC.2.6.2 If parallels P1 Q1 , P2 Q2 , P3 Q3 . . . fall across a pair of parallel lines L and M , what can we say about the lengths they cut from L and M ?2.7 Angles in a circleThe isosceles triangle theorem of Section 2.2, simple though it is, has aremarkable consequence.Invariance of angles in a circle. If A and B are two points on a circle,then, for all points C on one of the arcs connecting them, the angle ACB isconstant. To prove invariance we draw lines from A, B,C to the center of thecircle, O, along with the lines making the angle ACB (Figure 2.16). Because all radii of the circle are equal, |OA| = |OC|. Thus triangleAOC is isosceles, and the angles α in it are equal by the isosceles triangletheorem. The angles β in triangle BOC are equal for the same reason. Because the angle sum of any triangle is π (Section 2.1), it followsthat the angle at O in triangle AOC is π − 2α and the angle at O in triangleBOC is π − 2β . It follows that the third angle at O, angle AOB, is 2(α + β ),because the total angle around any point is 2π . But angle AOB is constant,so α + β is also constant, and α + β is precisely the angle at C. An important special case of this theorem is when A, O, and B lie in astraight line, so 2(α + β ) = π . In this case, α + β = π /2, and thus we havethe following theorem (which is also attributed to Thales).Angle in a semicircle theorem. If A and B are the ends of a diameter ofa circle, and C is any other point on the circle, then angle ACB is a rightangle.
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2.7 Angles in a circle 37 C β α π − 2α π − 2β O 2(α + β ) β B α A Figure 2.16: Angle α + β in a circle This theorem enables us to solve the problem left open at the end ofSection 2.5: Given a hypotenuse AB, how do we construct the right-angledtriangle whose other vertex C lies on a given line? Figure 2.17 shows how. C A B Figure 2.17: Constructing a right-angled triangle with given hypotenuse
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38 2 Euclid's approach to geometry The trick is to draw the semicircle on diameter AB, which can be doneby first bisecting AB to obtain the center of the circle. Then the point wherethe semicircle meets the given line (shown dashed) is necessarily the othervertex C, because the angle at C is a right angle. This construction completes the solution of the problem raised at theend of Section 2.5: finding a square equal in area to a given rectangle.In Section 2.8 we will show that Figure 2.17 also enables us to constructthe square root of an arbitrary length, and it gives a new proof of thePythagorean theorem.Exercises2.7.1 Explain how the angle in a semicircle theorem enables us to construct a right-angled triangle with a given hypotenuse AB.2.7.2 Then, by looking at Figure 2.13 from the bottom up, find a way to construct a square equal in area to a given rectangle.2.7.3 Given any two squares, we can construct a square that equals (in area) the sum of the two given squares. Why?2.7.4 Deduce from the previous exercises that any polygon may be "squared"; that is, there is a straightedge and compass construction of a square equal in area to the given polygon. (You may assume that the given polygon can be cut into triangles.) The possibility of "squaring" any polygon was apparently known to Greekmathematicians, and this may be what tempted them to try "squaring the circle":constructing a square equal in area to a given circle. There is no straightedge andcompass solution of the latter problem, but this was not known until 1882. Coming back to angles in the circle, here is another theorem about invarianceof angles:2.7.5 If a quadrilateral has its vertices on a circle, show that its opposite angles sum to π .2.8 The Pythagorean theorem revisitedIn Book VI, Proposition 31 of the Elements, Euclid proves a generalizationof the Pythagorean theorem. From it, we get a new proof of the ordinaryPythagorean theorem, based on the proportionality of similar triangles. Given a right-angled triangle with sides a, b, and hypotenuse c, wedivide it into two smaller right-angled triangles by the perpendicular to thehypotenuse through the opposite vertex (the dashed line in Figure 2.18).
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2.8 The Pythagorean theorem revisited 39 C β α b a α β A c1 D c2 B c Figure 2.18: Subdividing a right-angled triangle into similar triangles All three triangles are similar because they have the same angles α andβ . If we look first at the angle α at A and the angle β at B, then π α +β = 2because the angle sum of triangle ABC is π and the angle at C is π /2. Butthen it follows that angle ACD = β in triangle ACD (to make its angle sum= π ) and angle DCB = α in triangle DCB (to make its angle sum = π ). Now we use the proportionality of these triangles, calling the side op-posite α in each triangle "short" and the side opposite β "long" for conve-nience. Comparing triangle ABC with triangle ADC, we get long side b c1 = = , hence b2 = cc1 . hypotenuse c bComparing triangle ABC with triangle DCB, we get short side a c2 = = , hence a2 = cc2 . hypotenuse c aAdding the values of a2 and b2 just obtained, we finally get a2 + b2 = cc2 + cc1 = c(c1 + c2 ) = c2 because c1 + c2 = c,and this is the Pythagorean theorem.
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40 2 Euclid's approach to geometry This second proof is not really shorter than Euclid's first (given in Sec-tion 2.5) when one takes into account the work needed to prove the pro-portionality of similar triangles. However, we often need similar triangles,so they are a standard tool, and a proof that uses standard tools is generallypreferable to one that uses special machinery. Moreover, the splitting of aright-angled triangle into similar triangles is itself a useful tool—it enablesus to construct the square root of any line segment.Straightedge and compass construction of square rootsGiven any line segment l, construct the semicircle with diameter l + 1, andthe perpendicular to the diameter where the segments 1 and l meet (Figure √2.19). Then the length h of this perpendicular is l. h l 1 Figure 2.19: Construction of the square root To see why, construct the right-angled triangle with hypotenuse l + 1and third vertex where the perpendicular meets the semicircle. We knowthat the perpendicular splits this triangle into two similar, and hence pro-portional, triangles. In the triangle on the left, long side l = . short side hIn the triangle on the right, long side h = . short side 1Because these ratios are equal by proportionality of the triangles, we have l h = , h 1 √hence h2 = l; that is, h = l.
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2.8 The Pythagorean theorem revisited 41 This result complements the constructions for the rational operations√+, −, ×, and ÷ we gave in Chapter 1. The constructibility of these andwas first pointed out by Descartes in his book G´ om´ trie of 1637. Rational √ e eoperations and are in fact precisely what can be done with straightedgeand compass. When we introduce coordinates in Chapter 3 we will see thatany "constructible point" has coordinates obtainable from the unit length 1 √by +, −, ×, ÷, and .Exercises √Now that we know how to construct the +, −, ×, ÷, and of given lengths, wecan use algebra as a shortcut to decide whether certain figures are constructible bystraightedge and compass. If we know that a certain figure is constructible from √the length (1+ 5)/2, for example, then we know that the figure is constructible— √period—because the length (1 + 5)/2 is built from the unit length by the opera- √tions +, ×, ÷, and . This is precisely the case for the regular pentagon, which was constructedby Euclid in Book IV, Proposition 11, using virtually all of the geometry he haddeveloped up to that point. We also need nearly everything we have developed upto this point, but it fills less space than four books of the Elements! The following exercises refer to the regular pentagon of side 1 shown in Figure2.20 and its diagonals of length x. 1 x Figure 2.20: The regular pentagon2.8.1 Use the symmetry of the regular pentagon to find similar triangles implying x 1 = , 1 x−1 that is, x2 − x − 1 = 0.2.8.2 By finding the positive root of this quadratic equation, show that each diag- √ onal has length x = (1 + 5)/2.2.8.3 Now show that the regular pentagon is constructible.
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42 2 Euclid's approach to geometry2.9 DiscussionEuclid found the most important axiom of geometry—the parallel axiom—and he also identified the basic theorems and traced the logical connectionsbetween them. However, his approach misses certain fine points and is notlogically complete. For example, in his very first proof (the constructionof the equilateral triangle), he assumes that certain circles have a point incommon, but none of his axioms guarantee the existence of such a point.There are many such situations, in which Euclid assumes something is truebecause it looks true in the diagram. Euclid's theory of area is a whole section of his geometry that seemsto have no geometric support. Its concepts seem more like arithmetic—addition, subtraction, and proportion—but its concept of multiplication isnot the usual one, because multiplication of more than three lengths is notallowed. These gaps in Euclid's approach to geometry were first noticed in the19th century, and the task of filling them was completed by David Hilbertin his Grundlagen der Geometrie (Foundations of Geometry) of 1899. Onthe one hand, Hilbert introduced axioms of incidence and order, giving theconditions under which lines (and circles) meet. These justify the beliefthat "geometric objects behave as the pictures suggest." On the other hand,Hilbert replaced Euclid's theory of area with a genuine arithmetic, whichhe called segment arithmetic. He defined the sum and product of segmentsas we did in Section 1.4 and proved that these operations on segments havethe same properties as ordinary sum and product. For example, a + b = b + a, ab = ba, a(b + c) = ab + ac, and so on.In the process, Hilbert discovered that the Pappus and Desargues theorems(Exercises 1.4.3 and 1.4.4) play a decisive role. The downside of Hilbert's completion of Euclid is that it is lengthyand difficult. Nearly 20 axioms are required, and some key theorems arehard to prove. To some extent, this hardship occurs because Hilbert insistson geometric definitions of + and ×. He wants numbers to come from"inside" geometry rather than from "outside". Thus, to prove that ab = bahe needs the theorem of Pappus, and to prove that a(bc) = (ab)c he needsthe theorem of Desargues. Even today, the construction of segment arithmetic is an admirable feat.As Hilbert pointed out, it shows that Euclid was right to believe that the
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2.9 Discussion 43theory of proportion could be developed without new geometric axioms.Still, it is somewhat quixotic to build numbers "inside" Euclid's geometrywhen they are brought from "outside" into nearly every other branch ofgeometry. It is generally easier to build geometry on numbers than thereverse, and Euclidean geometry is no exception, as I hope to show inChapters 3 and 4. This is one reason for bypassing Hilbert's approach, so I will merelylist his axioms here. They are thoroughly investigated in Hartshorne's Ge-ometry: Euclid and Beyond or Hilbert's own book, which is available inEnglish translation. Hartshorne's book has the clearest available derivationof ordinary geometry and segment arithmetic from the Hilbert axioms, soit should be consulted by anyone who wants to see Euclid's approach takento its logical conclusion. There is another reason to bypass Hilbert's axioms, apart from theirdifficulty. In my opinion, Hilbert's greatest geometric achievement was tobuild arithmetic, not in Euclidean geometry, but in projective geometry.As just mentioned, Hilbert found that the keys to segment arithmetic arethe Pappus and Desargues theorems. These two theorems do not involvethe concept of length, and so they really belong to a more primitive kindof geometry. This primitive geometry (projective geometry) has only ahandful of axioms—fewer than the usual axioms for arithmetic—so it ismore interesting to build arithmetic inside it. It is also less trouble, becausewe do not have to prove the Pappus and Desargues theorems. We willexplain how projective geometry contains arithmetic in Chapters 5 and 6.Hilbert's axiomsThe axioms concern undefined objects called "points" and "lines," the re-lated concepts of "line segment," "ray," and "angle," and the relationsof "betweenness" and "congruence." Following Hartshorne, we simplifyHilbert's axioms slightly by stating some of them in a stronger form thannecessary. The first group of axioms is about incidence: conditions for points tolie on lines or for lines to pass through points. I1. For any two points A, B, a unique line passes through A, B. I2. Every line contains at least two points. I3. There exist three points not all on the same line.
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44 2 Euclid's approach to geometry I4. For each line L and point P not on L there is a unique line through P not meeting L (parallel axiom). The next group is about betweenness or order: a concept overlookedby Euclid, probably because it is too "obvious." The first to draw attentionto betweenness was the German mathematician Moritz Pasch, in the 1880s.We write A ∗ B ∗C to denote that B is between A and C. B1. If A ∗ B ∗C, then A, B,C are three points on a line and C ∗ B ∗ A. B2. For any two points A and B, there is a point C with A ∗ B ∗C. B3. Of three points on a line, exactly one is between the other two. B4. Suppose A, B,C are three points not in a line and that L is a line not passing through any of A, B,C. If L contains a point D between A and B, then L contains either a point between A and C or a point between B and C, but not both (Pasch's axiom). The next group is about congruence of line segments and congruenceof angles, both denoted by ∼ Thus, AB ∼ CD means that AB and CD =. =have equal length and ∠ABC ∼ ∠DEF means that ∠ABC and ∠DEF are =equal angles. Notice that C2 and C5 contain versions of Euclid's CommonNotion 1: "Things equal to the same thing are equal to each other." C1. For any line segment AB, and any ray R originating at a point C, there is a unique point D on R with AB ∼ CD. = C2. If AB ∼ CD and AB ∼ EF, then CD ∼ EF. For any AB, AB ∼ AB. = = = = C3. Suppose A ∗ B ∗ C and D ∗ E ∗ F. If AB ∼ DE and BC ∼ EF, then = = AC ∼ DF. (Addition of lengths is well-defined.) = −→ −→ C4. For any angle ∠BAC, and any ray DF, there is a unique ray DE on a −→ ∼ given side of DF with ∠BAC = ∠EDF. C5. For any angles α , β , γ , if α ∼ β and α ∼ γ , then β ∼ γ . Also, α ∼ α . = = = = ∼ ∼ C6. Suppose that ABC and DEF are triangles with AB = DE, AC = DF, and ∠BAC =∼ ∠EDF. Then, the two triangles are congruent, namely BC ∼ EF, ∠ABC ∼ ∠DEF, and ∠ACB ∼ ∠DFE. (This is SAS.) = = =
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2.9 Discussion 45 Then there is an axiom about the intersection of circles. It involves theconcept of points inside the circle, which are those points whose distancefrom the center is less than the radius. E. Two circles meet if one of them contains points both inside and out- side the other. Next there is the so-called Archimedean axiom, which says that nolength can be "infinitely large" relative to another. A. For any line segments AB and CD, there is a natural number n such that n copies of AB are together greater than CD. Finally, there is the so-called Dedekind axiom, which says that the lineis complete, or has no gaps. It implies that its points correspond to realnumbers. Hilbert wanted an axiom like this to force the plane of Euclideangeometry to be the same as the plane R2 of pairs of real numbers. D. Suppose the points of a line L are divided into two nonempty sub- sets A and B in such a way that no point of A is between two points of B and no point of B is between two points of A . Then, a unique point P, either in A or B, lies between any other two points, of which one is in A and the other is in B. Axiom D is not needed to derive any of Euclid's theorems. They donot involve all real numbers but only the so-called constructible numbersoriginating from straightedge and compass constructions. However, whocan be sure that we will never need nonconstructible points? One of themost important numbers in geometry, π , is nonconstructible! (Because thecircle cannot be squared.) Thus, it seems prudent to use Axiom D so thatthe line is complete from the beginning. In Chapter 3, we will take the real numbers as the starting point ofgeometry, and see what advantages this may have over the Euclid–Hilbertapproach. One clear advantage is access to algebra, which reduces manygeometric problems to simple calculations. Algebra also offers some con-ceptual advantages, as we will see.
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3CoordinatesP REVIEW Around 1630, Pierre de Fermat and Ren´ Descartes independently e discovered the advantages of numbers in geometry, as coordinates. Descartes was the first to publish a detailed account, in his book G´ om´ trie of 1637. For this reason, he gets most of the credit for e e the idea and the coordinate approach to geometry became known as Cartesian (from the old way of writing his name: Des Cartes). Descartes thought that geometry was as Euclid described it, and that numbers merely assist in studying geometric figures. But later math- ematicians discovered objects with "non-Euclidean" properties, such as "lines" having more than one "parallel" through a given point. To clarify this situation, it became desirable to define points, lines, length, and so on, and to prove that they satisfy Euclid's axioms. This program, carried out with the help of coordinates, is called the arithmetization of geometry. In the first three sections of this chap- ter, we do the main steps, using the set R of real numbers to define the Euclidean plane R2 and the points, lines, and circles in it. We also define the concepts of distance and (briefly) angle, and show how some crucial axioms and theorems follow. However, arithmeti- zation does much more. • It gives an algebraic description of constructibility by straight- edge and compass (Section 3.4), which makes it possible to prove that certain figures are not constructible. • It enables us to define what it means to "move" a geometric figure (Section 3.6), which provides justification for Euclid's proof of SAS, and raises a new kind of geometric question (Section 3.7): What kinds of "motion" exist? 46
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3.1 The number line and the number plane 473.1 The number line and the number planeThe set R of real numbers results from filling the √ in the set Q of gapsrational numbers with irrational numbers, such as 2. This innovationenables us to consider R as a line, because it has no gaps and the numbersin it are ordered just as we imagine points on a line to be. We say that R,together with its ordering, is a model of the line. One of our goals in thischapter is to use R to build a model for all of Euclidean plane geometry: astructure containing "lines," "circles," "line segments," and so on, with allof the properties required by Euclid's or Hilbert's axioms. The first step is to build the "plane," and in this we are guided by theproperties of parallels in Euclid's geometry. We imagine a pair of per-pendicular lines, called the x-axis and the y-axis, intersecting at a point Ocalled the origin (Figure 3.1). We interpret the axes as number lines, withO the number 0 on each, and we assume that the positive direction on thex-axis is to the right and that the positive direction on the y-axis is upward. y-axis b P = (a, b) x-axis O a Figure 3.1: Axes and coordinates Through any point P, there is (by the parallel axiom) a unique lineparallel to the y-axis and a unique line parallel to the x-axis. These twolines meet the x-axis and y-axis at numbers a and b called the x- and y-coordinates of P, respectively. It is important to remember which numberis on the x-axis and which is on the y-axis, because obviously the pointwith x-coordinate = 3 and y-coordinate = 4 is different from the point withx-coordinate = 4 and y-coordinate = 3 (just as the intersection of 3rd Streetand 4th Avenue is different from the intersection of 4th Street and 3rd Av-enue).
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48 3 Coordinates To keep the x-coordinate a and the y-coordinate b in their places, we usethe ordered pair (a, b). For example, (3, 4) is the point with x-coordinate= 3 and y-coordinate = 4, whereas (4, 3) is the point with x-coordinate = 4and y-coordinate = 3. The ordered pair (a, b) specifies P uniquely becauseany other point will have at least one different parallel passing through itand hence will differ from P in either the x- or y-coordinate. Thus, given the existence of a number line R whose points are realnumbers, we also have a number plane whose points are ordered pairs ofreal numbers. We often write this number plane as R × R or R 2 .3.2 Lines and their equationsAs mentioned in Chapter 2, one of the most important consequences ofthe parallel axiom is the Thales theorem and hence the proportionality ofsimilar triangles. When coordinates are introduced, this allows us to definethe property of straight lines known as slope. You know from high-schoolmathematics that slope is the quotient "rise over run" and, more impor-tantly, that the value of the slope does not depend on which two points ofthe line define the rise and the run. Figure 3.2 shows why. B β B β A α C A α C Figure 3.2: Why the slope of a line is constant In this figure, we have two segments of the same line: • AB, for which the rise is |BC| and the run is |AC|, and • A B , for which the rise is |B C | and the run is |A C |.
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3.2 Lines and their equations 49The angles marked α are equal because AC and A C are parallel, and theangles marked β are equal because the BC and B C are parallel. Also, theangles at C and C are both right angles. Thus, triangles ABC and A B C are similar, and so their correspondingsides are proportional. In particular, |BC| |B C | = , |AC| |A C |that is, slope = constant. Now suppose we are given a line of slope a that crosses the y-axis atthe point Q where y = c (Figure 3.3). If P = (x, y) is any point on this line,then the rise from Q to P is y − c and the run is x. Hence y−c slope = a = xand therefore, multiplying both sides by x, y − c = ax, that is, y = ax + c.This equation is satisfied by all points on the line, and only by them, so wecall it the equation of the line. y (x, y) = P y−c (0, c) = Q x x O Figure 3.3: Typical point on the line Almost all lines have equations of this form; the only exceptions arelines that do not cross the y-axis. These are the vertical lines, which alsodo not have a slope as we have defined it, although we could say they haveinfinite slope. Such a line has an equation of the form x = c, for some constant c.
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50 3 CoordinatesThus, all lines have equations of the form ax + by + c = 0, for some constants a, b, and c,called a linear equation in the variables x and y. Up to this point we have been following the steps of Descartes, whoviewed equations of lines as information deduced from Euclid's axioms (inparticular, from the parallel axiom). It is true that Euclid's axioms promptus to describe lines by linear equations, but we can also take the oppositeview: Equations define what lines and curves are, and they provide a model of Euclid's axioms—showing that geometry follows from properties of thereal numbers. In particular, if a line is defined to be the set of points (x, y) in thenumber plane satisfying a linear equation then we can prove the followingstatements that Euclid took as axioms: • there is a unique line through any two distinct points, • for any line L and point P outside L , there is a unique line through P not meeting L .Because these statements are easy to prove, we leave them to the exercises.ExercisesGiven distinct points P1 = (x1 , y1 ) and P2 = (x2 , y2 ), suppose that P = (x, y) is anypoint on a line through P1 and P2 .3.2.1 By equating slopes, show that x and y satisfy the equation y2 − y1 y − y1 = if x2 = x1 . x2 − x1 x − x13.2.2 Explain why the equation found in Exercise 3.2.1 is the equation of a straight line.3.2.3 What happens if x2 = x1 ?Parallel lines, not surprisingly, turn out to be lines with the same slope.3.2.4 Show that distinct lines y = ax + c and y = a x + c have a common point unless they have the same slope (a = a ). Show that this is also the case when one line has infinite slope.3.2.5 Deduce from Exercise 3.2.4 that the parallel to a line L is the unique line through P with the same slope as L .3.2.6 If L has equation y = 3x, what is the equation of the parallel to L through P = (2, 2)?
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3.3 Distance 513.3 DistanceWe introduce the concept of distance or length into the number plane R 2much as we introduce lines. First we see what Euclid's geometry suggestsdistance should mean; then we turn around and take the suggested meaningas a definition. Suppose that P1 = (x1 , y1 ) and P2 = (x2 , y2 ) are any two points in R2 .Then it follows from the meaning of coordinates that there is a right-angledtriangle as shown in Figure 3.4, and that |P1 P2 | is the length of its hy-potenuse. y P2 = (x2 , y2 ) y2 − y1 (x1 , y1 ) = P1 x2 − x1 x O Figure 3.4: The triangle that defines distance The vertical side of the triangle has length y2 − y1 , and the horizontalside has length x2 − x1 . Then it follows from the Pythagorean theorem that |P1 P2 |2 = (x2 − x1 )2 + (y2 − y1 )2 ,and therefore, |P1 P2 | = (x2 − x1 )2 + (y2 − y1 )2 . (*) Thus, it is sensible to define the distance |P1 P2 | between any two pointsP1 and P2 by the formula (*). If we do this, the Pythagorean theorem isvirtually "true by definition." It is certainly true when the right-angledtriangle has a vertical side and a horizontal side, as in Figure 3.4. Andwe will see later how to rotate any right-angled triangle to such a position(without changing the lengths of its sides).
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3.4 Intersections of lines and circles 53ExercisesAn interesting application of equidistant lines is the following.3.3.1 Show that any three points not in a line lie on a unique circle. (Hint: the center of the circle is equidistant from the three points.) The equations of lines and circles enable us to prove many geometric theo-rems by algebra, as Descartes realized. In fact, they greatly expand the scope ofgeometry by allowing many curves to be described by equations. But algebra isalso useful in proving that certain quantities are not equal. One example is thetriangle inequality.3.3.2 Consider any triangle, which for convenience we take to have one vertex at O = (0, 0), one at P = (x1 , 0) with x1 > 0, and one at Q = (x2 , y2 ). Show that |OP| = x1 , |PQ| = (x2 − x1 )2 + y2 , 2 |OQ| = x2 + y2 . 2 2The triangle inequality states that |OP| + |PQ| > |OQ| (any two sides of a triangleare together greater than the third). To prove this statement, it suffices to showthat (|OP| + |PQ|)2 > |OQ|2 .3.3.3 Show that (|OP| + |PQ|)2 − |OQ|2 = 2x1 (x2 − x1 )2 + y2 − (x2 − x1 ) . 23.3.4 Show that the term in square brackets in Exercise 3.3.3 is positive if y2 = 0, and hence that the triangle inequality holds in this case.3.3.5 If y2 = 0, why is this not a problem?Later we will give a more sophisticated approach to the triangle inequality, whichdoes not depend on choosing a special position for the triangle.3.4 Intersections of lines and circlesNow that lines and circles are defined by equations, we can give exactalgebraic equivalents of straightedge and compass operations: • Drawing a line through given points corresponds to finding the equa- tion of the line through given points (x1 , y1 ) and (x2 , y2 ). The slope between these two points is y2 −y1 , which must equal the slope x−x1 x2 −x1 y−y 1 between the general point (x, y) and the special point (x1 , y1 ), so the equation is y − y1 y2 − y1 = . x − x1 x2 − x1
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3.5 Angle and slope 55 operations +, −, ×, and Solving linear equations requires only the √ ÷, and the quadratic formula shows that is the only additional operation needed to solve quadratic equations. Thus, all intersection points involved in a straightedge and compass √ construction can be found with the operations +, −, ×, ÷, and .√ Now recall from Chapters 1 and 2 that the operations +, −, ×, ÷, and can be carried out by straightedge and compass. Hence, we get thefollowing result:Algebraic criterion for constructibility. A point is constructible (startingfrom the points 0 and 1) if and only if its coordinates are obtainable from √the number 1 by the operations +, −, ×, ÷, and . The algebraic criterion for constructibility was discovered by Descartes,and its greatest virtue is that it enables us to prove that certain figures orpoints are not constructible. For example, one can prove that the number√3 2 is not constructible by showing that it cannot be expressed by a finitenumber of square roots, and one can prove that the angle π /3 cannot be tri-sected by showing that cos π also cannot be expressed by a finite number 9of square roots. These results were not proved until the 19th century, byPierre Wantzel. Rather sophisticated algebra is required, because one hasto go beyond Descartes' concept of constructibility to survey the totality ofconstructible numbers.Exercises3.4.1 Find the intersections of the circles x2 + y2 = 1 and (x − 1)2 + (y − 2)2 = 4.3.4.2 Check the plausibility of your answer to Exercise 3.4.1 by a sketch of the two circles.3.4.3 The line x + 2y − 1 = 0 found by eliminating the x2 and y2 from the equa- tions of the circles should have some geometric meaning. What is it?3.5 Angle and slopeThe concept of distance is easy to handle in coordinate geometry becausethe distance between points (x1 , y1 ) and (x2 , y2 ) is an algebraic function oftheir coordinates, namely (x2 − x1 )2 + (y2 − y1 )2 .
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56 3 CoordinatesThis is not the case for the concept of angle. The angle θ between a liney = tx and the x-axis is tan−1 t, and the function tan−1 t is not an algebraicfunction. Nor is its inverse function t = tan θ or the related functions sin θ(sine) and cos θ (cosine). To stay within the world of algebra, we have to work with the slope trather than the angle θ . Lines make the same angle with the x-axis if theyhave the same slope, but to test equality of angles in general we need theconcept of relative slope: If line L1 has slope t1 and line L2 has slope t2 ,then the slope of L1 relative to L2 is defined to be t1 − t 2 ± . 1 + t 1 t2This awkward definition comes from the formula you have probably seenin trigonometry, tan θ1 − tan θ2 tan(θ1 − θ2 ) = , 1 + tan θ1 tan θ2by taking t1 = tan θ1 and t2 = tan θ2 . The reason for the ± sign and theabsolute value is that the slopes t1 , t2 alone do not specify an angle—theyspecify only a pair of lines and hence a pair of angles that add to a straightangle. (For more on using relative slope to discuss equality of angles, seeHartshorne's Geometry: Euclid and Beyond, particularly pp. 141–155.) At any rate, with some care it is possible to use the concept of rela-tive slope to test algebraically whether angles are equal. The concept alsomakes it possible to state the SAS and ASA axioms in coordinate geome-try, and to verify that all of Euclid's and Hilbert's axioms hold. We omit thedetails because they are laborious, and because we can approach SAS andASA differently now that we have coordinates. Specifically, it becomespossible to define the concept of "motion" that Euclid appealed to in hisproof of SAS! This will be done in the next section.ExercisesThe most useful instance of relative slope is where the lines are perpendicular.3.5.1 Show that lines of slopes t1 and t2 are perpendicular just in case t1t2 = −1.3.5.2 Use the condition for perpendicularity found in Exercise 3.5.1 to show that the line from (1, 0) to (3, 4) is perpendicular to the line from (0, 2) to (4, 0).
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3.6 Isometries 57 In the next section, we will define a rotation about O to be a transformationrc,s of R2 depending on two real numbers c and s such that c2 + s2 = 1. Thetransformation rc,s sends the point (x, y) to the point (cx − sy, sx + cy). It will beexplained in the next section why it is reasonable to call this a "rotation about O,"and why c = cos θ and s = sin θ , where θ is the angle of rotation. For the moment, suppose that this is the case, and consider the effect of tworotations rc1 ,s1 and rc2 ,s2 , where c1 = cos θ1 , s1 = sin θ1 ; c2 = cos θ2 , s2 = sin θ2 .This thought experiment leads us to proofs of the formulas for cos, sin, and tan ofθ1 + θ 2 :3.5.3 Show that the outcome of rc1 ,s1 and rc2 ,s2 is to send (x, y) to ((c1 c2 − s1 s2 )x − (s1 c2 + c1 s2 )y, (s1 c2 + c1 s2 )x + (c1 c2 − s1 s2 )y) .3.5.4 Assuming that rc1 ,s1 really is a rotation about O through angle θ1 , and rc2 ,s2 really is a rotation about O through angle θ2 , deduce from Exercise 3.5.3 that cos(θ1 + θ2 ) = cos θ1 cos θ2 − sin θ1 sin θ2 , sin(θ1 + θ2 ) = sin θ1 cos θ2 + cos θ1 sin θ2 .3.5.5 Deduce from Exercise 3.5.4 that tan θ1 + tan θ2 tan(θ1 + θ2 ) = , 1 − tan θ1 tan θ2 hence tan θ1 − tan θ2 tan(θ1 − θ2 ) = . 1 + tan θ1 tan θ23.6 IsometriesA possible weakness of our model of the plane is that it seems to singleout a particular point (the origin O) and particular lines (the x- and y-axes).In Euclid's plane, each point is like any other point and each line is likeany other line. We can overcome the apparent bias of R 2 by consideringtransformations that allow any point to become the origin and any line tobecome the x-axis. As a bonus, this idea gives meaning to the idea of"motion" that Euclid tried to use in his attempt to prove SAS. A transformation of the plane is simply a function f : R2 → R2 , in otherwords, a function that sends points to points.
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58 3 Coordinates A transformation f is called an isometry (from the Greek for "samelength") if it sends any two points, P1 and P2 , to points f (P1 ) and f (P2 ) thesame distance apart. Thus, an isometry is a function f with the property | f (P1 ) f (P2 )| = |P1 P2 |for any two points P1 , P2 . Intuitively speaking, an isometry "moves theplane rigidly" because it preserves the distance between points. There aremany isometries of the plane, but they can be divided into a few simpleand obvious types. We show examples of each type below, and, in the nextsection, we explain why only these types exist. You will notice that certain isometries (translations and rotations) makeit possible to move the origin to any point in the plane and the x-axis to anyline. Thus, R2 is really like Euclid's plane, in the sense that each pointis like any other point and each line is like any other line. This propertyentitles us to choose axes wherever it is convenient. For example, we areentitled to prove the triangle inequality, as suggested in the Exercises toSection 3.3, by choosing one vertex of the triangle at O and another on thepositive x-axis.TranslationsA translation moves each point of the plane the same distance in the samedirection. Each translation depends on two constants a and b, so we denoteit by ta,b . It sends each point (x, y) to the point (x + a, y + b). It is obviousthat a translation preserves the distance between any two points, but it isworth checking this formally—so as to know what to do in less obviouscases. So let P1 = (x1 , y1 ) and P2 = (x2 , y2 ). It follows that ta,b (P1 ) = (x1 + a, y1 + b), ta,b (P2 ) = (x2 + a, y2 + b)and therefore, |ta,b (P1 )ta,b (P2 )| = (x2 + a − x1 − a)2 + (y2 + b − y1 − b)2 = (x2 − x1 )2 + (y2 − y1 )2 = |P1 P2 |, as required.
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60 3 CoordinatesReflectionsThe easiest reflection to describe is reflection in the x-axis, which sendsP = (x, y) to P = (x, −y). Again it is obvious that this is an isometry, butwe can check by calculating the distance between reflected points P1 andP2 (Exercise 3.6.1). We can reflect the plane in any line, and we can do this by combiningreflection in the x-axis with translations and rotations. For example, reflec-tion in the line y = 1 (which is parallel to the x-axis) is the result of thefollowing three isometries: • t0,−1 , a translation that moves the line y = 1 to the x-axis, • reflection in the x-axis, • t0,1 , which moves the x-axis back to the line y = 1.In general, we can do a reflection in any line L by moving L to the x-axisby some combination of translation and rotation, reflecting in the x-axis,and then moving the x-axis back to L . Reflections are the most fundamental isometries, because any isometryis a combination of them, as we will see in the next section. In particular,any translation is a combination of two reflections, and any rotation is acombination of two reflections (see Exercises 3.6.2–3.6.4).Glide reflectionsA glide reflection is the result of a reflection followed by a translation in thedirection of the line of reflection. For example, if we reflect in the x-axis,sending (x, y) to (x, −y), and follow this with the translation t 1,0 of length1 in the x-direction, then (x, y) ends up at (x + 1, −y). A glide reflection with nonzero translation length is different from thethree types of isometry previously considered. • It is not a translation, because a translation maps any line in the di- rection of translation into itself, whereas a glide reflection maps only one line into itself (namely, the line of reflection). • It is not a rotation, because a rotation has a fixed point and a glide reflection does not. • It is not a reflection, because a reflection also has fixed points (all points on the line of reflection).
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3.7 The three reflections theorem 61Exercises3.6.1 Check that reflection in the x-axis preserves the distance between any two points.When we combine reflections in two lines, the nature of the outcome depends onwhether the lines are parallel.3.6.2 Reflect the plane in the x-axis, and then in the line y = 1/2. Show that the resulting isometry sends (x, y) to (x, y + 1), so it is the translation t0,1 .3.6.3 Generalize the idea of Exercise 3.6.2 to show that the combination of reflec- tions in parallel lines, distance d/2 apart, is a translation through distance d, in the direction perpendicular to the lines of reflection.3.6.4 Show, by a suitable picture, that the combination of reflections in lines that meet at angle θ /2 is a rotation through angle θ , about the point of intersec- tion of the lines. Another way to put the result of Exercise 3.6.4 is as follows: Reflectionsin any two lines meeting at the same angle θ /2 at the same point P give thesame outcome. This observation is important for the next three exercises (wherepictures will also be helpful).3.6.5 Show that reflections in lines L , M , and N (in that order) have the same outcome as reflections in lines L , M , and N , where M is perpendicular to N .3.6.6 Next show that reflections in lines L , M , and N have the same outcome as reflections in lines L , M , and N , where M is parallel to L and N is perpendicular to M .3.6.7 Deduce from Exercise 3.6.6 that the combination of any three reflections is a glide reflection.3.7 The three reflections theoremWe saw in Section 3.3 that the points equidistant from two points A andB form a line, which implies that isometries of the plane are very simple:An isometry f of R2 is determined by the images f (A), f (B), f (C) of threepoints A, B,C not in a line. The proof follows from three simple observations: • Any point P in R2 is determined by its distances from A, B,C. Be- cause if Q is another point with the same distances from A, B,C as P, then A, B,C lie in the equidistant line of P and Q, contrary to the assumption that A, B,C are not in a line.
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62 3 Coordinates • The isometry f preserves distances (by definition of isometry), so f (P) lies at the same respective distances from f (A), f (B), f (C) as P does from A, B,C. • There is only one point at given distances from f (A), f (B), f (C) be- cause these three points are not in a line—in fact they form a triangle congruent to triangle ABC, because f preserves distances.Thus, the image f (P) of any point P—and hence the whole isometry f —isdetermined by the images of three points A, B,C not in a line. This "three point determination theorem" gives us the:Three reflections theorem. Any isometry of R2 is a combination of one,two, or three reflections. Given an isometry f , we choose three points A, B,C not in a line, andwe look for a combination of reflections that sends A to f (A), B to f (B),and C to f (C). Such a combination is necessarily equal to f . We cancertainly send A to f (A) by reflection in the equidistant line of A and f (A).Call this reflection rA . Now rA sends B to rA (B), so if rA (B) = f (B) we need to do nothingmore for B. If rA (B) = f (B), we can send rA (B) to f (B) by reflection rB in theequidistant line of rA (B) and f (B). Fortunately, f (A) = rA (A) lies on thisline, because the distance from f (A) to f (B) equals the distance from r A (A)to rA (B) (because f and rA are isometries). Thus, rB does not move f (A),and the combination of rA followed by rB sends A to f (A) and B to f (B). The argument is similar for C. If C has already been sent to f (C), weare done. If not, we reflect in the line equidistant from f (C) and the pointwhere C has been sent so far. It turns out (by a check of equal distanceslike that made for f (A) above) that f (A) and f (B) already lie on this line,so they are not moved. Thus, we finally have a combination of no morethan three reflections that moves A to f (A), B to f (B), and C to f (C), asrequired. Now of course, one reflection is a reflection, and we found in the pre-vious exercise set that combinations of two reflections are translations androtations, and that combinations of three reflections are glide reflections(which include reflections). Thus, an isometry of R2 is either a transla-tion, a rotation, or a glide reflection.
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3.8 Discussion 63ExercisesGiven three points A, B,C and the points f (A), f (B), f (C) to which they are sentby an isometry f , it is possible to find three reflections that combine to form fby following the steps in the proof above. However, if one merely wants to knowwhat kind of isometry f is—translation, rotation, or glide reflection—then theanswer can be found more simply. To fix ideas, we take the initial three points to be A = (0, 1), B = (0, 0), andC = (1, 0). You will probably find it helpful to sketch the triples of points f (A),f (B), f (C) given in the following exercises.3.7.1 Suppose that f (A) = (1.4, 2), f (B) = (1.4, 1), and f (C) = (2.4, 1). Is f a translation or a rotation? How can you tell that f is not a glide reflection?3.7.2 Suppose that f (A) = (0.4, 1.8), f (B) = (1, 1), and f (C) = (1.8, 1.6). We can tell that f is not a translation or glide reflection (hence, it must be a rotation). How?3.7.3 Suppose that f (A) = (1.8, 1.6), f (B) = (1, 1), and f (C) = (0.4, 1.8). How do I know that this is a glide reflection?3.7.4 State a simple test for telling whether f is a translation, rotation, or glide reflection from the positions of f (A), f (B), and f (C).3.8 DiscussionThe discovery of coordinates is rightly considered a turning point in thedevelopment of mathematics because it reveals a vast new panorama ofgeometry, open to exploration in at least three different directions. • Description of curves by equations, and their analysis by algebra. This direction is called algebraic geometry, and the curves described by polynomial equations are called algebraic curves. Straight lines, described by the linear equations ax+by+c = 0, are called curves of degree 1. Circles, described by the equations (x−a)2 +(y−b)2 = r2 , are curves of degree 2, and so on. One can see that there are curves of arbitrarily high degree, so most of algebraic geometry is beyond the scope of this book. Even the curves of degree 3 are worth a book of their own, so for them, and other algebraic curves, we refer readers elsewhere. Two excellent books, which show how algebraic geometry relates to other parts of mathematics, are Elliptic Curves by H. P. McKean and V. Moll and Plane Algebraic Curves by E. Brieskorn and H. Knorrer. ¨
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64 3 Coordinates • Algebraic study of objects described by linear equations (such as lines and planes). Even this is a big subject, called linear algebra. Although it is technically part of algebraic geometry, it has a special flavor, very close to that of Euclidean geometry. We explore plane geometry from the viewpoint of linear algebra in Chapter 4, and later we make some brief excursions into three and four dimensions. The real strength of linear algebra is its ability to describe spaces of any number of dimensions in geometric language. Again, this investigation is beyond our scope, but we will recommend additional reading at the appropriate places. • The study of transformations, which draws on the special branch of algebra known as group theory. Because many geometric transfor- mations are described by linear equations, this study overlaps with linear algebra. The role of transformations was first emphasized by the German mathematician Felix Klein, in an address he delivered at the University of Erlangen in 1872. His address, known by its Ger- man name the Erlanger Programm, characterizes geometry as the study of transformation groups and their invariants. So far, we have seen only one transformation group and a handful ofinvariants—the group of isometries of R2 and what it leaves invariant(length, angle, straightness)—so the importance of Klein's idea can hardlybe clear yet. However, in Chapter 4 we introduce a very different group oftransformations and a very different invariant—the projective transforma-tions and the cross-ratio—so readers are asked to bear with us. In Chapters7 and 8, we develop Klein's idea in some generality and give another sig-nificant example, the geometry of the "non-Euclidean" plane.
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4Vectors and Euclidean spacesP REVIEW In this chapter, we process coordinates by linear algebra. We view points as vectors that can be added and multiplied by numbers, and we introduce the inner product of vectors, which gives an efficient algebraic method to deal with both lengths and angles. We revisit some theorems of Euclid to see where they fit in the world of vector geometry, and we become acquainted with some theorems that are particularly natural in this environment. For plane geometry, the appropriate vectors are ordered pairs (x, y) of real numbers. We add pairs according to the rule (u1 , u2 ) + (v1 , v2 ) = (u1 + v1 , u2 + v2 ), and multiply a pair by a real number a according to the rule a(u1 , u2 ) = (au1 , au2 ). These vector operations do not involve the concept of length or dis- tance; yet they enable us to discuss certain ratios of lengths and to prove the theorems of Thales and Pappus. The concept of distance is introduced through the concept of inner product u · v of vectors u and v. If u = (u1 , u2 ) and v = (v1 , v2 ), then u · v = u1 v1 + u2 v2 . The inner product gives us distance because u · u = |u|2 , where |u| is the distance of u from the origin 0. It also gives us angle because u · v = |u||v| cos θ , where θ is the angle between the directions of u and v from 0. 65
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4.1 Vectors 67 In fact, the rule for forming the sum of two vectors is often called the"parallelogram rule." Scalar multiplication by a is also geometrically interesting, because itrepresents magnification by the factor a. It magnifies, or dilates, the wholeplane by the factor a, transforming each figure into a similar copy of itself.Figure 4.2 shows an example of this with a = 2.5. aw au w av u v 0 Figure 4.2: Scalar multiplication as a dilation of the planeReal vector spacesIt seems that the operations of vector addition and scalar multiplicationcapture some geometrically interesting features of a space. With this inmind, we define a real vector space to be a set V of objects, called vectors,with operations of vector addition and scalar multiplication satisfying thefollowing conditions: • If u and v are in V , then so are u + v and au for any real number a. • There is a zero vector 0 such that u + 0 = u for each vector u. Each u in V has a additive inverse −u such that u + (−u) = 0. • Vector addition and scalar multiplication on V have the eight prop- erties listed at the beginning of this section.
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68 4 Vectors and Euclidean spaces It turns out that real vector spaces are a natural setting for Euclideangeometry. We must introduce extra structure, which is called the innerproduct, before we can talk about length and angle. But once the innerproduct is there, we can prove all theorems of Euclidean geometry, oftenmore efficiently than before. Also, we can uniformly extend geometry toany number of dimensions by considering the space Rn of ordered n-tuplesof real numbers (x1 , x2 , . . . , xn ). For example, we can study three-dimensional Euclidean geometry inthe space of ordered triples R3 = {(x1 , x2 , x3 ) : x1 , x2 , x3 ∈ R},where the sum of u = (u1 , u2 , u3 ) and v = (v1 , v2 , v3 ) is defined by (u1 , u2 , u3 ) + (v1 , v2 , v3 ) = (u1 + v1 , u2 + v2 , u3 + v3 )and the scalar multiple au is defined by a(u1 , u2 , u3 ) = (au1 , au2 , au3 ).ExercisesIt is obvious that R2 has the eight properties of a real vector space. However, itis worth noting that R2 "inherits" these eight properties from the correspondingproperties of real numbers. For example, the property u + v = v + u (called thecommutative law) for vector addition is inherited from the corresponding commu-tative law for number addition, u + v = v + u, as follows: u + v = (u1 , u2 ) + (v1 + v2 ) = (u1 + v1 , u2 + v2 ) by definition of vector addition = (v1 + u1 , v2 + u2 ) by commutative law for numbers = (v1 , v2 ) + (u1 , u2 ) by definition of vector addition = v + u.4.1.1 Check that the other seven properties of a vector space for R2 are inherited from corresponding properties of R.4.1.2 Similarly check that Rn has the eight properties of a vector space. The term "dilation" for multiplication of all vectors in R2 (or Rn for thatmatter) by a real number a goes a little beyond the everyday meaning of the wordin the case when a is smaller than 1 or negative.4.1.3 What is the geometric meaning of the transformation of R2 when every vector is multiplied by −1? Is it a rotation?4.1.4 Is it a rotation of R3 when every vector is multiplied by −1?
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4.2 Direction and linear independence 694.2 Direction and linear independenceVectors give a concept of direction in R2 by representing lines through 0.If u is a nonzero vector, then the real multiples au of u make up the linethrough 0 and u, so we call them the points "in direction u from 0." (Youmay prefer to say that −u is in the direction opposite to u, but it is simplerto associate direction with a whole line, rather than a half line.) Nonzero vectors u and v, therefore, have different directions from 0 ifneither is a multiple of the other. It follows that such u and v are linearlyindependent; that is, there are no real numbers a and b, not both zero, with au + bv = 0.Because, if one of a, b is not zero in this equation, we can divide by it andhence express one of u, v as a multiple of the other. The concept of direction has an obvious generalization: w has directionu from v (or relative to v) if w−v is a multiple of u. We also say that "w−vhas direction u," and there is no harm in viewing w − v as an abbreviationfor the line segment from v to w. As in coordinate geometry, we say thatline segments from v to w and from s to t are parallel if they have the samedirection; that is, if w − v = a(t − s) for some real number a = 0. Figure 4.3 shows an example of parallel line segments, from v to w andfrom s to t, both of which have direction u. w = v+ 3u 2 u t = s+ 1u 2 s v 0 Figure 4.3: Parallel line segments with direction u Here we have 3 1 w − v = u and t − s = u, so w − v = 3(t − s). 2 2
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70 4 Vectors and Euclidean spaces Now let us try out the vector concept of parallels on two important the-orems from previous chapters. The first is a version of the Thales theoremthat parallels cut a pair of lines in proportional segments.Vector Thales theorem. If s and v are on one line through 0, t and w areon another, and w − v is parallel to t − s, then v = as and w = at for somenumber a. If w − v is parallel to t − s, then w − v = a(t − s) = at − as for some real number a.Because v is on the same line through 0 as s, we have v = bs for some b,and similarly w = ct for some c (this is a good moment to draw a picture).It follows that w − v = ct − bs = at − as,and therefore, (c − a)t + (a − b)s = 0.But s and t are in different directions from 0, hence linearly independent,so c − a = a − b = 0. Thus, v = as and w = at, as required. As in axiomatic geometry (Exercise 1.4.3), the Pappus theorem followsfrom the Thales theorem. However, "proportionality" is easier to handlewith vectors.Vector Pappus theorem. If r, s, t, u, v, w lie alternately on two linesthrough 0, with u − v parallel to s − r and t − s parallel to v − w, then u − tis parallel to w − r. Figure 4.4 shows the situation described in the theorem. Because u − v is parallel to s − r, we have u = as and v = ar for somenumber a. Because t − s is parallel to v − w, we have s = bw and t = bvfor some number b. From these two facts, we conclude that u = as = abw and t = bv = bar,hence, u − t = abw − bar = ab(w − r),
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4.3 Midpoints and centroids 71 ar v =b t =b ar v= r 0 w s = bw u = as = abw Figure 4.4: The parallel Pappus configuration, labeled by vectorsand therefore, u − t is parallel to w − r. The last step in this proof, where we exchange ba for ab, is of coursea trifle, because ab = ba for any real numbers a and b. But it is a big stepin Chapter 6, where we try to develop geometry without numbers. Therewe have to build an arithmetic of line segments, and the Pappus theorem iscrucial in getting multiplication to behave properly.ExercisesIn Chapter 1, we mentioned that a second theorem about parallels, the Desarguestheorem, often appears alongside the Pappus theorem in the foundations of ge-ometry. This situation certainly holds in vector geometry, where the appropriateDesargues theorem likewise follows from the vector Thales theorem.4.2.1 Following the setup explained in Exercise 1.4.4, and the formulation of the vector Pappus theorem above, formulate a "vector Desargues theorem."4.2.2 Prove your vector Desargues theorem with the help of the vector Thales theorem.4.3 Midpoints and centroidsThe definition of a real vector space does not include a definition of dis-tance, but we can speak of the midpoint of the line segment from u to vand, more generally, of the point that divides this segment in a given ratio.
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72 4 Vectors and Euclidean spaces To see why, first observe that v is obtained from u by adding v − u,the vector that represents the position of v relative to u. More generally,adding any scalar multiple a(v − u) to u produces a point whose directionrelative to u is the same as that of v. Thus, the points u + a(v − u) areprecisely those on the line through u and v. In particular, the midpointof the segment between u and v is obtained by adding 1 (v − u) to u, and 2hence, 1 1 midpoint of line segment between u and v = u + (v − u) = (u + v). 2 2One might describe this result by saying that the midpoint of the line seg-ment between u and v is the vector average of u and v. This description of the midpoint gives a very short proof of the theoremfrom Exercise 2.2.1, that the diagonals of a parallelogram bisect each other.By choosing one of the vertices of the parallelogram at 0, we can assumethat the other vertices are at u, v, and u + v (Figure 4.5). u+v v u 0 Figure 4.5: Diagonals of a parallelogram Then the midpoint of the diagonal from 0 to u + v is 1 (u + v). And, by 2the result just proved, this is also the midpoint of the other diagonal—theline segment between u and v. The vector average of two or more points is physically significant be-cause it is the barycenter or center of mass of the system obtained by plac-ing equal masses at the given points. The geometric name for this vectoraverage point is the centroid. In the case of a triangle, the centroid has an alternative geometric de-scription, given by the following classical theorem about medians: the linesfrom the vertices of a triangle to the midpoints of the respective oppositesides.
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4.3 Midpoints and centroids 73Concurrence of medians. The medians of any triangle pass through thesame point, the centroid of the triangle. To prove this theorem, suppose that the vertices of the triangle are u, v, 1and w. Then the median from u goes to the midpoint 2 (v + w), and so on,as shown in Figure 4.6. w 1 2 (u + w) 1 2 (v + w) u 1 2 (u + v) v Figure 4.6: The medians of a triangle Looking at this figure, it seems likely that the medians meet at the point2/3 of the way from u to 1 (v + w), that is, at the point 2 2 1 1 2 1 u+ (v + w) − u = u + (v + w) − u = (u + v + w). 3 2 3 3 3Voil` ! This is the centroid, and a similar argument shows that it lies 2/3 aof the way between v and 1 (u + w) and 2/3 of the way between w and 212 (u + v). That is, the centroid is the common point of all three medians. You can of course check by calculation that 1 (u + v + w) lies 2/3 of 3the way between v and 1 (u + w) and also 2/3 of the way between w and 212 (u + v). But the smart thing is not to do the calculation but to predict theresult. We know that calculating the point 2/3 of the way between u and12 (v + w) gives 1 (u + v + w), 3a result that is unchanged when we permute the letters u, v, and w. Theother two calculations are the same, except for the ordering of the letters u,v, and w. Hence, they lead to the same result.
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74 4 Vectors and Euclidean spacesExercises4.3.1 Show that a square with vertices t, u, v, w has center 1 (t + u + v + w). 4The theorem about concurrence of medians generalizes beautifully to three di-mensions, where the figure corresponding to a triangle is a tetrahedron: a solidwith four vertices joined by six lines that bound the tetrahedron's four triangularfaces (Figure 4.7). Figure 4.7: A tetrahedron4.3.2 Suppose that the tetrahedron has vertices t, u, v, and w. Show that the cen- troid of the face opposite to t is 1 (u + v + w), and write down the centroids 3 of the other three faces.4.3.3 Now consider each line joining a vertex to the centroid of the opposite face. In particular, show that the point 3/4 of the way from t to the centroid of the opposite face is 1 (t + u + v + w)—the centroid of the tetrahedron. 44.3.4 Explain why the point 1 (t + u + v + w) lies on the other three lines from a 4 vertex to the centroid of the opposite face.4.3.5 Deduce that the four lines from vertex to centroid of opposite face meet at the centroid of the tetrahedron.4.4 The inner productIf u = (u1 , u2 ) and v = (v1 , v2 ) are vectors in R2 , we define their innerproduct u · v to be u1 v1 + u2 v2 . Thus, the inner product of two vectors isnot another vector, but a real number or "scalar." For this reason, u · v isalso called the scalar product of u and v.
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4.4 The inner product 75 It is easy to check, from the definition, that the inner product has thealgebraic properties u · v = v · u, u · (v + w) = u · v + u · w, (au) · v = u · (av) = a(u · v),which immediately give information about length and angle: • The length |u| is the distance of u = (u1 , u2 ) from 0, which is u2 + u2 1 2 by the definition of distance in R2 (Section 3.3). Hence, |u|2 = u2 + u2 = u · u. 1 2 It follows that the square of the distance |v − u| from u to v is |v − u|2 = (v − u) · (v − u) = |u|2 + |v|2 − 2u · v. • Vectors u and v are perpendicular if and only if u · v = 0. Because u has slope u2 /u1 and v has slope v2 /v1 , and we know from Section 3.5 that they are perpendicular if and only the product of their slopes is −1. That means u2 v1 =− and hence u2 v2 = −u1 v1 , u1 v2 multiplying both sides by u1 v2 . This equation holds if and only if 0 = u1 v1 + u2 v2 = u · v. We will see in the next section how to extract more information aboutangle from the inner product. The formula above for |v−u| 2 turns out to bethe "cosine rule" or "law of cosines" from high-school trigonometry. Buteven the criterion for perpendicularity gives a simple proof of a far-from-obvious theorem:Concurrence of altitudes. In any triangle, the perpendiculars from thevertices to opposite sides (the altitudes) have a common point. To prove this theorem, take 0 at the intersection of two altitudes, saythose through the vertices u and v (Figure 4.8). Then it remains to showthat the line from 0 to the third vertex w is perpendicular to the side v − u.
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76 4 Vectors and Euclidean spaces v 0 u w Figure 4.8: Altitudes of a triangle Because u is perpendicular to the opposite side w − v, we have u · (w − v) = 0, that is, u · w − u · v = 0. Because v is perpendicular to the opposite side u − w, we have v · (u − w) = 0, that is, v · u − v · w = 0.Adding these two equations, and bearing in mind that u · v = v · u, we get u · w − v · w = 0, that is, w · (v − u) = 0.Thus, w is perpendicular to v − u, as required.ExercisesThe inner product criterion for directions to be perpendicular, namely that theirinner product is zero, gives a neat way to prove the theorem in Exercise 2.2.2about the diagonals of a rhombus.4.4.1 Suppose that a parallelogram has vertices at 0, u, v, and u + v. Show that its diagonals have directions u + v and u − v.4.4.2 Deduce from Exercise 4.4.1 that the inner product of these directions is |u|2 − |v|2 , and explain why this is zero for a rhombus. The inner product also gives a concise way to show that the equidistant line oftwo points is the perpendicular bisector of the line connecting them (thus provingmore than we did in Section 3.3).
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4.5 Inner product and cosine 774.4.3 The condition for w to be equidistant from u and v is (w − u) · (w − u) = (w − v) · (w − v). Explain why, and show that this condition is equivalent to |u|2 − 2w · u = |v|2 − 2w · v.4.4.4 Show that the condition found in Exercise 4.4.3 is equivalent to u+v w− · (u − v) = 0, 2 and explain why this says that w is on the perpendicular bisector of the line from u to v.Having established that the line equidistant from u and v is the perpendicularbisector, we conclude that the perpendicular bisectors of the sides of a triangle areconcurrent—because this is obviously true of the equidistant lines of its vertices.4.5 Inner product and cosineThe inner product of vectors u and v depends not only on their lengths|u| and |v| but also on the angle θ between them. The simplest way toexpress its dependence on angle is with the help of the cosine function. Wewrite the cosine as a function of angle θ , cos θ . But, as usual, we avoidmeasuring angles and instead define cos θ as the ratio of sides of a right-angled triangle. For simplicity, we assume that the triangle has vertices 0,u, and v as shown in Figure 4.9. v |v| θ 0 u |u| Figure 4.9: Cosine as a ratio of lengths Then the side v is the hypotenuse, θ is the angle between the side uand the hypotenuse, and its cosine is defined by |u| cos θ = . |v|
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4.5 Inner product and cosine 79 A nice way to close this circle of ideas is to consider the special casein which u and v are the sides of a right-angled triangle and u − v is thehypotenuse. In this case, u is perpendicular to v, so u · v = 0, and the cosinerule becomes hypotenuse2 = |u − v|2 = |u|2 + |v|2—which is the Pythagorean theorem. This result should not be a surprise,however, because we have already seen how the Pythagorean theorem isbuilt into the definition of distance in R2 and hence into the inner product.ExercisesThe Pythagorean theorem can also be proved directly, by choosing 0 at the rightangle of a right-angled triangle whose other two vertices are u and v.4.5.1 Show that |v − u|2 = |u|2 + |v|2 under these conditions, and explain why this is the Pythagorean theorem. While on the subject of right-angled triangles, we mention a useful formulafor studying them.4.5.2 Show that (v + u) · (v − u) = |v|2 − |u|2 .This formula gives a neat proof of the theorem from Section 2.7 about the anglein a semicircle. Take a circle with center 0 and a diameter with ends u and −u asshown in Figure 4.11. Also, let v be any other point on the circle. v −u u 0 Figure 4.11: Points on a semicircle4.5.3 Show that the sides of the triangle meeting at v have directions v + u and v − u and hence show that they are perpendicular.
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82 4 Vectors and Euclidean spaces • u2 + u2 is the distance from 0 of (u1 , u2 , 0) (the hypotenuse of a 1 2 right-angled triangle with sides u1 and u2 ), • u2 + u2 + u2 is the distance from 0 of (u1 , u2 , u3 ) (the hypotenuse 1 2 3 of a right-angled triangle with sides u2 + u2 and u3 ). 1 2 All theorems proved in this chapter for vectors in the plane R 2 hold inRn . This fact is clear if we take the plane in Rn to consist of vectors ofthe form (x1 , x2 , 0, . . . , 0), because such vectors behave exactly the same asvectors (x1 , x2 ) in R2 . But in fact any given plane in Rn behaves the sameas the special plane of vectors (x1 , x2 , 0, . . . , 0). We skip the details, but itcan be proved by constructing an isometry of Rn mapping the given planeonto the special plane. As in R2 , any isometry is a product of reflections.In Rn , at most n + 1 reflections are required, and the proof is similar to theone given in Section 3.7.ExercisesA proof of Cauchy–Schwarz using only general properties of the inner product canbe obtained by an algebraic trick with quadratic equations. The general propertiesinvolved are the four listed at the beginning of Section 4.4 and the assumptionthat w · w = |w|2 ≥ 0 for any vector w (an inner product with the latter property iscalled positive definite).4.6.1 The Euclidean inner product for Rn defined above is positive definite. Why?4.6.2 For any real number x, and any vectors u and v, show that (u + xv) · (u + xv) = |u|2 + 2x(u · v) + x2 |v|2 , and hence that |u|2 + 2x(u · v) + x2 |v|2 ≥ 0 for any real number x.4.6.3 If A, B, and C are real numbers and A + Bx +Cx2 ≥ 0 for any real number x, explain why B2 − 4AC ≤ 0.4.6.4 By applying Exercise 4.6.3 to the inequality |u|2 + 2x(u · v) + x2 |v|2 ≥ 0, show that (u · v)2 ≤ |u|2 |v|2 , and hence |u · v| ≤ |u||v|.
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4.7 Rotations, matrices, and complex numbers 834.7 Rotations, matrices, and complex numbersRotation matricesIn Section 3.6, we defined a rotation of R2 as a function rc,s , where c and sare two real numbers such that c2 +s2 = 1. We described rc,s as the functionthat sends (x, y) to (cx − sy, sx + cy), but it is also described by the matrixof coefficients of x and y, namely c −s , where c = cos θ and s = sin θ . s cBecause most readers will already have seen matrices, it may be useful totranslate some previous statements about functions into matrix language,where they may be more familiar. (Readers not yet familiar with matriceswill find an introduction in Section 7.2.) Matrix notation allows us to rewrite (x, y) → (cx − sy, sx + cy) as c −s x cx − sy = s c y sx + cyThus, the function rc,s is applied to the variables x and y by multiplying the x c −scolumn vector on the left by the matrix . Functions are y s cthereby separated from their variables, so they can be composed withoutthe variables becoming involved—simply by multiplying matrices. This idea gives proofs of the formulas for cos(θ1 + θ2 ) and sin(θ1 + θ2 ),similar to Exercises 3.5.3 and 3.5.4, but with the variables x and y filteredout: cos θ1 − sin θ1 • Rotation through angle θ1 is given by the matrix . sin θ1 cos θ1 cos θ2 − sin θ2 • Rotation through angle θ2 is given by the matrix . sin θ2 cos θ2 • Hence, rotation through θ1 + θ2 is given by the product of these two matrices. That is,
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86 4 Vectors and Euclidean spaces4.7.5 Explain why any u with |u| = 1 can be written in the form cos θ + i sin θ for some angle θ , and conclude that multiplication by u rotates the point 1 (hence the whole plane) through angle θ . It follows, in particular, that multiplication by i = (0, 1) sends (1, 0) to (0, 1)and hence rotates the plane through π /2. This result in turn implies i2 = −1,because multiplication by i2 then rotates the plane through π , which is also theeffect of multiplication by −1.4.8 DiscussionBecause the geometric content of a vector space with an inner product ismuch the same as Euclidean geometry, it is interesting to see how manyaxioms it takes to describe a vector space. Remember from Section 2.9that it takes 17 Hilbert axioms to describe the Euclidean plane, or 16 if weare willing to drop completeness of the line. To define a vector space, we began in Section 4.1 with eight axioms forvector addition and scalar multiplication: u+v = v+u 1u = u u + (v + w) = (u + v) + w a(u + v) = au + av u+0 = u (a + b)u = au + bu u + (−u) = 0 a(bu) = (ab)u. Then, in Section 4.4, we added three (or four, depending on how youcount) axioms for the inner product: u · v = v · u, u · (v + w) = u · v + u · w, (au) · v = u · (av) = a(u · v), We also need relations among inner product, length, and angle—at aminimum the cosine formula, u · v = |u||v| cos θ ,so this is 12 or 13 axioms so far. But we have also assumed that the scalars a, b, . . . are real numbers, sothere remains the problem of writing down axioms for them. At the very
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4.8 Discussion 87least, one needs axioms saying that the scalars satisfy the ordinary rules ofcalculation, the so-called field axioms (this is usual when defining a vectorspace): −1 a + (−a) = 0, aa =1 (inverse laws) a(b + c) = ab + ac (distributive law) Thus, the usual definition of a vector space, with an inner product suit-able for Euclidean geometry, takes more than 20 axioms! Admittedly, thefield axioms and the vector space axioms are useful in many other parts ofmathematics, whereas most of the Hilbert axioms seem meaningful onlyin geometry. And, by varying the inner product slightly, one can changethe geometry of the vector space in interesting ways. For example, one canobtain the geometry of Minkowski space used in Einstein's special theoryof relativity. To learn more about the vector space approach to geometry,see Linear Algebra and Geometry, a Second Course by I. Kaplansky andMetric Affine Geometry by E. Snapper and R. J. Troyer. Still, one can dream of building geometry on a much simpler set ofaxioms. In Chapter 6, we will realize this dream with projective geometry,which we begin studying in Chapter 5.
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5PerspectiveP REVIEW Euclid's geometry concerns figures that can be drawn with straight- edge and compass, even though many of its theorems are about straight lines alone. Are there any interesting figures that can be drawn with straightedge alone? Remember, the straightedge has no marks on it, so it is impossible to copy a length. Thus, with a straightedge alone, we cannot draw a square, an equilateral triangle, or any figure involving equal line segments. Yet there is something interesting we can draw: a perspective view of a tiled floor, such as the one shown in Figure 5.1. Figure 5.1: Perspective view of a tiled floor This picture is interesting because it seems clear that all tiles in the view are of equal size. Thus, even though we cannot draw tiles that are actually equal, we can draw tiles that look equal. We will explain how to solve the problem of drawing perspective views in Section 5.2. The solution takes us into a new form of geometry—a geometry of vision—called projective geometry. 88
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5.1 Perspective drawing 895.1 Perspective drawingSometime in the 15th century, Italian artists discovered how to draw three-dimensional scenes in correct perspective. Figures 5.2 and 5.3 illustrate thegreat advance in realism this skill achieved, with pictures drawn before andafter the discovery. The "before" picture, Figure 5.2, is a drawing I foundin the book Perspective in Perspective by L. Wright. It is thought to datefrom the late 15th century, but it comes from England, where knowledgeof perspective had evidently not reached at that time. Figure 5.2: The birth of St Edmund, by an unknown artist The "after" picture, Figure 5.3, is the 1514 engraving Saint Jerome inhis study, by the great German artist Albrecht Durer (1471–1528). D¨ rer ¨ umade study tours of Italy in 1494 and 1505 and became a master of allaspects of drawing, including perspective. The simplest test of perspective drawing is the depiction of a tiled floor.The picture in Figure 5.2 clearly fails this test. All the tiles are drawn asrectangles, which makes the floor look vertical. We know from experiencethat a horizontal rectangle does not look rectangular—its angles are not allright angles because its sides converge to a common point on the horizon,as in the tabletop in D¨ rer's engraving. u
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90 5 Perspective Figure 5.3: St Jerome in his study, by Albrecht Durer ¨ The Italians drew tiles by a method called the costruzione legittima(legitimate construction), first published by Leon Battista Alberti in 1436.The bottom edge of the picture coincides with a line of tile edges, andany other horizontal line is chosen as the horizon. Then lines drawn fromequally spaced points on the bottom edge to a point on the horizon depictthe parallel columns of tiles perpendicular to the bottom edge (Figure 5.4).Another horizontal line, near the bottom, completes the first row of tiles. Figure 5.4: Beginning the costruzione legittima
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92 5 Perspective If each of the points 0, 1, 2, 3, . . . is sent to the next, then each of their perspec-tive images y = 0, 1 , 2 , 3 , . . . is sent to the next. 2 3 4 15.1.2 Show that the function f (y) = 2−y effects this move. 15.1.3 Which point on the y-axis is not moved by the function f (y) = 2−y , and what is the geometric significance of this point?5.2 Drawing with straightedge aloneThe construzione legittima takes advantage of something that is visuallyobvious but mathematically mysterious—the fact that parallel lines gen-erally do not look parallel, but appear to meet on the horizon. The pointwhere a family of parallels appear to meet is called their "vanishing point"by artists, and their point at infinity by mathematicians. The horizon itself,which consists of all the points at infinity, is called the line at infinity. However, the costruzione legittima does not take full advantage ofpoints at infinity. It involves some parallels that are really drawn parallel,so we need both straightedge and compass as used in Chapter 1. The con-struction also needs measurement to lay out the equally spaced points onthe bottom line of the picture, and this again requires a compass. Thus, thecostruzione legittima is a Euclidean construction at heart, requiring both astraightedge and a compass. Is it possible to draw a perspective view of a tiled floor with a straight-edge alone? Absolutely! All one needs to get started is the horizon and atile placed obliquely. The tile is created by the two pairs of parallel lines,which are simply pairs that meet on the horizon (Figure 5.7). horizon Figure 5.7: The first tile We then draw the diagonal of this tile and extend it to the horizon,obtaining the point at infinity of all diagonals parallel to this first one. Thisstep allows us to draw two more diagonals, of tiles adjacent to the first one.
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5.2 Drawing with straightedge alone 93These diagonals give us the remaining sides of the adjacent tiles, and wecan then repeat the process. The first few steps are shown in Figure 5.8.Figure 5.1 at the beginning of the chapter is the result of carrying out manysteps (and deleting the construction lines). Draw diagonal of first tile, extended to the horizon Extend diagonal of second tile to the horizon Draw side of second tile, through the new intersection Draw side of more tiles, through the new intersection Figure 5.8: Constructing the tiled floor This construction is easy and fun to do, and we urge the reader to geta straightedge and try it. Also try the constructions suggested in the exer-cises, which create pictures of floors with differently shaped tiles.
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94 5 PerspectiveExercisesConsider the triangular tile shown shaded in Figure 5.9. Notice that this trianglecould be half of the quadrangular tile shown in Figure 5.7 (this is a hint). Figure 5.9: A triangular tile5.2.1 Draw a perspective view of the plane filled with many copies of this tile.5.2.2 Also, by deleting some lines in your solution to Exercise 5.2.1, create a perspective view of the plane filled with congruent hexagons.5.3 Projective plane axioms and their modelsDrawing a tiled floor with straightedge alone requires a "horizon"—a lineat infinity. Apart from this requirement, the construction works becausecertain things remain the same in any view of the plane: • straight lines remain straight • intersections remain intersections • parallel lines remain parallel or meet on the horizon.Now parallel lines always meet on the horizon if you point yourself in theright direction, so if we could look in all directions at once we would seethat any two lines have a point in common. This idea leads us to believein a structure called a projective plane, containing objects called "points"and "lines" satisfying the following axioms. We write "points" and "lines"in quotes because they may not be the same as ordinary points and lines.Axioms for a projective plane 1. Any two "points" are contained in a unique "line." 2. Any two "lines" contain a unique "point." 3. There exist four "points", no three of which are in a "line."
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5.3 Projective plane axioms and their models 95 Notice that these are axioms about incidence: They involve only meet-ings between "points" and "lines," not things such as length or angle. Someof Euclid's and Hilbert's axioms are of this kind, but not many. Axiom 1 is essentially Euclid's first axiom for the construction of lines.Axiom 2 says that there are no exceptional pairs of lines that do not meet.We can define "parallels" to be lines that meet on a line called the "hori-zon," but this does not single out a special class of lines—in a projectiveplane, the "horizon" behaves the same as any other line. Axiom 3 says thata projective plane has "enough points to be interesting." We can think ofthe four points as the four vertices of a quadrilateral, from which one maygenerate the complicated structure seen in the pictures of a tiled floor at thebeginning of this chapter.The real projective planeIf there is such a thing as a projective plane, it should certainly satisfy theseaxioms. But does anything satisfy them? After all, we humans can neversee all of the horizon at once, so perhaps it is inconsistent to suppose thatall parallels meet. These doubts are dispelled by the following model, orinterpretation, of the axioms for a projective plane. The model is calledthe real projective plane RP2 , and it gives a mathematical meaning to theterms "point," "line," and "plane" that makes all the axioms true. Take "points" to be lines through O in R3 , "lines" to be planes throughO in R3 , and the "plane" to be the set of all lines through O in R3 . Then 1. Any two "points" are contained in a unique "line" because two given lines through O lie in a unique plane through O. 2. Any two "lines" contain a unique "point" because any two planes through O meet in a unique line through O. 3. There are four different "points," no three of which are in a "line": for example, the lines from O to the four points (1, 0, 0), (0, 1, 0), (0, 0, 1), and (1, 1, 1), because no three of these lines lie in the same plane through O.The last claim is perhaps a little hard to grasp by visualization, but it canbe checked algebraically because any plane through O has an equation ofthe form ax + by + cz = 0 for some real numbers a, b, c.
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96 5 PerspectiveIf, say, (1, 0, 0) and (0, 1, 0) are on this plane, then we find by substitutingthese values of x, y, z in the equation that a = 0 and b = 0, hence the plane is z = 0.But (0, 0, 1) and (1, 1, 1) do not lie on the plane z = 0. It can be checkedsimilarly that the plane through any two of the points does not contain theother two. It is no fluke that lines and planes through O in R3 behave as we want"points" and "lines" of a projective plane to behave, because they capturethe idea of viewing with an all-seeing eye. The point O is the position ofthe eye, and the lines through O connect the eye to points in the plane.Consider how the eye sees the plane z = −1, for example (Figure 5.10). z y x L1 L2 L3 P3 M P2 P1 Figure 5.10: Viewing a plane from O Points P1 , P2 , P3 , . . . in the plane z = −1 are joined to the eye by linesL1 , L2 , L3 , . . . through O, and as the point Pn tends to infinity, the line Lntends toward the horizontal. Therefore, it is natural to call the horizontallines through O the "points at infinity" of the plane z = −1, and to call theplane of all horizontal lines through O the "horizon" or "line at infinity" ofthe plane z = −1. Unlike the lines L1 , L2 , L3 , . . ., corresponding to points P1 , P2 , P3 , . . .of the Euclidean plane z = −1, horizontal lines through O have no coun-terparts in the Euclidean plane: They extend the Euclidean plane to a pro-jective plane. However, the extension arises in a natural way. Once we
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5.3 Projective plane axioms and their models 97replace the points P1 , P2 , P3 , . . . by lines in space, we realize that there areextra lines (the horizontal lines) corresponding to the points on the horizon. This model of the projective plane nicely captures our intuitive ideaof points at infinity, but it also makes the idea clearer. We can see, forexample, why it is proper for each line to have only one point at infinity,not two: because the lines L connecting O to points P along a line M inthe plane z = −1 tend toward the same horizontal line as P tends to infinityin either direction (namely, the parallel to M through O). It is hard to find a surface that behaves like RP2 , but it is easy tofind a curve that behaves like any "line" in it, a so-called real projectiveline. Figure 5.11 shows how. The "points" in a "line" of RP 2 , namely thelines through O in some plane through O, correspond to points of a circlethrough O. Each point P = O on the circle corresponds to the line throughO and P, and the point O itself corresponds to the tangent line at O. O P Figure 5.11: Modeling a projective line by a circleExercisesTo gain more familiarity with calculations in R3 , let us pursue the example of four"points" given above.5.3.1 Find the plane ax + by + cz = 0 through the points (0, 0, 1) and (1, 1, 1), and check that it does not contain the points (1, 0, 0) and (0, 1, 0).5.3.2 Show that RP2 has four "lines," no three of which have a common "point." Not only does RP2 contain four "lines," no three of which have a "point" incommon; the same is true of any projective plane, because this property followsfrom the projective plane axioms alone.5.3.3 Suppose that A, B,C, D are four "points" in a projective plane, no three of which are in a "line." Consider the "lines" AB, BC,CD, DA. Show that if AB and BC have a common point E, then E = B.5.3.4 Deduce from Exercise 5.3.3 that the three lines AB, BC,CD have no com- mon point, and that the same is true of any three of the lines AB, BC,CD, DA.
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98 5 Perspective5.4 Homogeneous coordinatesBecause "points" and "lines" of RP2 are lines and planes through O in R3 ,they are easily handled by methods of linear algebra. A line through O isdetermined by any point (x, y, z) = O, and it consists of the points (tx,ty,tz),where t runs through all real numbers. Thus, a "point" is not given by asingle triple (x, y, z), but rather by any of its nonzero multiples (tx,ty,tz).These triples are called the homogeneous coordinates of the "point." A plane through O has a linear equation of the form ax + by + cz = 0,called a homogeneous equation. The same plane is given by the equationtax+tby+tcz = 0 for any nonzero t. Thus, a "line" is likewise not given bya single triple (a, b, c), but by the set of all its nonzero multiples (ta,tb,tc). If (x1 , y1 , z1 ) and (x2 , y2 , z2 ) lie on different lines through O, then it isgeometrically obvious that they lie in a unique plane ax + by + cz = 0. Thecoordinates (a, b, c) of this plane can be found by solving the two equations ax1 + by1 + cz1 = 0, ax2 + by2 + cz2 = 0,for a, b, and c. Because there are more unknowns than equations, there isnot a single solution triple but a whole space of them—in this case, a set ofmultiples (ta,tb,tc), all representing the same homogeneous equation. This is the algebraic reason why two "points" lie on a unique "line" in 2RP . There is a similar reason why two "lines" have a unique "point" incommon. Two "lines" are given by two equations a1 x + b1 y + c1 z = 0, a2 x + b2 y + c2 z = 0,and we find their common "point" by solving these equations for x, y, andz. This problem is the same as above, but with the roles of a, b, c exchangedwith those of x, y, z. The solution in this case is a set of multiples (tx,ty,tz)representing the homogeneous coordinates of the common "point." The practicalities of finding the "line" through two "points" or the"point" common to two "lines" are explored in the next exercise set. Butfirst I want to make a theoretical point. It makes no algebraic differenceif the coordinates of "points" and "lines" are complex numbers. We candefine a complex projective plane CP2 , each "point" of which is a set oftriples of the form (tx,ty,tz), where x, y, z are particular complex numbers
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5.4 Homogeneous coordinates 99and t runs through all complex numbers. It remains true that any two"points" lie on a unique "line" and any two "lines" have unique commonpoint, simply because the algebraic properties of complex linear equationsare exactly the same as those of real linear equations. Similarly, one canshow there are four "points," no three of which are in a "line" of CP 2 . Thus, there is more than one model of the projective plane axioms.Later we shall look at other models, which enable us to see that certainproperties of RP2 are not properties of all projective planes and hence donot follow from the projective plane axioms.Projective spaceIt is easy to generalize homogeneous coordinates to quadruples (w, x, y, z)and hence to define the three-dimensional real projective space RP 3 . It has"points," "lines," and "planes" defined as follows (we use vector notationto shorten the definitions): • A "point" is a line through O in R4 , that is, a set of quadruples tu, where u = (w, x, y, z) is a particular quadruple of real numbers and t runs through all real numbers. • A "line" is a plane through O in R4 , that is, a set t1 u1 + t2 u2 where u1 and u2 are linearly independent points of R4 and t1 and t2 run through all real numbers. • A "plane" is a three-dimensional space through O in R4 , that is, a set t1 u1 + t2 u2 + t3 u3 , where u1 , u2 , and u3 are linearly independent points of R4 and t1 , t2 , and t3 run through all real numbers. Linear algebra then enables us to show various properties of the "points,""lines," and "planes" in RP3 , such as: 1. Two "points" lie on a unique "line." 2. Three "points" not on a "line" lie on a unique "plane." 3. Two "planes" have unique "line" in common. 4. Three "planes" with no common "line" have one common "point."These properties hold for any three-dimensional projective space, and RP 3is not the only one. There is also a complex projective space CP 3 , andmany others. RP3 has an unexpected influence on the geometry of thesphere, as we will see in Section 7.8.
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100 5 PerspectiveExercises5.4.1 Find the plane ax + by + cz = 0 that contains the points (1, 2, 3) and (1, 1, 1).5.4.2 Find the line of intersection of the planes x + 2y + 3z = 0 and x + y + z = 0.5.4.3 You can write down the solution of Exercise 5.4.2 as soon as you have solved Exercise 5.4.1. Why?5.5 ProjectionThe three-dimensional Euclidean space R3 , in which the lines through Oare the "points" of RP2 and the planes through O are the "lines" of RP2 ,also contains many other planes. Each plane P not passing through O canbe regarded as a perspective view of the projective plane RP 2 , a view thatcontains all but one "line" of RP2 . Each point P of P corresponds to a line ("of sight") through O, andhence to a "point" of RP2 . The only lines through O that do not meet Pare those parallel to P, and these make up the line at infinity or horizon ofP, as we have already seen in the case of the plane z = −1 in Section 5.3. If P1 and P2 are any two planes not passing through O we can projectP1 to P2 by sending each point P1 in P1 to the point P2 in P2 lying onthe same line through O as P1 (Figure 5.12). The geometry of RP2 iscalled "projective" because it encapsulates the geometry of a whole familyof planes related by projection. P2 P1 O P1 P2 Figure 5.12: Projecting one plane to another
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5.5 Projection 101Projections of projective linesProjection of one plane P1 onto another plane P2 produces an image ofP1 that is generally distorted in some way. For example, a grid of squareson P1 may be mapped to a perspective view of the grid that looks likeFigure 5.1. Nevertheless, straight lines remain straight under projection, sothere are limits to the amount of distortion in the image. To better under-stand the nature and scope of projective distortion, in this subsection weanalyze the mappings of the projective line obtainable by projection. An effective way to see the distortion produced by projection of oneline L1 onto another line L2 is to mark a series of equally spaced dots onL1 and the corresponding image dots on L2 . You can think of the imagedots as "shadows" of the dots on L1 cast by light rays from the point ofprojection P, except that we have projective lines through P, not rays, soit can seem as though the "shadow" on L2 comes ahead of the dot on L1 .(See Figure 5.15, but bear in mind that a projective line is really circular,so it is always possible to pass through P, to a point on L1 , then to a pointon L2 , in that order.) In the simplest cases, where L1 and L2 are parallel, the image dots arealso equally spaced. Figure 5.13 shows the case of projection from a pointat infinity, where the lines from the dots on L1 to their images on L2 areparallel and hence the dots on L1 are simply translated a constant distancel. If we choose an origin on each line and use the same unit of length oneach, then projection from infinity sends each x on L1 to x + l on L2 . L1 0 1 2 3 L2 0 l 1+l 2+l 3+l Figure 5.13: Projection from infinity When L1 is projected from a finite point P, then the distance betweendots is magnified by a constant factor k = 0. If we take P on a line through
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102 5 Perspectivethe zero points on L1 and L2 , then the projection sends each x on L1 tokx on L2 (Figure 5.14). Note also that this projection sends x on L2 to x/kon L1 , so the magnification factor can be any k = 0. P L1 0 x L2 0 kx Figure 5.14: Projection from a finite point When L1 and L2 are not parallel the distortion caused by projection ismore extreme. Figure 5.15 shows how the spacing of dots changes whenL1 is projected onto a perpendicular line L2 from a point O equidistantfrom both. Figure 5.16 is a closeup of the image line L2 , showing howthe image dots "converge" to a point corresponding to the horizontal linethrough O (which corresponds to the point at infinity on L1 ). L2 L1 O Figure 5.15: Example of projective distortion of the line
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5.5 Projection 103 1 1 1 1 1 1 L2 −1 −2 −3 5 4 3 2 1 Figure 5.16: Closeup of the image line We take O = (0, 0) as usual, and we suppose that L1 is parallel to thex-axis, that L2 is parallel to the y-axis, and that the dots on L1 are unitdistance apart. Then the line from O to the dot at x = n on L1 has slope1/n and hence it meets the line L2 at y = 1/n. Thus the map from L1to L2 is the function sending x to y = 1/x. This map exhibits the mostextreme kind of distortion induced by projection, with the point at infinityon L1 sent to the point y = 0 on L2 . Any combination of these projections is therefore a combination offunctions 1/x, kx, and x + l, which are called generating transformations.The combinations of generating transformations are precisely the functionsof the form ax + b f (x) = , where ad − bc = 0, cx + dthat we study in the next section.ExercisesBefore studying all these functions, it is useful to study the (simpler) subclassobtained by composing functions that send x to x + l or kx (for k = 0). The latterfunctions obviously include any function of the form f (x) = ax + b with a = 0,which is the result of multiplying by a, and then adding b.5.5.1 If f1 (x) = a1 x + b1 with a1 = 0 and f2 (x) = a2 x + b2 with a2 = 0, show that f1 ( f2 (x)) = Ax + B, with A = 0, and find the constants A and B.5.5.2 Deduce from Exercise 5.5.1 that the result of composing any number of functions that send x to x + l or kx (for k = 0) is a function of the form f (x) = ax + b with a = 0. We know that such functions represent combinations of certain projectionsfrom lines to parallel lines, but do they include any projection from a line to aparallel line?5.5.3 Show that projection of a line, from any finite point P, onto a parallel line is represented by a function of the form f (x) = ax + b.
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5.6 Linear fractional functions 105 We now wish to prove the converse: Any sequence of projections ofthe number line realizes a linear fractional function. From the previoussection, we know this is true for projection of a line onto a parallel line,so it suffices to find the function realized by projection of a line onto anintersecting line. We first take the case in which the lines are perpendic-ular (Figure 5.17). This case generalizes that of Figure 5.15, by allowingprojection from an arbitrary point (a, b). y (a, b) O x t f (t) Figure 5.17: Projecting a line onto a perpendicular line To find where the point t on the x-axis goes on the y-axis, we considerthe slope of the line through t and (a, b). Between these points, the rise is bb and the run is a − t, so the slope is a−t . Between t and the point f (t) onthe y-axis, the run is t and the rise is − f (t); hence, bt f (t) = , which is a linear fractional function. t −a For the general case of intersecting lines, we take one line to be thex-axis again, and the other to be the line y = cx. Again we project the pointt on the x-axis from (a, b) to the other line, and to find where t goes, wefirst find the equation of the line through t and (a, b). Equating the slopefrom t to (a, b) with the slope between an arbitrary point (x, y) on the lineand (a, b), we find the equation b b−y = . a−t a−xThis line meets the line y = cx where b b − cx = , a−t a−x
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106 5 Perspectiveand hence where bt x= , which is also a linear fractional function of t. ct − ac + b Thus, any single projection of a line can be represented by a linearfractional function of distance along the line. It is easy to check (Exercise5.6.2) that the result of composing linear fractional functions is linear frac-tional. Hence, any finite sequence of projections is represented by a linearfractional function.Dividing by zeroYou remember from high-school algebra that division by zero is not a validoperation, because it leads from true equations, such as 3 × 0 = 2 × 0, tofalse ones, such as 3 = 2. Nevertheless, in carefully controlled situations, itis permissible, and even enlightening, to divide by zero. One such situationis in projective mappings of the projective line. The linear fractional functions f (x) = ax+b we have used to describe cx+dprojective mappings of lines are actually defective if the variable x runsonly through the set R of real numbers. For example, the function f (x) =1/x we used to map points of the line L1 onto points of the line L2 asshown in Figure 5.15 does not in fact map all points. It cannot send thepoint x = 0 anywhere, because 1/0 is undefined; nor can it send any pointto y = 0, because 0 = 1/x for any real x. This defect is neatly fixed byextending the function f (x) = 1/x to a new object x = ∞, and declaringthat 1/∞ = 0 and 1/0 = ∞. The new object ∞ is none other than the pointat infinity of the line L1 , which is supposed to map to the point 0 on L2 .Likewise, if 1/0 = ∞, the point 0 on L1 is sent to the point ∞ on L2 , as itshould be. Thus, the function f (x) = 1/x works properly, not on the real line R,but on the real projective line R ∪ {∞}—a line together with a point atinfinity. The rules 1/∞ = 0 and 1/0 = ∞ simply reflect this fact. It is much the same with any linear fractional function f (x) = ax+b . cx+dThe denominator of the fraction is 0 when x = −d/c, and the correct valueof the function in this case is ∞. Conversely, no real value of x gives f (x)the value a/c, but x = ∞ does. For this reason, any function f (x) = ax+b cx+dwith ad − bc = 0 maps the real projective line R ∪ {∞} onto itself. The mapis also one-to-one, as may seen in the exercises below.
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5.6 Linear fractional functions 107The real projective line RP1We can now give an algebraic definition of the object we called the "realprojective line" in Section 5.3. It is the set R ∪ {∞} together with all thelinear fractional functions mapping R ∪ {∞} onto itself. We call this set,with these functions on it, the real projective line RP1 . The set R∪{∞} certainly has the points we require for a projective line;the functions are to give R ∪ {∞} the "elasticity" of a line that undergoesprojection. The ordinary line R is not very "elastic" in this sense. Oncewe have decided which point is 0 and which point is 1, the numerical valueof every point on R is uniquely determined. In contrast, the position of apoint on RP1 is not determined by the positions of 0 and 1 alone. For example, there is a projection that sends 0 to 0, 1 to 1, but 2 to 3.Nevertheless, there is a constraint on the "elasticity" of RP 1 . If 0 goes to 0,1 goes to 1, and 2 goes to 3, say, then the destination of every other point xis uniquely determined. In the next two sections, we will see why.Exercises ax+bThe formula cx+d = a + c(cx+d) gives an inkling why the condition ad − bc = 0 is c bc−adpart of the definition of a linear fractional function: If ad − bc = 0, then ax+b = a cx+d cis a constant function, and hence it maps the whole line onto one point. If we want to map the line onto another line, it is therefore necessary to havead − bc = 0. It is also sufficient, because we can solve the equation y = ax+b for cx+dx in that case. ax+b5.6.1 Solve the equation y = cx+d for x, and note where your solution assumes ad − bc = 0.5.6.2 If f1 (x) = a1 x+d1 and f2 (x) = c 1 x+b1 a2 x+b2 c2 x+d2 , compute f1 ( f2 (x)), and verify that it Ax+B is of the form Cx+D . A B a1 b1 a2 b25.6.3 Verify also that = . C D c1 d1 c2 d2 Thus, linear fractional functions behave like 2 × 2 matrices. Moreover, thecondition ad − bc = 0 corresponds to having nonzero determinant, which explainswhy this is the condition for an inverse function to exist.5.6.4 It also guarantees that if a1 d1 − b1 c1 = 0 for f1 (x) and a2 d2 − b2 c2 = 0 for f2 (x) in Exercise 5.5.2, then AD − BC = 0 for f 1 ( f2 (x)). Why?
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108 5 Perspective5.7 The cross-ratio You might say it was a triumph of algebra to invent this quan- tity that turns out to be so valuable and could not be imagined geometrically. Or if you are a geometer at heart, you may say it is an invention of the devil and hate it all your life. Robin Hartshorne, Geometry: Euclid and Beyond, p. 341. It is visually obvious that projection can change lengths and even theratio of lengths, because equal lengths often appear unequal under projec-tion. And yet we can recognize that Figure 5.1 is a picture of equal tiles,even though they are unequal in size and shape. Some clue to their equalitymust be preserved, but what? It cannot be length; it cannot be a ratio oflengths; but, surprisingly, it can be a ratio of ratios, called the cross-ratio. The cross-ratio is a quantity associated with four points on a line. Ifthe four points have coordinates p, q, r, and s, then their cross-ratio is thefunction of the ordered 4-tuple (p, q, r, s) defined by (r − p)/(s − p) (r − p)(s − q) , which can also be written as . (r − q)/(s − q) (r − q)(s − p)The cross-ratio is preserved by projection. To show this, it suffices to showthat it is preserved by the three generating transformations from which wecomposed all linear fractional maps in the previous section: 1. The map sending x to x + l. Here the numbers p, q, r, s are replaced by p + l, q + l, r + l, s + l, respectively. This does not change the cross-ratio because the l terms cancel by subtraction. 2. The map sending x to kx. Here the numbers p, q, r, s are replaced by kp, kq, kr, ks, respectively. This does not change the cross-ratio because the k terms cancel by division. 3. The map sending x to 1/x. Here the numbers p, q, r, s are replaced by 1 , 1 , 1 , 1 , respectively, so p q r s the cross-ratio (r − p)(s − q) (r − q)(s − p)
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5.7 The cross-ratio 109 is replaced by p−r q−s ( 1 − 1 )( 1 − 1 ) r p s q pr · qs = q−r p−s taking common denominators, ( 1 − 1 )( 1 − 1 ) r q s p qr · ps (p − r)(q − s) = multiplying through by pqrs, (q − r)(p − s) (r − p)(s − q) = changing the sign in all factors, (r − q)(s − p) and thus, the cross-ratio is unchanged in this case also.Is the cross-ratio visible?If we take the four equally spaced points p = 0, q = 1, r = 2, and s = 3 onthe line, then their cross-ratio is (r − p)(s − q) 2 × 2 4 = = . (r − q)(s − p) 1 × 3 3It follows that any projective image of these points also has cross-ratio 4/3.Do four points on a line look equally spaced if their cross-ratio is 4/3? Testyour eye on the quadruples of points in Figure 5.18, and then do Exercise5.7.2 to find the correct answer. Figure 5.18: Which is a projective image of equally spaced points?ExercisesWe will see in the next section that any three points on RP1 can be projected toany three points. Hence, there cannot be an invariant involving just three points. However, the invariance of the cross-ratio tells us that, once the images ofthree points are known, the whole projection map is known (compare with the"three-point determination" of isometries of the plane in Section 3.7).
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110 5 Perspective5.7.1 Show that there is only one point s that has a given cross-ratio with given points p, q, and r.In particular, if we have the points p = 0, q = 2, r = 3 (which we do in the threequadruples in Figure 5.18), there is exactly one s that gives the cross-ratio 4/3required for "equally spaced" points.5.7.2 Find the value of s that gives the cross-ratio 4/3, and hence find the "equally spaced" quadruple in Figure 5.18. Before the discovery of perspective, artists sometimes attempted to draw atiled floor by making the width of each row of tiles a constant fraction e of the onebefore.5.7.3 Show that this method is not correct by computing the cross-ratio of four points separated by the distances 1, e, and e2 .5.8 What is special about the cross-ratio?In the remainder of this book, we use the abbreviation (r − p)(s − q) [p, q; r, s] = (r − q)(s − p)for the cross-ratio of the four points p, q, r, s, taken in that order. We have shown that the cross-ratio is an invariant of linear fractionaltransformations, but it is obviously not the only one. Examples of other in-variants are (cross-ratio)2 and cross-ratio +1. The cross-ratio is special be-cause it is the defining invariant of linear fractional transformations. Thatis, the linear fractional transformations are precisely the transformationsof RP1 that preserve the cross-ratio. (Thus, the cross-ratio defines linearfractional transformations the way that length defines isometries.) We prove this fact among several others about linear fractional trans-formations and the cross-ratio.Fourth point determination. Given any three points p, q, r ∈ RP1 , anyother point x ∈ RP1 is uniquely determined by its cross-ratio [p, q; r, x] = ywith p, q, r. This statement holds because we can solve the equation (r − p)(x − q) y= (r − q)(x − p)uniquely for x in terms of p, q, r, and y.
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5.8 What is special about the cross-ratio? 111Existence of three-point maps. Given three points p, q, r ∈ RP1 and threepoints p , q , r ∈ RP1 , there is a linear fractional transformation f sendingp, q, r to p , q , r , respectively. This statement holds because there is a projection sending any threepoints p, q, r to any three points p , q , r , and any projection is linear frac-tional by Section 5.6. The way to project is shown in Figure 5.19. P p q r p q r Figure 5.19: Projecting three points to three points Without loss of generality, we can place the two copies of RP 1 so thatp = p . Then the required projection is from the point P where the linesqq and rr meet.Uniqueness of three-point maps. Exactly one linear fractional functionsends three points p, q, r to three points p , q , r , respectively. A linear fractional f sending p, q, r to p , q , r , respectively, must sendany x = p, q, r to x satisfying [p, q; r, x] = [p , q ; r , x ], because f preservesthe cross-ratio by Section 5.7. But x is unique by fourth point determina-tion, so there is exactly one such function f .Characterization of linear fractional maps. These are precisely the mapsof RP1 that preserve the cross-ratio. By Section 5.7, any linear fractional map f preserves the cross-ratio.That is, [ f (p, f (q); f (r), f (s)] = [p, q; r, s] for any four points p, q, r, s. Conversely, suppose that f is a map of RP1 with [ f (p), f (q); f (r), f (s)] = [p, q; r, s] for any four points p, q, r, s.By the existence of three-point maps, we can find a linear fractional g thatagrees with f on p, q, r. But then, because f preserves the cross-ratio, gagrees with f on s also, by unique fourth point determination. Thus, g agrees with f everywhere, so f is a linear fractional map.
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112 5 Perspective The existence of three-point maps says that any three points on RP 1can be sent to any three points by a linear fractional transformation. Thus,any invariant of triples of points must have the same value for any triple,and so it is trivial. A nontrivial invariant must involve at least four points,and the cross-ratio is an example. It is in fact the fundamental example, inthe following sense.The fundamental invariant. Any invariant of four points is a function ofthe cross-ratio. To see why, suppose I(p, q, r, s) is a function, defined on quadruplesof distinct points, that is invariant under linear fractional transformations.Thus, I ( f (p), f (q), f (r), f (s)) = I(p, q, r, s) for any linear fractional f .In other words, I has the same value on all quadruples (p , q , r , s ) that re-sult from (p, q, r, s) by a linear fractional transformation. But more is true: Ihas the same value on all quadruples (p , q , r , s ) with the same cross-ratioas (p, q, r, s), because such a quadruple (p , q , r , s ) results from (p, q, r, s)by a linear fractional transformation. This follows from the existence anduniqueness of three-point maps: • by existence, we can send p, q, r to p , q , r , respectively, by a linear fractional transformation f , and • by uniqueness, f also sends s to s , the unique point that makes [p, q; r, s] = [p , q ; r , s ].Because I has the same value on all quadruples with the same cross-ratio,it is meaningful to view I as a function J of the cross-ratio, defined by J([p, q; r, s]) = I(p, q, r, s).ExercisesThe following exercises illustrate the result above about invariant functions ofquadruples. They show that the invariants obtained by permuting the variables inthe cross-ratio y = [p, q; r, s] are simple functions of y, such as 1/y and y − 1.5.8.1 If [p, q; r, s] = y, show that [p, q; s, r] = 1/y.5.8.2 If [p, q; r, s] = y, show that [q, p; r, s] = 1/y.
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5.9 Discussion 1135.8.3 Prove that [p, q; r, s] + [p, r; q, s] = 1, so it follows that if [p, q; r, s] = y, then [p, r; q, s] = 1 − y. The transformations y → 1/y and y → 1 − y obtained in this way generate alltransformations of the cross-ratio obtained by permuting its variables. There aresix such transformations (even though there are 24 permutations of four variables).5.8.4 Show that the functions of y obtained by combining 1/y and 1 − y in all ways are 1 1 1 y y, , 1 − y, 1 − , , . y y 1−y y−15.8.5 Explain why any permutation of four variables may be obtained by ex- changes: either of the first two, the middle two, or the last two variables.5.8.6 Deduce from Exercises 5.8.1–5.8.5 that the invariants obtained from the cross-ratio y by permuting its variables are precisely the six listed in Exer- cise 5.8.4. The six linear fractional functions of y obtained in Exercise 5.8.4 constitutewhat is sometimes called the cross-ratio group. It is an example of a concept wewill study in Chapter 7: the concept of a group of transformations. Unlike mostof the groups studied there, this group is finite.5.9 DiscussionThe plane RP2 studied in this chapter is the most important projectiveplane, but it is far from being the only one. Many other projective planescan be constructed by imitating the construction of RP2 , which is basedon ordered triples (x, y, z) and linear equations ax + by + cz = 0. It is notessential for x, y, z to be real numbers. As noted earlier, they could be com-plex numbers, but more generally they could be elements of any field. Afield is any set with + and × operations satisfying the nine field axiomslisted in Section 4.8. If F is any field, we can consider the space F3 of ordered triples (x, y, z)with x, y, z ∈ F. Then the projective plane FP2 has • "points," each of which is a set of triples (kx, ky, kz), where x, y, z ∈ F are fixed and k runs through the elements of F, • "lines," each of which consists of the "points" satisfying an equation of the form ax + by + cz = 0 for some fixed a, b, c ∈ F.
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114 5 Perspective The projective plane axioms can be checked for FP2 just as they werefor RP2 . The same calculations apply, because the field axioms ensure thatthe same algebraic operations work in F (solving equations, for example).This gives a great variety of planes FP2 , because there are a great varietyof fields F. Perhaps the most familiar field, after R and C, is the set Q of rationalnumbers. QP2 is not unlike RP2 , except that all of its points have rationalcoordinates, and all of its lines are full of gaps, because they contain onlyrational points. More surprising examples arise from taking F to be a finite field, ofwhich there is one with pn elements for each power pn of each prime p.The simplest example is the field F2 , whose members are the elements 0and 1, with the following addition and multiplication tables. + 0 1 × 0 1 0 0 1 0 0 0 1 1 0 1 0 1 The projective plane F2 P2 has seven points, corresponding to the sevennonzero points in F3 : 2 (1, 0, 0), (0, 1, 0), (0, 0, 1), (0, 1, 1), (1, 0, 1), (1, 1, 0), (1, 1, 1).These points are arranged in threes along the seven lines in Figure 5.20,one of which is drawn as a circle so as to connect its three points. (0,0,1) (1,0,1) (0,1,1) (1,1,1) (1,0,0) (0,1,0) (1,1,0) Figure 5.20: The smallest projective plane
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5.9 Discussion 115 Notice that the lines satisfy the linear equations x = 0, y = 0, z = 0, x + y = 0, y + z = 0, z + x = 0, x + y + z = 0.For example, the points on the circle satisfy x + y + z = 0. (Of course, thecoordinates have nothing to do with position in the plane of the diagram.The figure is mainly symbolic, while attempting to show "points" collectedinto "lines.") This structure is called the Fano plane, and it is the smallest projectiveplane. Despite being small, it is well-behaved, because its "lines" satisfylinear equations, just as lines do in the traditional geometric world. Thanksto finite fields, linear algebra works well in many finite structures. It hasled to the wholesale development of finite geometries, many of which haveapplications in the mathematics of information and communication. However, the three axioms for a finite projective plane do not ensurethat the plane is of the form FP2 , with coordinates for points and linearequations for lines. They can be satisfied by bizarre "nonlinear" structures,as we will see in the next chapter. A fourth axiom is needed to engendera field F of coordinates, and the axiom is none other than the theorem ofPappus that we met briefly in Chapters 1 and 4. This state of affairs will beexplained in Chapter 6.The invariance of the cross-ratioThe invariance of the cross-ratio was discovered by Pappus around 300 CEand rediscovered by Desargues around 1640. It appears (not very clearly)as Proposition 129 in Book VII of Pappus' Mathematical Collection andagain in Mani` re universelle de Mr Desargues in 1648. The latter is a epamphlet on perspective by written by Abraham Bosse, a disciple of De-sargues. It also contains the first published statement of the Desarguestheorem mentioned in Chapters 1 and 4. Because of this, and the fact thathe wrote the first book on projective geometry, Desargues is considered tobe the founder of the subject. Nevertheless, projective geometry was littleknown until the 19th century, when geometry expanded in all directions.In the more general 19th century geometry (which often included use ofcomplex numbers), the cross-ratio continued to be a central concept.
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116 5 Perspective One of the reasons we now consider the appropriate generalization ofclassical projective geometry to be projective geometry with coordinates ina field is that the cross-ratio continues to make sense in this setting. Linear fractional transformations and the cross-ratio make sense whenR is replaced by any field F. The transformations x → x + l and x → kxmake sense on F, and x → 1/x makes sense on F ∪ {∞} if we set 1/0 = ∞and 1/∞ = 0. Then the transformations ax + b x→ , where a, b, c, d ∈ F and ad − bc = 0, cx + dmake sense on the "F projective line" FP1 = F ∪ {∞}. The cross-ratiois invariant by the same calculation as in Section 5.7, thanks to the fieldaxioms, because the usual calculations with fractions are valid in a field.
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6Projective planesP REVIEW In this chapter, geometry fights back against the forces of arithme- tization. We show that coordinates need not be brought into geom- etry from outside—they can be defined by purely geometric means. Moreover, the geometry required to define coordinates and their arithmetic is simpler than Euclid's geometry. It is the projective geometry introduced in the previous chapter, but we have to build it from scratch using properties of straight lines alone. We started this project in Section 5.3 by stating the three axioms for a projective plane. However, these axioms are satisfied by many structures, some of which have no reasonable system of coordinates. To build coordinates, we need at least one additional axiom, but for convenience we take two: the Pappus and Desargues properties that were proved with the help of coordinates in Chapter 4. Here we proceed in the direction opposite to Chapter 4: Take Pappus and Desargues as axioms, and use them to define coordinates. The coordinates are points on a projective line, and we add and multiply them by constructions like those in Chapter 1. But instead of using parallel lines as we did there, we call lines "parallel" if they meet on a designated line called the "horizon" or the "line at infinity." The main problem is to prove that our addition and multiplication operations satisfy the field axioms. This is where the theorems of Pappus and Desargues are crucial. Pappus is needed to prove the commutative law of multiplication, ab = ba, whereas Desargues is needed to prove the associative law, a(bc) = (ab)c. 117
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118 6 Projective planes6.1 Pappus and Desargues revisitedThe theorems of Pappus and Desargues stated in Chapters 1 and 4 hada similar form: If two particular pairs of lines are parallel, then a thirdpair is parallel. Because parallel lines meet on the horizon, the Pappusand Desargues theorems also say that if two particular pairs of lines meeton the horizon, then so does a third pair. And because the horizon is notdifferent from any other line, these theorems are really about three pairs oflines having their intersections on the same line. In this projective setting, the Pappus theorem takes the form shown inFigure 6.1. The six vertices of the hexagon are shown as dots, and theopposite sides are shown as a black pair, a gray pair, and a dotted pair.The line on which each of the three pairs meet is labeled L , and we haveoriented the figure so that L is horizontal (but this is not at all necessary).Projective Pappus theorem. Six points, lying alternately on two straightlines, form a hexagon whose three pairs of opposite sides meet on a line. L Figure 6.1: The projective Pappus configuration This statement of the Pappus theorem is called projective because itinvolves only the concepts of points, lines, and meetings between them.Meetings between geometric objects are called incidences, and, for thisreason, the Pappus theorem is also called an incidence theorem. The threeaxioms of a projective plane, given in Section 5.3, are the simplest exam-ples of incidence theorems.
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6.1 Pappus and Desargues revisited 119 The projective Desargues theorem is another incidence theorem. Itconcerns the pairs of corresponding sides of two triangles, shown in solidgray in Figure 6.2. The triangles are in perspective from a point P, whichmeans that each pair of corresponding vertices lies on a line through P.The three corresponding pairs of sides are again shown as black, gray, anddotted, and each pair meets on a line labeled L .Projective Desargues theorem. If two triangles are in perspective from apoint, then their pairs of corresponding sides meet on a line. L P Figure 6.2: The projective Desargues configuration An important special case of the Desargues theorem has the center ofprojection P on the line L where the corresponding sides of the trianglesmeet. This special case is called the little Desargues theorem, and it isshown in Figure 6.3.Little Desargues theorem. If two triangles are in perspective from a pointP, and if two pairs of corresponding sides meet on a line L through P,then the third pair of corresponding sides also meets on L . Because the projective Pappus and Desargues theorems involve onlyincidence concepts, one would like proofs of them that involve only thethree axioms for a projective plane given in Section 5.3. Unfortunately,this is not possible, because there are examples of projective planes notsatisfying the Pappus and Desargues theorems. What we can do, however,is take the Pappus and Desargues theorems as new axioms. Together withthe original three axioms for projective planes, these two new axioms applyto a broad class of projective planes called Pappian planes.
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120 6 Projective planes L P Figure 6.3: The little Desargues configuration The Pappian planes include RP2 and many other planes, but not all.They turn out to be the planes with coordinates satisfying the same laws ofalgebra as the real numbers—the field axioms. The object of this chapteris to show how coordinates arise when the Pappus and Desargues theoremshold, and why they satisfy the field axioms. In doing so, we will see thatprojective geometry is simpler than algebra in a certain sense, because weuse only five geometric axioms to derive the nine field axioms.ExercisesIn some projective planes, the Desargues theorem is false. Here is one example,which is called the Moulton plane. Its "points" are ordinary points of R2 , togetherwith a point at infinity for each family of parallel "lines." However, the "lines"of the Moulton plane are not all ordinary lines. They include the ordinary linesof negative, horizontal, or vertical slope, but each other "line" is a broken lineconsisting of a half line of slope k > 0 below the x-axis, joined to a half line ofslope k/2 above the x-axis. Figure 6.4 shows some of the "lines." y x O Figure 6.4: Lines of the Moulton plane
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6.2 Coincidences 1216.1.1 Find where the "line" from (0, −1) to (2, 1/2) meets the x-axis.6.1.2 Explain why any two "points" of the Moulton plane lie on a unique "line."6.1.3 Explain why any two "lines" of the Moulton plane meet in a unique "point." (Parallel "lines" have a common "point at infinity" by definition, so do not worry about them.)6.1.4 Give four "points," no three of which lie on the same "line."6.1.5 Thus, the Moulton plane satisfies the three axioms of a projective plane. But it does not satisfy even the little Desargues theorem, as Figure 6.5 shows. Explain. y L x O Figure 6.5: Failure of the little Desargues theorem in the Moulton plane6.2 CoincidencesTwo points A, B always lie on a line. But it is accidental, so to speak, if athird point C lies on the line through A and B. Such an accidental meetingis called a "coincidence" in everyday life, and this is a good name for it inprojective geometry too: coincidence = two incidences together—in thiscase the incidence of A and B with a line, and the incidence of C with thesame line. The theorems of Pappus and Desargues state that certain coincidencesoccur. In fact, they are coincidences of the type just described, in which twopoints lie on a line and a third point lies on the same line. The perspectivepicture of the tiled floor also involves certain coincidences, as becomesclear when we look again at the first few steps in its construction (Figure6.6).
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122 6 Projective planes Draw diagonal of first tile, extended to the horizon Extend diagonal of second tile to the horizon Draw side of second tile, through the new intersection Draw second column of tiles, through the new intersection Figure 6.6: Constructing the tiled floor At this step, a coincidence occurs. Three of the points we have con-structed lie on a straight line, which is shown dashed in Figure 6.7. Figure 6.7: A coincidence in the tiled floor
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6.2 Coincidences 123 This coincidence can be traced to a special case of the little Desarguestheorem, which involves the two shaded triangles shown in Figure 6.8. P Figure 6.8: A little Desargues configuration in the tiled floor This case of little Desargues says that the two dotted lines (diagonalsof "double tiles") meet on the horizon. These lines give us a second littleDesargues configuration, shown in Figure 6.9, from which we concludethat the dashed diagonals also meet on the horizon, as required to explainthe coincidence in Figure 6.7. P Figure 6.9: A second little Desargues configurationExercisesThe occurrence of the little Desargues configuration in the tiled floor may be easierto see if we draw the lines meeting on the horizon as actual parallels. The littleDesargues theorem itself is easier to state in terms of actual parallels (Figure 6.10).6.2.1 Formulate an appropriate statement of the little Desargues theorem when one has parallels instead of lines meeting on L .
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124 6 Projective planes Figure 6.10: The parallel little Desargues configuration6.2.2 Now redraw Figures 6.7, 6.8, and 6.9 so that the lines meeting on L are shown as actual parallels.6.2.3 What is the nature of the "coincidence" in Figure 6.7 now?6.2.4 Find occurrences of the little Desargues configuration in your diagrams. Hence, explain why the "coincidence" in Exercise 6.2.3 follows from your statement of the little Desargues theorem in Exercise 6.2.1. The theorem that proves the coincidence in the drawing of the tiled floor isactually a special case of the little Desargues theorem: the case in which a vertexof one triangle lies on a side of the other. Thus, it is not clear that the coincidenceis false in the Moulton plane, where we know only that the general little Desarguestheorem is false by the exercises in Section 6.1.6.2.5 By placing an x-axis in a suitable position on Figure 6.11, show that the tiled floor coincidence fails in the Moulton plane. Figure 6.11: A coincidence that fails in the Moulton plane
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6.3 Variations on the Desargues theorem 1256.3 Variations on the Desargues theoremIn Section 6.1, we stated the Desargues theorem in the form: If two trian-gles are in perspective from a point, then their three pairs of correspond-ing sides meet on a line. The Desargues theorem is a very flexible theorem,which appears in many forms, and two that we need later are the following.(We need these theorems only as consequences of the Desargues theorem,but they are actually equivalent to it.)Converse Desargues theorem. If corresponding sides of two trianglesmeet on a line, then the two triangles are in perspective from a point. To deduce this result from the Desargues theorem, let ABC and A B Cbe two triangles whose corresponding sides meet on the line L . Let P bethe intersection of AA and BB , so we want to prove that P lies on CC aswell. Suppose that PC meets the line B C at C (Figure 6.12 shows C ,hypothetically, unequal to C ). L Q A A B P B C C C Figure 6.12: The converse Desargues theorem Then the triangles ABC and A B C are in perspective from P and there-fore, by the Desargues theorem, their corresponding sides meet on a line.We already know that AB meets A B on L , and that BC meets B C on L .Hence, AC meets A C on L , necessarily at the point Q where AC meetsL . It follows that QA goes through C . But we also know that QA meetsB C at C . Hence, C = C . Thus, C is indeed on the line PC, so ABC and A B C are in perspectivefrom P, as required.
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126 6 Projective planes The second consequence of the Desargues theorem is called the "scis-sors theorem." I do not know how common this name is, but it is used onp. 69 of the book Fundamentals of Mathematics II. Geometry, edited byBehnke, Bachmann, Fladt, and Kunle. In any case, it is an apt name, asyou will see from Figure 6.13.Scissors theorem. If ABCD and A B C D are quadrilaterals with verticesalternately on two lines, and if AB is parallel to A B , BC to B C , and ADto A D , then also CD is parallel to C D . C A C A E E P B D B D Figure 6.13: The scissors theorem To prove this theorem, let E be the intersection of AD and BC and letE be the intersection of A D and B C , as shown in Figure 6.13. Then thetriangles ABE and A B E have corresponding sides parallel. Hence, theyare in perspective from the intersection P of AA and BB , by the converseDesargues theorem. But then the triangles CDE and C D E are also in perspective from P.Because their sides CE and C E , DE and D E , are parallel by assumption,it follows from the Desargues theorem that CD and C D are also parallel,as required. The scissors theorem just proved says that if the black, gray, and dashedlines in Figure 6.13 are parallel, then so are the dotted lines. What if theblack, gray, and dotted lines are parallel: Are the dashed lines again paral-lel? The answer is yes, and the proof is similar, but with a slightly differentpicture (Figure 6.14).
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6.3 Variations on the Desargues theorem 127 P Figure 6.14: Second case of the scissors theorem We have extended the black and dotted lines until they meet, formingtriangles with their corresponding black, gray, and dotted sides parallel.Then it follows from the converse Desargues theorem that these trianglesare in perspective from P. But then so are the triangles with dashed, black,and dotted sides. Hence their dashed sides are parallel by the Desarguestheorem.Remark. In practice, the scissors theorem is often used in the followingway. We have a pair of scissors ABCD and another figure D A B C F withparallel pairs of black, gray, dashed, and dotted lines as shown in Figure6.15. We want to prove that D = F (so the ends of the gray and dottedlines coincide, and the second figure is also a pair of scissors). C A C A P B D B D F Figure 6.15: Applying the scissors theorem This coincidence happens because the line C D is parallel to CD bythe scissors theorem, so C D is the same line as C F , and hence D = F .
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128 6 Projective planesExercisesBecause the Desargues theorem implies its converse, another way to show that theDesargues theorem fails in the Moulton plane is to show that its converse fails.This plan is easily implemented with the help of Figure 6.16. (Moulton himselfused this figure when he introduced the Moulton plane in 1902.) y P x OFigure 6.16: The converse Desargues theorem fails in the Moulton plane6.3.1 Explain why Figure 6.16 shows the failure of the converse Desargues theo- rem in the Moulton plane.6.3.2 Formulate a converse to the little Desargues theorem, and show that it fol- lows from the little Desargues theorem.6.3.3 Show that the converse little Desargues theorem implies a "little scissors theorem" in which the quadrilaterals have their vertices on parallel lines.6.3.4 Design a figure that directly shows the failure of the little scissors theorem in the Moulton plane.6.4 Projective arithmeticIf we choose any two lines in a projective plane as the x- and y-axes, wecan add and multiply any points on the x-axis by certain constructions. Theconstructions resemble constructions of Euclidean geometry, but they usestraightedge only, so they make sense in projective geometry. To keep themsimple, we use lines we call "parallel," but this merely means lines meetingon a designated "line at infinity." The real difficulty is that the constructionof a + b, for example, is different from the construction of b + a, so it is a"coincidence" if a + b = b + a. Similarly, it is a "coincidence" if ab = ba,or if any other law of algebra holds. Fortunately, we can show that therequired coincidences actually occur, because they are implied by certaingeometric coincidences, namely, the Pappus and Desargues theorems.
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6.4 Projective arithmetic 129AdditionTo construct the sum a + b of points a and b on the x-axis, we take any lineL parallel to the x-axis and construct the lines shown in Figure 6.17: 1. A line from a to the point where L meets the y-axis. 2. A line from b parallel to the y-axis. 3. A parallel to the first line through the intersection of the second lineand L . y L x O a b a+b Figure 6.17: Construction of the sum This construction is similar in spirit to the construction of the sum inSection 1.1. There we "copied a length" by moving it from one place toanother by a compass. The spirit of the compass remains in the projec-tive construction: the black line and the gray line form a "compass" that"copies" the length Oa to the point b. We need the line L to construct a + b, but we get the same point a + bfrom any other line L parallel to the x-axis. This coincidence followsfrom the little Desargues theorem as shown in Figure 6.18. y L L x O a b a+b Figure 6.18: Why the sum is independent of the choice of L The black sides of the solid triangles are parallel by construction, asare the gray sides, one of which ends at the point a + b constructed fromL . Then it follows from the little Desargues theorem that the dotted sidesare also parallel, and one of them ends at the point a + b constructed fromL . Hence, the same point a + b is constructed from both L and L .
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130 6 Projective planesMultiplicationTo construct the product ab of two points a and b on the x-axis, we firstneed to choose a point = O on the x-axis to be 1. We also choose a point= O to be the 1 on the y-axis. The point ab is constructed by drawing theblack and gray lines from 1 and a on the x-axis to 1 on the y-axis, and thendrawing their parallels as shown in Figure 6.19. This construction is theprojective version of "multiplication by a" done in Section 1.4. y 1 O x 1 a b ab Figure 6.19: Construction of the product ab Choosing the 1 on the x-axis means choosing a unit of length on the x-axis, so the position of ab definitely depends on it. For example, ab = b ifa = 1 but ab = b if a = 1. However, the position of ab does not depend onthe choice of 1 on the y-axis, as the scissors theorem shows (Figure 6.20). y 1 1 O x 1 a b ab Figure 6.20: Why the product is independent of the 1 on the y-axis If we choose 1 instead of 1 to construct ab, the path from b to abfollows the dashed and the dotted line instead of the black and the grayline. But it ends in the same place, because the dotted line to ab is parallelto the dotted line to a, by the scissors theorem.
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6.4 Projective arithmetic 131Interchangeability of the axesOnce we have chosen points called 1 on both the x- and y-axes, it is naturalto let each point a on the x-axis correspond to the point on the y-axis ob-tained by drawing the line through a parallel to the line through the points1 on both axes (Figure 6.21). y a 1 O x 1 a Figure 6.21: Corresponding points It is also natural to define sum and product on the y-axis by construc-tions like those on the x-axis. But then the question arises: Do the y-axissum and product correspond to the x-axis sum and product? To show that sums correspond, we need to construct a + b on the x-axis, and then show that the corresponding point a + b on the y-axis is they-axis sum of the y-axis a and b. Figure 6.22 shows how this constructionis done. y a+b M b a L x O a b a+b Figure 6.22: Corresponding sums We construct a + b on the x-axis using the line L through a on they-axis. That is, draw the line M through b on the x-axis and parallel tothe y-axis, and then draw the line (dashed) from the intersection of M andL parallel to the line from a on the x-axis to the intersection of L and they-axis. This dashed line meets the x-axis at a + b, and (because it is parallelto the line from a to a) it also meets the y-axis at a + b.
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132 6 Projective planes Now we construct a + b on the y-axis using the line M (as on the x-axis, the sum does not depend on line chosen, as long as it is parallel to they-axis). That is, draw the line through b on the y-axis parallel to the x-axis,and then draw the line (dotted) parallel to the line from a on the y-axis to bon the x-axis (because this b is the intersection of the x-axis with M ). But then, as is clear from Figure 6.22, we have a Pappus configurationof gray, dashed, and dotted lines between the y-axis and M , hence thedotted line (leading to the y-axis sum) and the dashed line (leading to thepoint corresponding to the x-axis sum) end at the same point, as required. To show that products correspond, we use the scissors theorem fromSection 6.3. Figure 6.23 shows the corresponding points 1, a, b, and ab onboth axes. The gray lines construct ab on the x-axis, and the dotted linesconstruct ab on the y-axis. y ab b a 1 x O 1 a b ab Figure 6.23: Construction of the product ab on both axes It follows from the scissors theorem that the dotted line on the rightends at the same point as the black line from ab on the x-axis parallel to thelines connecting the corresponding points a and the corresponding pointsb. Hence, the product of a and b on the y-axis (at the end of the dotted line)is indeed the point corresponding to ab on the x-axis.ExercisesThese definitions of sum and product lead immediately to some of the simpler lawsof algebra, namely, those concerned with the behavior of 0 and 1. The completelist of algebraic laws is given in the Section 6.5.
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6.5 The field axioms 1336.4.1 Show that a + O = a for any a, so O functions as the zero on the x-axis.6.4.2 Show that, for any a, there is a point b that serves as −a; that is, a + b = O. (Warning: Do not be tempted to use measurement to find b. Work backward from O = a + b, reversing the construction of the sum.)6.4.3 Show that a1 = a for any a.6.4.4 Show that, for any a = O, there is a b that serves as a−1 ; that is, ab = 1. (Again, do the construction of the product in reverse.)You will notice that we have not attempted to define sums or products involvingthe point at infinity ∞ on the x-axis.6.4.5 What happens when we try to construct a + ∞?6.4.6 What is −∞?You should find that the answers to Exercises 6.4.5 and 6.4.6 are incompatiblewith ordinary arithmetic. This is why we do not include ∞ among the points weadd and multiply.6.5 The field axiomsIn calculating with numbers, and particularly in calculating with symbols("algebra"), we assume several things: that there are particular numbers 0and 1; that each number a has a additive inverse, −a; that each numbera = 0 has a reciprocal, a−1 ; and that the following field axioms hold. (Weintroduced these in the discussion of vector spaces in Section 4.8.) a + (−a) = 0, aa−1 = 1 (inverse laws) a(b + c) = ab + ac (distributive law) We generally use these laws unconsciously. They are used so often,and they are so obviously true of numbers, that we do not notice them. Butfor the projective sum and product of points, they are not obviously true.It is not even clear that a + b = b + a, because the construction of a + bis different from the construction of b + a. It is truly a coincidence thata + b = b + a in projective geometry, the result of a geometric coincidenceof the type discussed in Section 6.2.
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134 6 Projective planes In this chapter, we show that just two coincidences—the theorems ofPappus and Desargues—imply all nine field axioms. In fact, it is knownthat Pappus alone is sufficient, because it implies Desargues. We do notprove this fact here, partly because it is difficult, and partly because theDesargues theorem itself is interesting: It implies all the field axioms ex-cept ab = ba. Thus, the theorems of Pappus and Desargues have algebraiccontent that can be measured accurately by the field axioms they imply.Pappus implies all nine, and Desargues only eight—all but ab = ba.Proof of the commutative laws y 1 x O 1 a b ba Figure 6.24: Construction of the product ba We begin with the law ab = ba, which is the most important conse-quence of the Pappus theorem. Figure 6.24 shows the construction of bafrom a and b, lying at the end of the second dotted line. It is different fromthe construction of ab, and Figure 6.25 shows the constructions of both aband ba on the same diagram. y 1 x O 1 a b ab = ba Figure 6.25: Construction of both ab and ba Then ab = ba because the gray and dotted lines end at the same place,by the Pappus theorem. The Pappus configuration in Figure 6.25 consistsof all the lines except the line joining 1 on the x-axis to 1 on the y-axis.
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6.5 The field axioms 135 There is a similar proof that a + b = b + a. Remember from Section6.4 that a + b is the result of attaching the segment Oa at b. Thus, b + a isthe result of attaching Ob at a, which is different from the construction ofa + b. Looking at both constructions together (Figure 6.26), we see that thegray line leads to a + b and the dotted line leads to b + a. However, both ofthese lines end at the same point, thanks to the Pappus theorem. y L x O a b a+b = b+a Figure 6.26: Construction of both a + b and b + aExercises6.5.1 Look back to the vector proof of the Pappus theorem in Section 4.2, and point out where it uses the assumption ab = ba. The Pappus configuration that proves a + b = b + a is actually a special one,because the vertices of the hexagon lie on parallel lines. The same special config-uration also occurs in the proof in Section 6.4 that sums correspond on the x- andy-axes. The special configuration corresponds to a special Pappus theorem, some-times called the "little Pappus theorem." It is usually stated without mention ofparallel lines; in which case, one has to talk about opposite sides of the hexagonmeeting on a line L .6.5.2 Given that the assumptions of the little Pappus theorem are a hexagon with vertices on two lines that meet at a point P, and two pairs of opposite sides meeting on a line L that goes through P, what is the conclusion? It is known that the little Desargues theorem implies the little Pappus theorem;a proof is in Fundamentals of Mathematics, II by Behnke et al., p. 70. Thus, theresults deduced here from the little Pappus theorem can also be deduced from thelittle Desargues theorem (although generally not as easily).
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6.6 The associative laws 137 One can clearly see two pairs of scissors, each consisting of a dashedline, a dotted line, a black line, and a gray line. In the scissors on the right,the gray line ends at a + (b + c) and the dotted line at (a + b) + c. But theends of these lines coincide, by the scissors theorem. Hence a + (b + c) =(a + b) + c. Because the scissors in this proof lie between parallel lines, we needonly the little scissors theorem (and hence only the little Desargues theo-rem, by the remark in the previous exercise set). Next we consider the associative law of multiplication, a(bc) = (ab)c.The diagram (Figure 6.30) is similar, except that the pairs of scissors liebetween nonparallel lines (the x- and y-axes), so now we need the fullDesargues theorem. y xO 1 a b ab c bc a(bc) = (ab)c Figure 6.30: Why a(bc) and (ab)c coincide The gray line ends at a(bc) and the dotted line ends at (ab)c. But theends of these lines coincide, by the scissors theorem, so a(bc) = (ab)c.ExercisesThere is an algebraic system that satisfies all of the field axioms except the com-mutative law of multiplication. It is called the quaternions and is denoted by H,after Sir William Rowan Hamilton, who discovered the quaternions in 1843. In 1845, Arthur Cayley showed that the quaternions could be defined as 2 × 2complex matrices of the form a + ib c + id q= . −c + id a − ibMost of their properties follow from general properties of matrices. In fact, allthe laws of algebra are immediate except the existence of q−1 and commutativemultiplication.
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138 6 Projective planes6.6.1 Show that q has determinant a2 + b2 + c2 + d 2 and hence that q−1 exists for any nonzero quaternion q.6.6.2 Find specific quaternions s and t such that st = ts. We can write any quaternion as q = a1 + bi + cj + dk, where 1 0 i 0 0 1 0 i 1= , i= , j= , k= . 0 1 0 −i −1 0 i 06.6.3 Verify that i2 = j2 = k2 = ijk = −1 (This is Hamilton's description of the quaternions). It is possible to define the quaternion projective plane HP2 using quaternioncoordinates. HP2 satisfies the Desargues theorem because it is possible to do thenecessary calculations without using commutative multiplication. But it does notsatisfy the Pappus theorem, because this implies commutative multiplication forH. HP2 is therefore a non-Pappian plane—probably the most natural example.6.7 The distributive lawTo prove the distributive law a(b + c) = ab + ac, we take advantage of theability to do addition and multiplication on both axes. We construct b + cfrom b and c on the x-axis, and then map b, c, and b + c to ab, ac, anda(b + c) on the y-axis by lines parallel to the line from 1 on the x-axis toa on the y-axis. Then we use addition on the y-axis to construct ab + acthere, and finally, use the Pappus theorem to show that ab + ac and a(b + c)are the same point. Here are the details. First, observe that we can map any b on the x-axis to ab on the y-axisby sending it along a line parallel to the line from 1 on the x-axis to a onthe y-axis (Figure 6.31). y ab b a 1 O x 1 b Figure 6.31: Multiplication via parallels
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6.7 The distributive law 139 This is the same as constructing ab from a and b on the y-axis, becausethe line from b to b is parallel to the line from 1 to 1, as required by thedefinition of multiplication. Next we add b and c on the x-axis, using a special choice of line L :the parallel through ab on the y-axis. We also connect b, c, and b + c,respectively, to ab, ac, and a(b + c) on the y-axis by parallel lines, showndashed in Figure 6.32. The line through c that constructs b + c, namely theparallel M to the y-axis, is used in turn to add ab and ac on the y-axis. y a(b + c) M ab + ac ac ab L O x b c b+c Figure 6.32: Why a(b + c) = ab + ac This figure has the same structure as Figure 6.22; only the labels havechanged. Now the dashed line ends at a(b + c), and the dotted line ends atab + ac. But again the endpoints coincide by the theorem of Pappus, andso a(b + c) = ab + ac.ExercisesWe need not prove the other distributive law, (b + c)a = ba + bc, because we areassuming Pappus, so multiplication is commutative.6.7.1 Explain in this case why a(b + c) = ab + ac implies (b + c)a = ba + bc.However, in some important systems with noncommutative multiplication, bothdistributive laws remain valid.6.7.2 Explain why both distributive laws are valid for the quaternions.6.7.3 More generally, show that both distributive laws are valid for matrices.
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140 6 Projective planes6.8 DiscussionThe idea of developing projective geometry without the use of numberscomes from the German mathematician Christian von Staudt in 1847. Hiscompatriots Hermann Wiener and David Hilbert took the idea further inthe 1890s, and it reached a high point with the publication of Hilbert'sbook, Grundlagen der Geometrie (Foundations of geometry), in 1899. Itwas Hilbert who first established a clear correlation between geometric andalgebraic structure: • Pappus with commutative multiplication, • Desargues with associative multiplication.The correlation is significant because some important algebraic systemssatisfy all the field axioms except commutative multiplication. The best-known example is the quaternions, which has been known since 1843, but,for some reason, Hilbert did not mention it. To construct a non-Pappianplane, he created a rather artificial noncommutative coordinate system. It is perhaps a lucky accident of history that Hilbert discovered the roleof the Desargues theorem at all. He was forced to use it because, in 1899,it was still not known that Pappus implies Desargues. This implicationwas first proved by Gerhard Hessenberg in 1904. Even then the proof wasfaulty, and the mistake was not corrected until years later. The whole circle of ideas was neatly tied up by yet another Germanmathematician, Ruth Moufang, in 1930. She found that the little Desarguestheorem also has algebraic significance. In a projective plane satisfyingthe little Desargues theorem, with addition and multiplication defined asin Section 6.4, one can prove all the field axioms except commutativityand associativity. One can even prove a partial associativity law calledcancellation or alternativity: a−1 (ab) = b = (ba)a−1 (alternativity) The commutative, associative, and alternative laws are beautifully ex-emplified by the possible multiplication operations that can be defined"reasonably" on the Euclidean spaces Rn . ("Reasonably" means respectingat least the dimension of the space. For more on the problem of general-izing the idea of number to n dimensions, see the book Numbers by D.Ebbinghaus et al.)
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6.8 Discussion 141 • Commutative multiplication is possible only on R1 and R2 , and it yields the number systems R and C. • Associative, but noncommutative, multiplication is possible only on R4 , and it yields the quaternions H. • Alternative, but nonassociative, multiplication is possible only on R8 , and it yields a system called the octonions O. The octonions were discovered by a friend of Hamilton called John Graves, in 1843, and they were discovered independently by Cayley in 1845. Ruth Moufang was the first to recognize the importance of quaternionsand octonions in projective geometry. She pointed out the quaternion pro-jective plane, as a natural example of a non-Pappian plane, and was thefirst to discuss the octonion projective plane OP2 . OP2 is the most naturalexample of a plane that satisfies little Desargues but not Desargues. In Section 5.4, we sketched the construction of the real projective space 3RP by means of homogeneous coordinates. This idea is easily general-ized to obtain the n-dimensional real projective space RP n , and one canobtain CPn and HPn in precisely the same way. Surprisingly, the idea doesnot work for the octonions. The only octonion projective spaces are theoctonion projective line OP1 = O ∪ {∞} and the octonion projective planeOP2 discovered by Moufang. The reason for the nonexistence of OP3 is extremely interesting andhas to do with the nature of the Desargues theorem in three dimensions.Remember that the Desargues theorem assumes a pair of triangles in per-spective and concludes that the intersections of corresponding sides lie on aline. We know (because of the example of the Moulton plane) that the con-clusion does not follow by basic incidence properties of points and lines.But if the triangles lie in three-dimensional space, the conclusion followsby basic incidence properties of points, lines, and planes. The spatial Desargues theorem is clear from a picture that emphasizesthe placement of the triangles in three dimensions, such as Figure 6.33.The planes containing the two triangles meet in a line L , where the pairsof corresponding sides necessarily meet also. The argument is a little trickier if the two triangles lie in the sameplane. But, provided the plane lies in a projective space, it can be carriedout (one shows that the planar configuration is a "shadow" of a spatialconfiguration).
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142 6 Projective planes L Figure 6.33: The spatial Desargues configuration Thus, the Desargues theorem holds in any projective space of at leastthree dimensions. This is why OP3 cannot exist. If it did, the Desarguestheorem would hold in it, and we could then show that O is associative—which it is not. Q. E. D.
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7TransformationsP REVIEW In this book, we have seen at least three geometries: Euclidean, vector, and projective. In Euclidean geometry, the basic concept is length, but angle and straightness derive from it. In vector geome- try, the basic concepts are vector sums and scalar multiples, but from these we derive others, such as midpoints of line segments. Finally, in projective geometry, the basic concept is straightness. Length and angle have no meaning, but a certain combination of lengths—the cross-ratio—is meaningful because it is unchanged by projection. We found the cross-ratio as an invariant of projective transforma- tions. The concept of length was not discovered this way, but never- theless, it is an invariant of certain transformations. It is an invariant of the isometries, for the simple reason that isometries are defined to be transformations of the plane that preserve length. These examples are two among many that suggest geometry is the study of invariants of groups of transformations. This definition of geometry was first proposed by the German mathematician Felix Klein in 1872. Klein's concept of geometry is perhaps still not broad enough, but it does cover the geometry in this book. In this chapter we look again at Euclidean, vector, and projective geometry from Klein's viewpoint, first explaining precisely what "transformation" and "group" mean. It turns out that the appro- priate transformations for projective geometry are linear, and that linear transformations also play an important role in Euclidean and vector geometry. Linear transformations also pave the way for hyperbolic geometry, a new "non-Euclidean" geometry that we study in Chapter 8. 143
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144 7 Transformations7.1 The group of isometries of the planeIn Chapter 3, we took up Euclid's idea of "moving" geometric figures,and we made it precise in the concept of an isometry of the plane R 2 . Anisometry is defined to be a function f : R2 → R2 that preserves distance;that is, | f (P1 ) f (P2 )| = |P1 P2 | for any points P1 , P2 , ∈ R2 ,where |P1 P2 | = (x2 − x1 )2 + (y2 − y1 )2 denotes the distance between thepoints P1 = (x1 , y1 ) and P2 = (x2 , y2 ). It follows immediately from this definition that, when f and g areisometries, so is their composite or product f g (the result of applying g,then f ). Namely, | f (g(P1 )) f (g(P2 ))| = |g(P1 )g(P2 )| because f is an isometry = |P1 P2 | because g is an isometry. What is less obvious is that any isometry f has an inverse, f −1 , whichis also an isometry. To prove this fact, we use the result from Section 3.7that any isometry of R2 is the product of one, two, or three reflections. First suppose that f = r1 r2 r3 , where r1 , r2 , and r3 are reflections. Then,because a reflection composed with itself is the identity function, we find f r3 r2 r1 = r 1 r2 r3 r3 r2 r1 = r 1 r2 r2 r1 because r3 r3 is the identity function = r 1 r1 because r2 r2 is the identity function = identity function,and therefore, r3 r2 r1 = f −1 . This calculation also shows that f −1 is anisometry, because it is a product of reflections. The proof is similar (butshorter) when f is the product of one or two reflections. These properties of isometries are characteristic of a group of trans-formations. A transformation of a set S is a function from S to S, and acollection G of transformations forms a group if it has the two properties: • If f and g are in G, then so is f g. • If f is in G, then so is its inverse, f −1 .It follows that G includes the identity function f f −1 , which can be writtenas 1. This notation is natural when we write the composite of two functionsf , g as the "product" f g.
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7.1 The group of isometries of the plane 145What is a geometry?In 1872, the German mathematician Felix Klein pointed out that variouskinds of geometry go with various groups of transformations. For exam-ple, the Euclidean geometry of R2 goes with the group of isometries ofR2 . The meaningful concepts of the geometry correspond to propertiesthat are left unchanged by transformations in the group. Isometries of R 2leave distance or length unchanged, so distance is a meaningful conceptof Euclidean geometry. It is called an invariant of the isometry group ofR2 . This invariance is no surprise, because isometries are defined as thetransformations that preserve distance. However, it is interesting that other things are also invariant underisometries, such as straightness of lines and circularity of circles. It is notentirely obvious that a length-preserving transformation preserves straight-ness, but it can be proved by showing first that any reflection preservesstraightness, and then using the theorem of Section 3.7, that any isometryis a product of reflections. An example of a concept without meaning in Euclidean geometry is"being vertical," because a vertical line can be transformed to a nonverticalline by an isometry (for example, by a rotation). We can do without theconcept of "vertical" in geometry because we have the concept of "beingrelatively vertical," that is, perpendicular. A concept that is harder to dowithout is "clockwise order on the circle." This concept has no meaning inEuclidean geometry because the points A = (−1, 0), B = (0, 1), C = (1, 0),and D = (0, −1) have clockwise order on the circle, but their respectivereflections in the x-axis do not. However, we can define oriented Euclidean geometry, in which clock-wise order is meaningful, by using a smaller group of transformations.Instead of the group Isom(R2 ) of all isometries of R2 , take Isom+ (R2 ),each member of which is the product of an even number of reflections.Isom+ (R2 ) is a group because • If f and g are products of an even number of reflections, so is f g. • If f = r1 r2 · · · r2n is the product of an even number of reflections, then so is f −1 . In fact, f −1 = r2n · · · r2 r1 , by the argument used above to invert the product of any number of reflections.And any transformation in Isom+ (R2 ) preserves clockwise order becauseany product of two reflections does: The first reflection reverses the order,and then the second restores it.
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146 7 Transformations This example shows how a geometry of R2 with more concepts comesfrom a group with fewer transformations. In R3 , one has the concept of"handedness"—which distinguishes the right hand from the left—which isnot preserved by all isometries of R3 . However, it is preserved by productsof an even number of reflections in planes. Thus, the geometry of Isom(R 3 )does not have the concept of handedness, but the geometry of Isom + (R3 )does. Restricting the transformations to those that preserve orientation—asit is generally called—is a common tactic in geometry. However, the main goal of this book is to show that there are inter-esting geometries with fewer concepts than Euclidean geometry. Thesegeometries are obtained by taking larger groups of transformations, whichwe study in the remainder of this chapter.Exercises7.1.1 Use the results of Section 3.7 to show that each member of Isom+ (R2 ) is either a translation or a rotation.7.1.2 Why does an isometry map any circle to a circle? We took care to write the inverse of the isometry r1 r2 r3 as r3 r2 r1 because onlythis ordering of terms will always give the correct result.7.1.3 Give an example of two reflections r1 and r2 such that r1 r2 = r2 r1 .7.2 Vector transformationsIn Chapter 4, we viewed the plane R2 as a real vector space, by consideringits points to be vectors that can be added and multiplied by scalars. Ifu = (u1 , u2 ) and v = (v1 , v2 ), we defined the sum of u and v by u + v = (u1 + v1 , u2 + v2 )and the scalar multiple au of u by a real number a by au = (au1 , au2 ).A transformation f of R2 preserves these two operations on vectors if f (u + v) = f (u) + f (v) and f (au) = a f (u), (*)and such a transformation is called linear.
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7.2 Vector transformations 147 One reason for calling the transformation "linear" is that it preservesstraightness of lines. A straight line is a set of points of the form a + tb,where a and b are constant vectors and t runs through the real numbers.Figure 7.1 shows the role of the vectors a and b: a is one point on the line,and b gives the direction of the line. a + tb a b 0 Figure 7.1: Points on a line If we apply a linear transformation f to this set of points, we get the setof points f (a + tb). And by the linearity conditions (*), this set consists ofpoints of the form f (a) + t f (b), which is another straight line: f (a) is onepoint on it, and f (b) gives the direction of the line. It follows from this calculation that, if L1 and L2 are two lines withdirection b and f is a linear transformation, then f (L1 ) and f (L2 ) aretwo lines with direction f (b). In other words, a linear transformation alsopreserves parallels.Matrix representationAnother consequence of the linearity conditions (*) is that each lineartransformation f of R2 can be specified by four real numbers a, b, c, d:any point (x, y) of R2 is sent by f to the point (ax + by, cx + dy). Certainly, there are numbers a, b, c, d that give the particular values f ((1, 0)) = (a, c) and f ((0, 1)) = (b, d).But the value of f ((x, y)) follows from these particular values by linearity: (x, y) = x(1, 0) + y(0, 1),
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7.2 Vector transformations 149Examples of linear transformationsAny 2 × 2 real matrix M represents a linear transformation, because it fol-lows from the definition of matrix multiplication that M(u + v) = Mu + Mv and M(au) = aMufor any vectors u and v (written in column form). Among the invertible linear transformations are certain isometries, suchas rotations and reflections in lines through the origin. Recall from Section3.6 that a rotation is a transformation of the form c −s(x, y) → (cx−sy, sx+cy), hence given by the matrix R = . s cThe numbers c and s satisfy c2 + s2 = 1 (they are actually cos θ and sin θ ,where θ is the angle of rotation); hence, c s det R = 1 and therefore R−1 = . −s cLikewise, reflection in the x-axis is the linear transformation 1 0 (x, y) → (x, −y), given by the matrix X = . 0 −1 We can reflect R2 in any line L through O with the help of the rotationR that sends the x-axis to L : • First apply R−1 to send L to the x-axis. • Then carry out the reflection by applying X. • Then send the line of reflection back to L by applying R.In other words, to reflect the point u in L , we find the value of RXR −1 u.Hence, reflection in L is represented by the matrix RXR−1 . Thus, the linear transformations of R2 include the isometries that areproducts of reflection on lines through O. But this is not all. An exampleof a linear transformation that is not an isometry is the stretch by factor kin the x-direction, k 0 (x, y) → (kx, y), given by the matrix S = . 0 1It can be shown that any invertible linear transformation of R 2 is a productof reflections in lines through O and stretches in the x-direction (by factorsk = 0).
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150 7 TransformationsAffine transformationsLinear transformations preserve geometrically natural properties such asstraightness and parallelism, but they also preserve the origin, which reallyis not geometrically different from any other point. To abolish the specialposition of the origin, we allow linear transformations to be composed withtranslations, obtaining what are called affine transformations. If we writean arbitrary linear transformation of (column) vectors u in the form f (u) = Mu, where M is an invertible matrix,then an arbitrary affine transformation takes the form g(u) = Mu + c, where c is a constant vector.Because translations preserve everything except position, affine transfor-mations preserve everything that linear transformations do, except position.In effect, they allow any point to become the origin. The geometry of affine transformations is called affine geometry. Itstheorems include those in the first few sections of Chapter 4, such as thefact that diagonals of a parallelogram bisect each other, and the concur-rence of the medians of a triangle. These theorems belong to affine geom-etry because they are concerned only with quantities, such as the midpointof a line segment, that are preserved by affine transformations.Exercises a b7.2.1 Compute MM −1 for the general 2 × 2 matrix M = , and verify c d 1 0 that it equals the identity matrix . 0 17.2.2 Write down the matrix for clockwise rotation through angle π /4.7.2.3 Write down the matrix for reflection in the line y = x, and check that it equals RXR−1 , where R is the matrix for rotation through π /4 found in Exercise 7.2.2. k 07.2.4 The matrix M = represents a dilation of the plane by factor k 0 k (also known as a similarity transformation). Explain geometrically why this transformation is a product of reflections in lines through O and of stretches by factor k in the x-direction.
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7.3 Transformations of the projective line 1517.2.5 Show that the midpoint of any line segment is preserved by linear transfor- mations and hence by affine transformations.7.2.6 More generally, show that the ratio of lengths of any two segments of the same line is preserved by affine transformations.7.3 Transformations of the projective lineLooking back at our approach to the projective line in Chapter 5, we seethat we were following Klein's idea. First we found the transformations ofthe projective line, and then a quantity that they leave invariant—the cross-ratio. In this section we look more closely at projective transformations,and show that they too can be viewed as linear transformations. In Sections 5.5 and 5.6, we showed that the transformations of the pro-jective line R ∪ {∞} are precisely the linear fractional functions ax + b f (x) = where ad − bc = 0. cx + dWe did this by showing: • Any linear fractional function is a product of functions sending x to x + l, x to kx, and x to 1/x, and that each of the latter functions can be realized by projection of one line onto another. • Conversely, any projection of one line onto another is represented by a linear fractional function of x, with the understanding that 1/0 = ∞ and 1/∞ = 0.In the exercises to Section 5.6 you were asked to show that linear fractional a bfunctions f (x) = ax+b behave like the matrices M = cx+d , by show- c ding that composition of the functions corresponds to multiplication of thecorresponding matrices. In this section, we explain the connection by rep-resenting mappings of the projective line directly by linear transformationsof the plane. We begin by defining the projective line in the manner of Section 5.4.There we defined the real projective plane RP2 . Its "points" are the linesthrough O in R3 , and its "lines" are the planes through O. Here we needonly one projective line, which we can take to be the real projective lineRP1 , whose "points" are the lines through O in the ordinary plane R 2 .
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152 7 Transformations We label each line through O, if it meets the line y = 1, by the x-coordinate s of the point of intersection (Figure 7.2). The single line thatdoes not meet y = 1, namely, the x-axis, naturally gets the label ∞. y s 1 s O xFigure 7.2: Correspondence between lines through O and points on y = 1 Figure 7.3 shows some lines through O with their labels. 0 −1 1 −2 2 −3 3 ∞ Figure 7.3: Labeling of lines through O Now a projective map of the ordinary line y = 1 sends the point with x-coordinate s to the point with x-coordinate f (s), for some linear fractionalfunction as + b f (s) = . cs + d
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7.3 Transformations of the projective line 153This function corresponds to a map of the plane R2 sending the line withlabel s to the line with label as+b . Bearing in mind that the label represents cs+dthe "reciprocal slope" ("run over rise") of the line, we find that one suchmap is the linear map of the plane given by the matrix a b M= . c d To see why, we apply this linear map to a typical point (sx, x) on theline with label s. We find where M sends it by writing (sx, x) as a columnvector and multiplying it on the left by M: a b sx asx + bx = . c d x csx + dx asx + bxThe column vector represents the point (asx + bx, csx + dx), csx + dxwhich lies on the line with reciprocal slope asx + bx as + b = . csx + dx cs + dThe latter line is therefore independent of x and it is the line with labelas+b as+bcs+d . Thus, M maps the line with label s to the line with label cs+d , asrequired. Because a "point" of RP1 is a whole line through O, we care only thatthe matrix a b M= c dsends the line with label s to the line with label as+b . It does not matter cs+dhow M moves individual points on the line. It is not generally the case thatM sends the particular point (s, 1) on the line with label s to the particularpoint ( as+b , 1) on the line with label as+b . Indeed, it is clearly impossible cs+d cs+dwhen M represents the map s → 1/s of RP1 . A matrix M sends each pointof R2 to another point of R2 . Hence it cannot send (0, 1) to (1/0, 1) =(∞, 1), because the latter is not a point of R2 . However, M can send theline with label 0 (the y-axis) to the line with label ∞ (the x-axis), and thisis exactly what happens (see Exercises 7.3.1 and 7.3.2).
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154 7 Transformations It should also be pointed out that the representation of linear fractionalfunctions by matrices is not unique. The fraction as + b kas + kb is equal to for any k = 0. cs + d kcs + kd as+bHence, the function f (s) = cs+d is represented not only by a b ka kb M= but also by kM = for any k = 0. c d kc kdThe linear transformations kM combine the transformation M with dilationby factor k, so they are all different. Thus, the same linear fractional func-tion f is represented by infinitely many different transformations kM of R 2 .The message, again, is that we care only that each of these transformationssends the line with label s to the line with label f (s).Exercises7.3.1 Write down a matrix M that represents the map s → 1/s of RP1 .7.3.2 Verify that your matrix M in Exercise 7.3.1 maps the y-axis onto the x-axis.7.3.3 Sketch a picture of the lines with labels 1/2, −1/2, 1/3, and −1/3. The nonuniqueness of the matrix M corresponding to the linear fractionalfunction f raises the question: Is there a natural way to choose one matrix foreach linear fractional function? Actually, no, but there is a natural way to choosetwo matrices.7.3.4 Given that a b M= and ad − bc = 0, c d show that the determinant of kM has absolute value 1 for exactly two of the matrices kM, where k = 0.7.4 Spherical geometryThe unit sphere in R3 consists of all points at unit distance from O, that is,all points (x, y, z) satisfying the equation x2 + y2 + z2 = 1.
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7.4 Spherical geometry 155This surface is also called the 2-sphere, or S2 , because its points can bedescribed by two coordinates—latitude and longitude, for example. Itsgeometry is essentially two-dimensional, like that of the Euclidean planeR2 or the real projective plane RP2 , and indeed the fundamental objects ofspherical geometry are "points" (ordinary points on the sphere) and "lines"(great circles on the sphere). However, like the projective plane RP2 , the sphere S2 is best under-stood via properties of the three-dimensional space R3 . In particular, the"lines" on S2 are the intersections of S2 with planes through O in R3 —thegreat circles—and the isometries of S2 are precisely the isometries of R3that leave O fixed. By definition, an isometry f of R3 preserves distance. Hence, if fleaves O fixed, it sends each point at distance 1 from O to another point atdistance 1 from O. In other words, an isometry f of R3 that fixes O alsomaps S2 into itself. The restriction of f to S2 is therefore an isometry ofS2 , because f preserves distances on S2 as it does everywhere else. Thisstatement is true whether one uses the straight-line distance between pointsof S2 or, as is more natural, the great-circle distance along the curved sur-face of S2 (Figure 7.4). The isometries of S2 are the maps of S2 into itselfthat preserve great circle distance, and we will see next why they are allrestrictions of isometries of R3 . O 1 Q θ P Figure 7.4: Great-circle distance
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156 7 TransformationsThe isometries of S2The simplest isometries of R3 that fix O are reflections in planes throughO. The corresponding isometries of S2 are the reflections in great circles.Two planes P1 and P2 meet in a line L through O, and the product ofreflections in P1 and P2 is a rotation about L (through twice the anglebetween P1 and P2 ). This situation is completely analogous to that inR2 , where the product of reflections through O is a rotation (through twicethe angle between the lines). Finally, there are products of reflections in three planes that are differ-ent from products of reflections in one or two planes. One such isome-try is the antipodal map sending each point (x, y, z) to its antipodal point(−x, −y, −z). This map is the product of • reflection in the (y, z)-plane, which sends (x, y, z) to (−x, y, z), • reflection in the (z, x)-plane, which sends (x, y, z) to (x, −y, z), • reflection in the (x, y)-plane, which sends (x, y, z) to (x, y, −z). As in R2 , there is a "three reflections theorem" that any isometry ofS2 is the product of one, two, or three reflections. The proof is similar tothe proof for R2 in Sections 3.3 and 3.7 (see the exercises below). Thisthree reflections theorem shows why all isometries of S2 are restrictions ofisometries of R3 , namely, because this is true of reflections in great circles.ExercisesThe proof of the three reflections theorem begins, as it did for R2 , by consideringthe equidistant set of two points.7.4.1 Show that the equidistant set of two points in R3 is a plane. Show also that the plane passes through O if the two points are both at distance 1 from O.7.4.2 Deduce from Exercise 7.4.1 that the equidistant set of two points on S2 is a "line" (great circle) on S2 .Next, we establish that there is a unique point on S2 at given distances from threepoints not in a "line."7.4.3 Suppose that two points P, Q ∈ S2 have the same distances from three points A, B,C ∈ S2 not in a "line." Deduce from Exercise 7.4.2 that P = Q.7.4.4 Deduce from Exercise 7.4.3 that an isometry of S2 is determined by the images of three points A, B,C not in a "line."
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7.5 The rotation group of the sphere 157Thus, it remains to show the following. Any three points A, B,C ∈ S2 not in a"line" can be mapped to any other three points A , B ,C ∈ S2 , which are separatedby the same respective distances, by one, two, or three reflections.7.4.5 Complete this proof of the three reflections theorem by imitating the argu- ment in Section 3.7.7.5 The rotation group of the sphereThe group Isom(S2 ) of all isometries of S2 has a subgroup Isom+ (S2 ) con-sisting of the isometries that are products of an even number of reflections.Like Isom+ (R2 ), this is the "orientation-preserving" subgroup. But, unlikeIsom+ (R2 ), Isom+ (S2 ) includes no "translations"—only rotations. We al-ready know that the product of two reflections of S2 is a rotation. Hence,to show that the product of any even number of reflections is a rotation, itremains to show that the product of any two rotations of S2 is a rotation. Suppose that the two rotations of S2 are • a rotation through angle θ about point P (that is, a rotation with axis through P and its antipodal point −P), • a rotation through angle ϕ about point Q.We have established that a rotation through θ about P is the product ofreflections in "lines" (great circles) through P. Moreover, they can be any"lines" L and M through P as long as the angle between L and Mis θ /2. In particular, we can take the line M to go through P and Q.Similarly, a rotation through ϕ about Q is the product of reflections in anylines through Q meeting at angle ϕ /2, so we can take the first "line" tobe M . The second "line" of reflection through Q is then the "line" N atangle ϕ /2 from M (Figure 7.5). If rL , rM , rN denote the reflections in L , M , N , respectively, then rotation through θ about P = r M rL , rotation through ϕ about Q = r N rM .(Bear in mind that products of transformations are read from right to left,as this is the order in which functions are applied.) Hence, the product ofthese rotations is rN rM rM rL = rN rL , because rM rM is the identity.
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158 7 Transformations L P R θ /2 M N ϕ /2 Q Figure 7.5: Reflection "lines" on the sphereAnd it is clear from Figure 7.5 that r N rL is a rotation (about the point Rwhere L meets N ).Some special rotationsBefore trying to obtain an overview of the rotation group of the sphere, itis helpful to look at the rotation group of the circle, which is analogous butconsiderably simpler. The circle can be viewed as the unit one-dimensional sphere S 1 inR2 , and its rotations are products of reflections in lines through O. Thiscircumstance is what makes the rotation group of the circle similar to therotation group of the sphere. What makes it a lot simpler is the fact thateach rotation of S1 corresponds to a point of S1 , because each rotation ofS1 is determined by the point to which it sends the specific point (1, 0). Inother words, each rotation of the circle corresponds to an angle, namely theangle between the initial and final positions of any line through O. Also,rotations of S1 commute, because rotation through θ followed by rotationthrough ϕ results in rotation through θ + ϕ , which is also the result ofrotation through ϕ followed by rotation through θ .
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7.6 Representing space rotations by quaternions 159 In contrast to S1 , a rotation of S2 depends on three numbers: two anglesthat give the direction of its axis, and the amount of turn about this axis.Thus, the rotations of S2 cannot correspond to the points of S2 , althoughthey do correspond to the points of an interesting three-dimensional space,as we shall see in Section 7.6. Rotations of S2 generally do not commute, as can be seen by combininga quarter turn z1/4 around the z-axis with a half-turn x1/2 around the x-axis.Supposing that the quarter turn is in the direction that takes (1, 0, 0) to(0, 1, 0), we have z1/4 x1/2 (1, 0, 0) −→ (0, 1, 0) −→ (0, −1, 0),whereas x1/2 z1/4 (1, 0, 0) −→ (1, 0, 0) −→ (0, 1, 0).ExercisesIn the Euclidean plane R2 , the product of a rotation about a point P and a rotationabout a point Q is not necessarily a rotation.7.5.1 Give an example of two rotations of R2 whose product is a translation.7.5.2 By imitating the construction of rotations of S2 via reflections, explain how to decide whether the product of two rotations of R2 is a rotation and, if so, how to find its center and angle. The group of all isometries of R2 , unlike the group of rotations of R2 aboutO, is not commutative.7.5.3 Find a rotation and reflection of R2 that do not commute.7.6 Representing space rotations by quaternionsThe most elegant (and practical) way to describe rotations of R 3 or S2 iswith the help of the quaternions, which were introduced in Section 6.6.Because they appeared there only in exercises, we now review their basicproperties for the sake of completeness. A quaternion is a 2 × 2 matrix of the form a + ib c + id q= , where a, b, c, d ∈ R and i2 = −1. −c + id a − ib
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160 7 TransformationsWe also write q in the form q = a1 + bi + cj + dk, where 1 0 i 0 0 1 0 i1= , i= , j= , k= . 0 1 0 −i −1 0 i 0The various products of i, j, and k are easily worked out by matrix multi-plication, and one finds for example that ij = k = −ji and i2 = −1. Because q corresponds to the quadruple (a, b, c, d) of real numbers, wecan view q as a point in R4 . If p is an arbitrary point in R4 then the mapsending p → pq multiplies all distances in R4 by |q|, the distance of q fromthe origin. To see why, notice that det q = a2 + b2 + c2 + d 2 = |q|2 .Then it follows from the multiplicative property of determinants that |pq|2 = det(pq) = (det p)(det q) = |p|2 |q|2 and hence |pq| = |p||q|.It follows that, for any points p1 , p2 ∈ R4 , |p1 q − p2 q| = |(p1 − p2 )q| = |p1 − p2 ||q|.Hence, the distance |p1 − p2 | between any two points is multiplied by theconstant |q|. In particular, if |q| = 1, then the map p → pq is an isometryof R4 . The map p → qp (which is not necessarily the same as the map p →pq, because quaternion multiplication is not commutative) is likewise anisometry when |q| = 1. These maps are useful for studying rotations of R 4but, more surprisingly, also for studying rotations of R3 .Rotations of (i, j, k)-spaceIf p is any quaternion in (i, j, k)-space, p = xi + yj + zk, where x, y, z ∈ R,and if q is any nonzero quaternion, then it turns out that qpq−1 also liesin (i, j, k)-space. Thus, if |q| = 1, then the map p → qpq−1 defines anisometry of R3 , because (i, j, k)-space is just the space of real triples (x, y, z)and hence a copy of R3 .
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162 7 Transformations Thus, the isometry p → qpq−1 of (i, j, k)-space leaves the i-axis fixedand rotates the (j, k)-plane through angle θ , so it is a rotation through θabout the i-axis. It should be emphasized that if the quaternion q represents a certainrotation of R3 , then so does the opposite quaternion −q, because qpq−1 =(−q)p(−q)−1 . Thus, rotations of R3 actually correspond to pairs of quater-nions ±q with |q| = 1. This has interesting consequences when we try tointerpret the group of rotations of R3 as a geometric object in its own right(Section 7.8).ExercisesThe representation of space rotations by quaternions is analogous to the represen-tation of plane rotations by complex numbers, which was described in Section 4.7.As a warmup for the study of a finite group of space rotations in Section 7.7, welook here at some finite groups of plane rotations and the geometric objects theypreserve. We take the plane to be C, the complex numbers.7.6.1 Consider the square with vertices 1, i, −1, and −i. There is a group of four rotations of C that map the square onto itself. These rotations correspond to multiplying C by which four numbers?7.6.2 The cyclic group Cn is the group of n rotations that maps a regular n-gon onto itself. These rotations correspond to multiplying C by which n com- plex numbers? The noncommutative multiplication of quaternions is a blessing when wewant to use them to represent space rotations, because we know that productsof space rotations do not generally commute. Nevertheless, one wonders whetherthere is a reasonable commutative "product" operation on any Rn , for any n ≥ 3."Reasonable" here includes the property |uv| = |u||v| that holds for products on Rand R2 (the real and complex numbers), and the field axioms from Section 6.5. In particular, there should be a multiplicative identity: a point 1 such that|1| = 1 and u1 = u for any point u. Moreover, because n ≥ 3, we can find points iand j, also of absolute value 1, such that 1, i, and j are in mutually perpendiculardirections from O. Figure 7.6 shows these points, together with their negatives. √7.6.3 Show that |1 + i| = 2 = |1 − i|, and deduce from the assumptions about the product operation that 2 = |1 − i2 |, which means that the point 1 − i2 is at distance 2 from O.7.6.4 Show also that |i2 | = 1, so the point 1 − i2 is at distance 1 from 1. Conclude from this and Exercise 7.6.3 that i2 = −1.
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7.7 A finite group of space rotations 163 j −1 i O −i 1+i 1 1−i −j Figure 7.6: Points in perpendicular directions from O7.6.5 Show similarly that u2 = −1 for any point u whose direction from O is perpendicular to the direction of 1, and whose absolute value is 1.7.6.6 Given that i and j are in perpendicular directions, show (multiplying the whole space by i) that so are i2 = −1 and ij, and hence so too are 1 and ij.7.6.7 Thus, ij is one point u for which u2 = −1, by Exercise 7.6.5. Deduce that −1 = (ij)2 = (ij)(ij) = jiij by the commutative and associative laws and show that this leads to the contradiction −1 = 1.Therefore, when n ≥ 3, there is no product on Rn that satisfies all the field axioms.7.7 A finite group of space rotationsR3 is home to the regular polyhedra, the remarkable symmetric objectsdiscussed in Section 1.6 and shown in Figure 1.19. The best known ofthem is the cube, and the simplest of them is the tetrahedron, which fitsinside the cube as shown in Figure 7.7. Also shown in this figure are some rotations of the tetrahedron thatare called symmetries because they preserve its appearance. If we choose afixed position of the tetrahedron—a "tetrahedral hole" in space as it were—then these rotations bring the tetrahedron to positions where it once again
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164 7 Transformations 1/2 turn 1/3 turn Figure 7.7: The tetrahedron and axes of rotationfits in the hole. Altogether there are 12 such rotations. We can choose anyone of the four faces to match a fixed face of the hole, say, the front face.Each of the four faces that can go in front has three edges that can matcha given edge, say, the bottom edge, in the front face of the hole. Thus,we have 4 × 3 = 12 ways in which the tetrahedron can occupy the sameposition, each corresponding to a different symmetry. But once we havechosen a particular face to go in front, and a particular edge of that faceto go on the bottom, we know where everything goes, so the symmetry iscompletely determined. Hence, there are exactly 12 rotational symmetries. Each symmetry can be obtained, from a given initial position, by ro-tations like those shown in Figure 7.7. First there is the trivial rotation,which gives the identity symmetry, obtained by rotation through angle zero(about any axis). Then there are 11 nontrivial rotations, divided into twodifferent types: • The first type is a 1/2 turn about an axis through centers of opposite edges of the tetrahedron (which also goes through opposite face cen- ters of the cube). There are three such axes. Hence, there are three rotations of this type. • The second type is a 1/3 turn about an axis through a vertex and the center of the face opposite to it (which also goes through opposite
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7.7 A finite group of space rotations 165 vertices of the cube). There are four such axes, and hence eight rotations of this type—because the 1/3 turn clockwise is different from the 1/3 turn anticlockwise.Notice also that each 1/2 turn moves all four vertices, whereas each 1/3turn leaves one vertex fixed and moves the remaining three. Thus, the 11nontrivial rotations are all different. Therefore, together with the trivialrotation, they account for all 12 symmetries of the tetrahedron.The quaternions representing rotations of the tetrahedronAs explained in Section 7.6, a rotation of (i, j, k)-space through angle θabout axis li + mj + nk corresponds to a quaternion pair ±q, where θ θ q = cos + (li + mj + nk) sin . 2 2If we choose coordinate axes so that the sides of the cube in Figure 7.7are parallel to the i, j, and k axes, then the axes of rotation are virtuallyimmediate, and the corresponding quaternions are easy to work out. • We can take the lines through opposite face centers of the cube to be the i, j, and k axes. For a 1/2 turn, the angle θ = π , and hence θ /2 = π /2. Therefore, because cos π = 0 and sin π = 1, the 1/2 2 2 turns about the i, j, and k axes are given by the quaternions i, j, and k themselves. Thus, the three 1/2 turns are represented by the three quaternion pairs ±i, ±j, ±k. • Given the choice of i, j, and k axes, the four rotation axes through opposite vertices of the cube correspond to four quaternion pairs, which together make up the eight combinations 1 √ (±i ± j ± k) (independent choices of + or − sign). 3 1 The factor √3 is to give each of these quaternions the absolute value 1, as specified for the representation of rotations.
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166 7 Transformations For each 1/3 turn, we have θ = ±2π /3. Hence, √ θ π 1 θ π 3 cos = cos = , sin = ± sin = ± . 2 3 2 2 3 2 √ √ The 3 in sin π neatly cancels the factor 1/ 3 in the axis of rotation, 3 and we find that the eight 1/3 turns are represented by the eight pairs of opposites among the 16 quaternions 1 i j k ± ± ± ± . 2 2 2 2Finally, the identity rotation is represented by the pair ±1, and thus the 12symmetries of the tetrahedron are represented by the 24 quaternions 1 i j k ±1, ±i, ±j, ±k, ± ± ± ± . 2 2 2 2The 24-cellThese 24 quaternions all lie at distance 1 from O in R4 , and they are dis-tributed in a highly symmetrical manner. In fact, they are the vertices of afour-dimensional figure analogous to a regular polyhedron—called a reg-ular polytope. This particular polytope is called the 24-cell. Because wecannot directly perceive four-dimensional figures, the best we can do isstudy the 24-cell via projections of it into R3 (just as we often study poly-hedra, such as the tetrahedron and cube, via projections onto the plane suchas Figure 7.7). One such projection is shown in Figure 7.8 (which of courseis a projection of a three-dimensional figure onto the plane—but it is easyto visualize what the three-dimensional figure is). This superb drawing istaken from Hilbert and Cohn-Vossen's Geometry and the Imagination.ExercisesThe vertices of the 24-cell include the eight unit points (positive and negative) onthe four axes in R4 , but the other 16 points have some less obvious properties. j7.7.1 Verify directly that the 16 points ± 1 ± 2 ± 2 ± k are all at distance 1 from 2 i 2 the origin in R 4.7.7.2 Deduce that the distance from the center to any vertex of a four-dimensional cube is equal to the length of its side. j 1 i7.7.3 Show also that each of the points ± 2 ± 2 ± 2 ± k is at distance 1 from the 2 four nearest unit points on the axes.
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7.8 The groups S3 and RP3 167 Figure 7.8: The 24-cell7.8 The groups S3 and RP3The rotations of the tetrahedron, which were discussed in Section 7.7,vividly show that a group of rotations is itself a geometric object. Thisstatement is just as true of the group of all rotations of S2 . In fact, thisgroup is closely related to two important geometric objects: the 3-sphereS3 and the three-dimensional real projective space RP3 . Just as the 1- and 2-spheres are the sets of points at unit distance fromO in R2 and R3 respectively, the 3-sphere is the set of points in R4 at unitdistance from O: S3 = {(a, b, c, d) ∈ R4 : a2 + b2 + c2 + d 2 = 1}.The points (a, b, c, d) on S3 correspond to quaternions q = a1+bi+cj+dkwith |q| = 1, because |q|2 = a2 + b2 + c2 + d 2 . Hence, rotations of S2 ,which correspond to pairs ±q of such quaternions, correspond to pointpairs ±(a, b, c, d) on S3 .
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168 7 Transformations And to what else do the point pairs ±(a, b, c, d) correspond? Well,remember from Section 5.4 how we described the real projective spaceRP3 . Its "points" are lines through O in R4 . But a line through O in R4meets S3 in a pair of antipodal points ±(a, b, c, d). Thus, it is also valid toview the point pairs ±(a, b, c, d) on S3 as single "points" of RP3 . Hence,rotations of S2 correspond to points of RP3 , and the group of all rotationsof S2 is in some sense the "same" as the geometric object RP3 . To explain what we mean by "sameness" here, we have to say some-thing about groups in general, what it means for two groups to be the"same", and what it means for a geometric object to acquire the structureof a group.Abstract groups and isomorphismsWe began this chapter with the idea of a group of transformations: a col-lection G of functions on a space S with the properties that • if f , g ∈ G, then f g ∈ G, • if f ∈ G, then f −1 ∈ G.The "product" f g of f and g here is the composite function, which is de-fined by f g(x) = f (g(x)). However, we have found it convenient to rep-resent certain functions, such as rotations, by algebraic objects, such asmatrices, whose "product" is defined algebraically. It is therefore desirable to have a more general concept of group, whichdoes not presuppose that the product operation is function composition. Wedefine an abstract group to be a set G, which contains a special element 1and for each g an element g−1 , with a "product" operation satisfying thefollowing axioms: g1 (g2 g3 ) = (g1 g2 )g3 (associativity) g1 = g (identity) −1 gg =1 (inverse)The associative axiom is automatically satisfied for function composition,because if g1 , g2 , g3 are functions, then g1 (g2 g3 ) and (g1 g2 )g3 both meanthe same thing, namely, the function g1 (g2 (g3 (x))). It is also satisfied whenthe group consists of numbers, because the product of numbers is wellknown to be associative.
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7.8 The groups S3 and RP3 169 In other cases, the easiest way to prove associativity is to show, if pos-sible, that the group operation corresponds to function composition. Forexample, the matrix product operation is associative because matrices be-have like linear transformations under composition. It follows in turn thatthe quaternion product operation is associative, because quaternions can beviewed as matrices. When we say that the elements of a certain group G "correspond to"or "behave like" or "can be viewed as" elements of another group G , wehave in mind a precise relationship called an isomorphism of G onto G .The word "isomorphism" comes from the Greek for "same form," and itmeans that there is a one-to-one correspondence between G and G thatpreserves products. That is, an isomorphism is a function ϕ :G→G such that ϕ (g1 g2 ) = ϕ (g1 )ϕ (g2 ).For example, the group G of rotations of the circle, under composition ofisometries, is isomorphic to group G of complex numbers of the form cos θ + i sin θ , under multiplication.If rθ denotes the rotation through angle θ , then the isomorphism ϕ is de-fined by ϕ (rθ ) = cos θ + i sin θ . Sometimes there is a natural one-to-one correspondence ϕ between agroup G and a set S. In that case, we can use ϕ to transfer the groupstructure from G to S. That is, we define the product of elements ϕ (g 1 ) andϕ (g2 ) to be ϕ (g1 g2 ). Here are some examples. • The complex numbers cos θ + i sin θ form a group, and they corre- spond to the points (cos θ , sin θ ) of the unit circle S1 . Therefore, we can define the product of points (cos θ1 , sin θ1 ) and (cos θ2 , sin θ2 ) on S1 to be the point corresponding to the product of the corresponding complex numbers. This point is (cos(θ1 + θ2 ), sin(θ1 + θ2 )). • Likewise, the quaternions q = a1 + bi + cj + dk with |q| = 1 form a group, and they correspond to the points (a, b, c, d) of the 3-sphere S3 . Hence, we can define the product of points (a1 , b1 , c1 , d1 ) and (a2 , b2 , c2 , d2 ) corresponding to the quaternions q1 and q2 , say, to be the point corresponding to the quaternion q1 q2 . Under this product operation, S3 is a group.
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170 7 Transformations • Finally, pairs of opposite quaternions ±q with |q| = 1 form a group under the operation defined by (±q1 )(±q2 ) = ±q1 q2 . We know that these pairs are in one-to-one correspondence with the points of RP3 . Hence, we can transfer the group structure of these quaternion pairs (which is also the group structure of the rotations of S2 ) to RP3 . Under the transferred operation, RP3 is a group. The group structures on S1 , S3 , and RP3 obtained in this way are par-ticularly interesting because they are continuous. That is, if g 1 is near tog1 and g2 is near to g2 , then g1 g2 is near to g1 g2 . It is known that S2 doesnot have a continuous group structure, and in fact S1 and S3 are the onlyspheres with continuous group structures on them.7.9 DiscussionThe word "geometry" comes from the Greek for "earth measurement," andlegend has it that the subject grew from the land measurement concerns offarmers whose land was periodically flooded by the river Nile. As recentlyas the 18th century, one finds carpenters and other artisans listed amongthe subscribers to geometry books, so there is no doubt that Euclidean ge-ometry is the geometry of down-to-earth measurement. It continues to bea tactile subject today, when one talks about "translating," "rotating," and"moving objects rigidly." The most visual branch of geometry is projective geometry, because itis more concerned with how objects look than with what they actually are.It is no surprise that projective geometry originated from the concerns ofartists, and that many of its practitioners today work in the fields of videogames and computer graphics. Affine geometry occupies a position in the middle. It also originatesfrom an artistic tradition, but from one less radical than that of Renais-sance Italy—the classical art of China and Japan. Chinese and Japanesedrawings often adopt unusual viewpoints, where one might expect per-spective, but they generally preserve parallels. Typically, the picture isa "projection from infinity," which is an affine map. Figure 7.9 shows anexample, a woodblock print by the Japanese artist Suzuki Harunobu fromaround 1760.
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7.9 Discussion 171 Figure 7.9: Harunobu's Courtesan on veranda Notice that all the parallel lines are shown as parallel, with the resultthat the (obviously rectangular) panels on the screen appear as identicalparallelograms. Likewise, the planks on the veranda appear with paral-lel edges and equal widths, which creates a certain "flatness" because allparts of the picture seem to be the same distance away from us. Speakingmathematically, they are—because the view is what one would see frominfinity with infinite magnification. A similar effect occurs in photographsof distant buildings taken with a large amount of zoom. Affine maps are also popular in engineering drawing, in which the so-called "axonometric projection" is often used to depict an object in threedimensions while retaining correct proportions in a given direction. The
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172 7 Transformationsaffine picture gives a good compromise between a realistic view and anaccurate plan. See Figure 7.10, which shows an axonometric projection ofa cube. Figure 7.10: Affine view of the cubeThe fourth dimensionThe discovery of quaternions in 1843 was the first of a series of discoveriesthat drew attention to spaces of more than three dimensions and to theremarkable properties of R4 in particular. From around 1830, the Irish mathematician William Rowan Hamiltonhad been searching in vain for "n-dimensional number systems" analogousto the real numbers R and the complex numbers C. Because C can beviewed as R2 under vector addition (u1 , u2 ) + (v1 , v2 ) = (u1 + v1 , u2 + v2 )and the multiplication operation (u1 , u2 )(v1 , v2 ) = (u1 v1 − u2 v2 , u1 v2 + u2 v1 ),Hamilton thought that R3 could also be viewed as a number system bysome clever choice of multiplication rule. He took a "number system" tobe what we now call a field, together with an absolute value |u| = |(u1 , u2 , u3 )| = u2 + u2 + u2 1 2 3which is multiplicative: |uv| = |u||v|.
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7.9 Discussion 173 We now know (for example, by Exercise 7.6.7) that such a system isimpossible in any Rn with n ≥ 3. But, luckily for Hamilton, it is almostpossible in R4 . The quaternions satisfy all the field axioms except commu-tative multiplication, and their absolute value is multiplicative. The onlyother Rn that comes close is R8 , where the octonions O satisfy all the fieldproperties except the commutative and associative laws. (Recall from Sec-tion 6.8 that the quaternions and octonions also play an important role inprojective geometry.) Hamilton knew that quaternions give a nice representation of rotationsin R3 , but the first to work out the quaternions for the symmetries of regularpolyhedra was Cayley in 1863. Cayley's enumeration of these quaternionsmay be found in his Mathematical Papers, volume 5, p. 529. The fiveregular polyhedra actually exhibit only three types of symmetry—becausethe cube and octahedron have the same symmetry type, as do the dodec-ahedron and the icosahedron—which therefore correspond to three highlysymmetric sets of quadruples in R4 . Cayley did not investigate the geometric properties of these point setsin R4 , but in fact they were already known to the Swiss geometer LudwigSchl¨ fli in 1852. As we have seen, the 12 rotations of the tetrahedron acorrespond to the 24 vertices of a figure called the 24-cell. It gets this namebecause it is bounded by 24 identical regular octahedra. It is one of sixregular figures in R4 , analogous to the regular polyhedra in R3 , called theregular polytopes. They were discovered by Schl¨ fli, who also proved that athere are regular figures analogous to the tetrahedron, cube, and octahedronin each Rn , but that R3 and R4 are the only Rn containing other regularfigures. The 24-cell is the simplest of the exceptional regular figures in R 4 ; theother two are the 120-cell (bounded by 120 regular dodecahedra) and the600-cell (bounded by 600 regular tetrahedra). The 120-cell has 600 ver-tices, which correspond to the cell centers of the 600-cell, and vice versa,so the two are related "dually" like the dodecahedron and the icosahedron. Moreover, the 600-cell arises from the icosahedron in the same waythat the 24-cell arises from the tetrahedron. Its 120 vertices correspond to60 pairs of opposite quaternions, each representing a rotational symmetryof the icosahedron. For more on these amazing objects, see my article Thestory of the 120-cell, which can be read online at
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8Non-Euclidean geometryP REVIEW In previous chapters, we have seen several reasons why there is such a subject as "foundations of geometry." Geometry is fundamentally visual; yet it can be communicated by nonvisual means: by logic, linear algebra, or group theory, for example. The several ways to communicate geometry give several foundations. But also, there is more than one geometry. Section 7.4 gave a hint of this when we briefly discussed the geometry of the sphere in the language of "points" and "lines." It seems reasonable to call great circles "lines" because they are the straightest curves on the sphere; but they certainly do not have all of the properties of Euclid's lines. This characteristic makes geometry on a sphere a non-Euclidean geometry—one that has been known since ancient times. But it was never seen as a challenge to Euclid, probably because the geometry of the sphere is simply a part of three-dimensional Euclidean geom- etry, where great circles coexist with genuine straight lines. The real challenge to Euclid emerged from disquiet over the parallel axiom. Many people found it inelegant and wished that it was a consequence of Euclid's other axioms. It is not, because there is a geometry that satisfies all of Euclid's axioms except the parallel axiom. This is the geometry of a surface called the non-Euclidean plane. The non-Euclidean plane is not an artificial construct built only to show that the parallel axiom cannot be proved. It arises in many places, and today one can hardly discuss differential geometry, the theory of complex numbers, and projective geometry without it. In this chapter, we will see how it arises from the real projective line. 174
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8.1 Extending the projective line to a plane 1758.1 Extending the projective line to a planeIn this book, we have been concerned mainly with the geometry of lines,partly because lines are the foundation of geometry and partly becauselines are remarkably interesting. From the regular polygons to the Pappusand Desargues configurations, figures built from lines reveal beautiful con-nections between geometry and other parts of mathematics. We have seensome of these connections but have barely begun to explore them in depth. In fact we have not yet gone far toward understanding the geometry ofeven one line—the real projective line RP1 . In Chapter 5, we arrived atan algebraic summary of RP1 by representing its transformations as linearfractional functions ax + b f (x) = , where a, b, c, d ∈ R and ad − bc = 0, (*) cx + dand by uncovering the cross-ratio, a "ratio of ratios" left invariant by alllinear fractional transformations. But this summary is not as geometric asone would like. It is hard (although perhaps not impossible) to "see" thecross-ratio, and indeed it is hard to see geometric phenomena on the lineat all. If only we could extend the projective line in another dimension sothat we could see it as a plane! Amazingly, this is possible, and the presentchapter shows how. The idea is to let RP1 be the boundary ("at infinity") of a plane whosetransformations extend the linear fractional transformations of RP 1 in anatural way. Algebra suggests how this should be done. It suggests re-placing the real variable x in the linear fractional transformations (*) by acomplex variable z and interpreting az + b f (z) = cz + das a transformation of the plane C of complex numbers. This idea needs a little modification. We should really use transforma-tions of the upper half plane of complex numbers z = x + iy with y > 0,because the line of real numbers divides C into two halves. Either half canbe taken as the "plane" bounded by the real line, but, when we want totransform one particular half, the extension from x to z is not always theobvious one. For example, the transformation x → −x of the line shouldnot be extended to the transformation z → −z, because the latter maps the
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176 8 Non-Euclidean geometryupper half plane onto the lower. The correct extension is described in Sec-tion 8.2. Nevertheless, the extension from real to complex numbers works amaz-ingly well. The extended transformations leave invariant a geometric quan-tity that is clearly visible, namely angle. The cross-ratio is also invariant(well, almost), for much the same algebraic reasons as before. And it givesus an invariant length in the half plane—something we certainly do nothave in the projective line RP1 . The concept of length that emerges in this way is a little subtle, and itis not as easily visible as angle. It is a non-Euclidean measure of length,and it gives rise to non-Euclidean lines, which turn out to be the ordinarylines of the form x = constant and the semicircles with their centers onthe x-axis. These "lines" are the curves of shortest non-Euclidean lengthbetween given points, and they have all the properties of "lines" in Euclid'sgeometry except the parallel property. That is, if L is any non-Euclideanline and P is a point of the upper half plane outside L then there is morethan one non-Euclidean line through P that does not meet L . Figure 8.1shows an example. P L Figure 8.1: Failure of the parallel axiom for non-Euclidean "lines" The dotted line represents the real line y = 0 so the half plane y > 0consists of the points strictly above it. The semicircle L is one "line" inthe half plane, and the two semicircles passing through the point P clearlydo not meet L , so they are two "parallels" of L . In the remainder of thischapter we explain in more detail why these semicircles should be regardedas "lines," and why they satisfy all of Euclid's axioms for lines except theparallel axiom.
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8.1 Extending the projective line to a plane 177 Thus, the complex half plane not only allows us to visualize the ge-ometry of the projective line; it also answers a fundamental question inthe foundations of geometry by showing that the parallel axiom does notfollow from Euclid's other axioms.The non-Euclidean "line" through two pointsOne property of non-Euclidean "lines" can be established immediately.There is a unique non-Euclidean "line" through any two points (and hencenon-Euclidean "lines" satisfy the first of Euclid's axioms). • If the two points lie on the same vertical line x = l, then x = l is a non- Euclidean "line" containing them. And it is the only one, because a semicircle with its endpoints on the x-axis has at most one point on each line x = l. • If the two points P and Q do not lie on the same vertical line, there is a unique point R on the x-axis equidistant for both of them, namely, where the equidistant line of P and Q meets the x-axis. Then the semicircle with center R through P and Q is the unique non-Euclidean "line" through P and Q.ExercisesOne can begin to understand the geometric significance of linear fractional trans-formations of the half plane by studying the simplest ones, z → z + l and z → kzfor real k and l.8.1.1 Show that the transformations z → z + l and z → kz (for k > 0) map the upper half plane onto itself and that they map "lines" to "lines."8.1.2 Explain why this is not the case when k and l are not real.8.1.3 Show how to map the semicircle x2 + y2 = 1, y > 0, onto the semicircle (x − 1)2 + y2 = 4, y > 0, by a combination of transformations z → z + l and z → kz.8.1.4 More generally, explain why any semicircle with center on the x-axis can be mapped onto any other by a combination of transformations z → z + l and z → kz.What is not yet clear is why semicircles should be regarded as "lines." Their"linelike" behavior stems from the transformation z → 1/z, which we study inSection 8.2.
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178 8 Non-Euclidean geometry8.2 Complex conjugationWe know from Section 5.6 that all linear fractional transformations of RP 1are products of the transformations x → x + l, x → kx, and x → 1/x for realconstants k = 0 and l. We called these the generating transformations ofRP1 . The transformation x → x + l obviously extends to the transformationz → z + l, which maps the upper half plane onto itself for any real l, but theappropriate extension of x → kx is z → kz only for k > 0, because z → kzdoes not map the upper half plane onto itself when k < 0. In particular,what is the appropriate extension of x → −x to a map of the upper halfplane? Geometrically, the answer is obvious. The transformation x → −x isreflection of the line in O, so its most appropriate extension is reflection ofthe half plane in the y-axis, that is, the transformation x + iy → −x + iy.This transformation can be expressed more simply with the help of thecomplex conjugate z, which is defined as follows. If z = x + iy, then z =x − iy. Then the reflection of z in the y-axis is −z because −z = −(x + iy) = −(x − iy) = −x + iy.Thus, the appropriate extension of x → −x is z → −z (Figure 8.2). y −z z −x O x Figure 8.2: Extending reflection from the line to the half plane More generally, the appropriate extension of x → kx when k < 0 isz → kz, the product of the reflection z → −z with the map z → |k|z (dilationby factor |k|).
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180 8 Non-Euclidean geometryEquations of non-Euclidean "lines"Complex conjugation not only enables us to express reflection in lines andcircles; it also enables us to write the equations of non-Euclidean "lines"very simply as equations in z. • First consider "lines" that are actual Euclidean lines, namely those of the form x = a, where a is a real number. An arbitrary point on this line is of the form z = a + iy, so z = a − iy, and z therefore satisfies the equation z + z = 2a. (*) • Next consider "lines" that are semicircles with centers on the x-axis. If the center is c and the radius is r, then any z on the circle satisfies |z − c| = r, or equivalently, |z − c|2 = r2 . But now notice that for any complex number x + iy we have |x + iy|2 = x2 + y2 = (x + iy)(x − iy) = (x + iy)(x + iy). Hence, |z − c|2 = (z − c)(z − c) = (z − c)(z − c) and the equation |z − c|2 = r2 becomes (z − c)(z − c) = r 2 , that is, zz − cz − cz + cc = r 2 . Finally, because c is a real number, we have c = c, so the equation can be written as zz − c(z + z) + c2 − r2 = 0. (**) The equations (*) and (**) are both of the form Azz + B(z + z) +C = 0 for some A, B,C ∈ R. (***)Conversely, if A and B are not both zero, then (***) reduces to one of theequations (*) or (**) above, if it is satisfied by any points z at all.
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8.2 Complex conjugation 181 • If A = 0, then (***) becomes z + z + C/B = 0, which is (*) with 2a = −C/B. • If A = 0, then (***) becomes zz + (z + z)B/A + C/A = 0, which is (**) with c = −B/A and c2 − r2 = C/A if r2 = B2 /C2 − C/A ≥ 0. If r2 < 0, then no points z satisfy the equation (***), because the equation is equivalent to |z − c|2 = r2 and |z − c|2 is necessarily > 0.Thus, equations of non-Euclidean "lines" are the satisfiable equations Azz + B(z + z) +C = 0, where A, B,C ∈ R are not all zero.ExercisesI expect that most readers of this book are familiar with the complex numbers,but it still seems worthwhile to review the properties of the complex conjugate.Its role in geometric transformations may not be familiar, so we develop the basicfacts from first principles.8.2.1 Writing z1 as x1 + iy1 and z2 as x2 + iy2 , show that z1 + z2 = z1 + z2 and z1 z2 = z 1 z2 .8.2.2 Similarly, show that 1/z = 1/z.8.2.3 Deduce from Exercises 8.2.1 and 8.2.2 that, for any a, b, c, d ∈ R and z ∈ C, the complex conjugate of az+b is az+b . cz+d cz+d With these facts established, we are in a position to determine the extension tothe half plane of each linear fractional transformation x → ax+b of RP1 . What we cx+dknow so far is that the extension of x → x + l is z → z + l, the extension of x → kxis z → kz when k > 0 and z → kz when k < 0, and that the extension of x → 1/xis z → 1/z. We also know that any transformation x → ax+b is a product of these cx+dgenerating transformations. Hence, the extension of x → ax+b to the half plane is cx+dthe product of the corresponding extensions. It seems likely that the latter productis either z → az+b or z → az+b , so the main problem is to decide when the product cz+d cz+dis z → az+b and when it is z → az+b . cz+d cz+d8.2.4 Write each generating transformation of RP1 in the form x → ax+b , and cx+d hence, show that those whose extension involves z are precisely those for which ad − bc < 0.8.2.5 Deduce from Exercise 8.2.4 and Exercise 5.6.3 that the extension of a prod- uct, of transformations x → a1 x+d1 and x → a2 x+d2 , is the product of their c 1 x+b1 c 2 x+b2 extensions. ax+b8.2.6 Deduce from Exercise 8.2.5, or otherwise, that the extension of x → cx+d is z → az+b when ad − bc > 0 and z → az+b otherwise. cz+d cz+d
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182 8 Non-Euclidean geometry It may seem unfortunate that the extension of x → ax+b is one of two dif- cx+dferent types: a function of z or a function of z. However, these two algebraictypes are inevitable because they reflect a geometric distinction: The functionsof z are orientation-preserving, and the functions of z are not. In particular, thelinear fractional transformations z → az+b with ad − bc > 0 are precisely the cz+dorientation-preserving transformations of the half plane.8.3 Reflections and M¨ bius transformations oThe extensions of the transformations x → ax+b from RP1 to the half plane cx+dcould be called "linear fractional," but this would be confusing, becauseone half of them are linear fractional functions of z and the other half arelinear fractional functions of z. Instead they are called Mobius transfor- ¨mations, after the German mathematician August Ferdinand M obius. In ¨1855, M¨ bius introduced a theory of transformations generated by reflec- otions in circles, using the obvious generalization from reflection in the unit ¨circle to reflection in an arbitrary circle. We will see below that all M obiustransformations of the half plane are products of reflections. One advantage of the reflection idea is that it makes sense in three (ormore) dimensions, where reflection in a sphere is meaningful but "linearfractional transformation" generally is not. It is also revealing to view thetransformations of RP1 as the restrictions of M¨ bius transformations of the ohalf plane, as this brings to light a concept of "projective reflection". Reflection in an arbitrary circle is defined by generalizing the relation-ship between z and 1/z shown in Figure 8.3. We say that points Q andQ are reflections of each other in the circle with center P and radius r ifP, Q, Q lie in a straight line and |PQ||PQ | = r2 (Figure 8.4). Q Q r P Figure 8.4: Reflection in an arbitrary circle
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8.3 Reflections and M¨ bius transformations o 183 If the circle (or, rather, its upper half) is a non-Euclidean line, then thecenter P lies on the x-axis, and reflection in this circle can be composedfrom generating transformations of the half plane as follows: • translate P to O, • reduce the radius to 1 by dilating by 1/r, • reflect in the unit circle, • restore the radius to r by dilating by r, • translate the center from O back to P. Likewise, reflection in an arbitrary vertical line, say x = a, can be com-posed from generating transformations of the half plane as follows: • translate the line x = a to the y-axis, • reflect in the y-axis, • translate the y-axis to the line x = a.Thus, all reflections in non-Euclidean lines are products of generatingtransformations of the half plane. Conversely, we now show that every generating transformation of thehalf plane is a product of reflections (and hence so is every transformationof the half plane). The generating transformations z → −z and z → 1/z arereflections by definition, so it remains to deal with the remaining generatingtransformations. • the horizontal translation z → z + l: this is a Euclidean translation, and it is the product of reflections in the lines x = 0 and x = l/2. • the dilation z → kz, where k > 0: this is the product of the reflection z → 1/z in the unit circle and the map z → k/z, which is reflection in √ the circle with center O and radius k. It should be mentioned that ordinary reflection—reflection in a straightline—is the limiting case of reflection in a circle obtained by letting Pand r tend to infinity in such a way that the circle tends to a straight line.Because Euclidean lines are the fixed point sets of ordinary reflections, itis natural that the "lines" of the half plane should be the fixed point sets ofits "reflections."
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184 8 Non-Euclidean geometryProjective reflectionsLooking back from the half plane to its boundary line RP 1 , we realize thatwe now know more about projective transformations of the line than wedid before. Any projective transformation of RP1 is a product of projectivereflections, where a projective reflection is the restriction, to RP 1 , of areflection of the half plane. There is a "three reflections theorem" for RP1 , analogous to the threereflections theorem for isometries of the Euclidean plane (Section 3.7).This follows from a three reflections theorem for the half plane, similar tothe one for the Euclidean plane, that we will prove in Section 8.8.ExercisesThe simplest reflections of RP1 are ordinary reflection in O, x → −x, and therestriction of reflection in the unit circle, x → 1/x. The map x → 1/x might becalled "reflection in the point-pair {−1, 1}," and it generalizes to "reflection inthe point-pair {a, b}." (A point-pair {a, b} is a "0-dimensional sphere," because itconsists of the points at constant distance (b − a)/2 from the "center" (a + b)/2.)8.3.1 Write down the formula for ordinary reflection in the point x = a.8.3.2 Explain why the map x → c2 /x is reflection in the point-pair {−c, c}.8.3.3 Using Exercise 8.3.2, or otherwise, show that reflection in the point-pair {a, b} is given by the linear fractional function x(a + b) − 2ab f (x) = . 2x − (a + b)8.3.4 Show that, as b → ∞, the function for reflection in the point-pair {a, b} tends to the function for ordinary reflection in the point x = a.8.4 Preserving non-Euclidean linesWe have now extended the projective transformations x → ax+b of RP1 to cx+dM¨ bius transformations z → az+b or z → az+b of the half plane, but are o cz+d cz+dM¨ bius transformations of the half plane any easier to understand? We ointend to show that they are, by showing that they have more easily visibleinvariants than the transformations of RP1 . First we show the invarianceof non-Euclidean lines, which we now define officially as the vertical linesx = constant and the semicircles with centers on the x-axis.
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8.4 Preserving non-Euclidean lines 185 Each M¨ bius transformation of the half plane maps non-Euclidean olines to non-Euclidean lines. For the generating transformations z → z + l, z → kz for k > 0, andz → −z, this is easy to see. Each of these transformations sends verticallines to vertical lines, circles to circles, and the x-axis to the x-axis because: • z → z + l is a horizontal translation of the half plane. • z → kz with k > 0 is a dilation of the half plane by k. • z → −z is the Euclidean reflection of the half plane in the y-axis. Thus, any product of the three transformations just listed sends verticallines to vertical lines and semicircles with centers on the x-axis to semicir-cles with centers on the x-axis. Hence, all products of the transformationsz → z + l, z → kz for k > 0, and z → −z preserve non-Euclidean lines. To show that all M¨ bius transformations preserve non-Euclidean lines, oit therefore remains to show that reflection in the unit circle, z → 1/z, pre-serves non-Euclidean lines. This is less obvious, because reflection in acircle can send a vertical line to a semicircle and vice versa. We prove thatnon-Euclidean lines are preserved by using their equations (***) derivedin Section 8.3. Given a non-Euclidean line, whose points z satisfy an equation Azz + B(z + z) +C = 0 for some A, B,C ∈ R, (***)we wish to find the equation satisfied by the points of its reflection in theunit circle. These are the points w = 1/z, so we seek the equation satisfiedby w. The required equation is likely to involve w = 1/z as well, so we arelooking for an equation connecting 1/z and 1/z. Such an equation is easyto find: just divide the equation (***) by zz. Division yields the equation 1 1 C A+B + + = 0, z z zzthat is, Cww + B(w + w) + A = 0. (****)Equation (****) is satisfied by the reflections w = 1/z of the points z sat-isfying (***), and (****) has the same form as (***), because A, B,C ∈ R.Hence, (****) also represents a non-Euclidean line.
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186 8 Non-Euclidean geometryExercisesAn example in which reflection in the unit circle sends a vertical line to a semicir-cle is shown in Figure 8.5. O 1 Figure 8.5: Reflection of the line x = 18.4.1 Give intuitive reasons why the reflection of the line x = 1 in the unit circle should have one end at 1 on the x-axis and the other end at O.8.4.2 Show that the line x = 1 has equation z + z = 2, and that its reflection in the unit circle has equation w + w = 2ww.8.4.3 Verify that w + w = 2ww is the equation of the semicircle with ends O and 1 on the x-axis.8.5 Preserving angle ¨Next to non-Euclidean lines, the most visible invariant of M obius transfor-mations is angle. Because non-Euclidean lines are not necessarily straight,the angle between two of them is really the angle between their tangents atthe point of intersection. Nevertheless, it is easy to see the angle betweennon-Euclidean lines. Figure 8.6 shows an example. L C M 0 1 Figure 8.6: Some non-Euclidean lines and the angles between them
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8.5 Preserving angle 187 The three non-Euclidean lines are the unit circle C , the vertical Lwhere x = 1/2, and its reflection M in the unit circle, which happens to bethe semicircle with endpoints 0 and 2 on the x-axis. At the point where the three non-Euclidean lines meet, they divide thespace around the point into six equal angles, so each angle is 2π /6 = π /3.This equality is confirmed by the tangents, which are shown as dashedlines. Notice that any two of C , L , and M are reflections of each otherin the third non-Euclidean line, so the figure shows numerous instances ofan angle equal to its reflection. To show that any angle is preserved byany M¨ bius transformation, we look once again at the properties of the ogenerating transformations.The effect of M¨ bius transformations oThe M¨ bius transformations z → z + l and z → −z are Euclidean isome- otries; hence, they certainly preserve angle (along with length, area, and soon). The M¨ bius transformations z → kz for k > 0 are dilations; hence, othey too preserve angle. Thus, it suffices to prove that angle is preservedby the remaining generator of Mobius transformations: reflection in the ¨unit circle, z → 1/z. The latter transformation is the composite of z → −zand z → −1/z, so it suffices in turn to prove that z → −1/z preserves angle. ¨ We therefore concentrate our attention on the Mobius transformationz → −1/z. This transformation does not in general preserve Euclideanlines, because it may map them to circles. Thus, we need to be aware that"angle" generally means the angle between curves and hence the anglebetween the tangents. However, we can avoid computing the position oftangents by taking the infinitesimal view of angle. That is, we study whatbecomes of the direction between two points, z and z + ∆z, when we sendthem to −1/z and −1/(z + ∆z), respectively, and let ∆z tend to zero. If ∆z is the point at distance ε from O in direction θ , then ∆z = ε (cos θ + i sin θ ),because cos θ + i sin θ is the point at distance 1 from O in direction θ . Itfollows that the point at distance ε from z in direction θ is z + ∆z = z + ε (cos θ + i sin θ ),and that the point z + ∆z tends to z in the constant direction θ as ε tends tozero.
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188 8 Non-Euclidean geometry The difference between the image points −1/(z + ∆z) and −1/z istherefore 1 1 z + ε (cos θ + i sin θ ) − z − = z z + ε (cos θ + i sin θ ) z(z + ε (cos θ + i sin θ )) ε (cos θ + i sin θ ) = . z(z + ε (cos θ + i sin θ ))Now as ε tends to zero, this difference is ever more closely approximatedby ε (cos θ + i sin θ ) . z2To be precise, the direction from −1/z to −1/(z+∆z) tends to the directionof ε (cos θ + i sin θ )z−2 , which is θ +constant. The constant is the argument(angle) of z−2 , recalling from Section 4.7 that the argument of a product ofcomplex numbers is the sum of their arguments. The angle between two smooth curves meeting at z (approximated bythe difference in directions from z to points z + ∆1 z and z + ∆2 z on therespective curves) is therefore the angle between the images of these curvesunder the map z → −1/z. This is because a smooth curve is one for whichthe direction from z to z + ∆z tends to a constant θ as z + ∆z tends to z alongthe curve. Non-Euclidean lines are smooth, so the angle between them ispreserved by the transformation z → −1/z, as required.Tilings of the half planeIf one takes a triangle with angles π /p, π /q, π /r, for some natural numbersp, q, r, then any reflection of that triangle will have angles π /p, π /q, π /r.Reflecting the reflections causes the space around each vertex to be exactlyfilled with corners of triangles. For example, the space around the vertex ofangle π /p becomes filled with 2p corners of angle π /p. In fact, the wholehalf plane becomes filled, or tiled, by copies of the original triangle. Anexample is shown in Figure 8.7, where the basic tile has angles π /2, π /3,and π /7. Notice that the angle sum π /2 + π /3 + π /7 is less than π . In fact, theangle sum of any triangle bounded by non-Euclidean lines is less than π ,and the quantity (π − angle sum) is proportional to the area of the triangle.This elegant result is less surprising when one learns that the area of spher-ical triangle is also proportional to π − angle sum (see exercises below).
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8.5 Preserving angle 189 Figure 8.7: Tiling by repeated reflectionsHowever, it does reveal a limitation in the half plane view of non-Euclideangeometry: All the triangles in Figure 8.7 have equal non-Euclidean area,but they certainly do not look equal! One should think of the half plane as a kind of "perspective view" of thenon-Euclidean plane with the x-axis as a horizon. The x-axis is infinitelydistant, because there are infinitely many identical triangles between anypoint of the half plane and the x-axis. In this respect, the half plane is like aperspective view of a Euclidean tiled floor, except that ordinary perspectivepreserves straightness and distorts angle, whereas this "non-Euclidean per-spective" distorts straightness and preserves angle. There are other viewsof the non-Euclidean plane that make non-Euclidean lines look straight(see Section 8.9), but any such view has a curved horizon! Another way in which a tiling of the half plane resembles a perspectiveview is that one can estimate the length of a line by counting the numbers oftiles that lie along it. There is indeed a non-Euclidean measure of distancethat is invariant under M¨ bius transformations, and we will see exactly owhat it is in Section 8.6.
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190 8 Non-Euclidean geometryExercisesThe English mathematician Thomas Harriot discovered that the area of a sphericaltriangle is proportional to (angle sum − π ) in 1603. His argument is based on thetwo views of a spherical triangle shown in Figure 8.8. C A B α B ∆αβ γ A β γ C (a) α ∆αβ γ ∆α (b) Figure 8.8: Area of a spherical triangle
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8.6 Non-Euclidean distance 191 View (a) shows all sides of the spherical triangle extended to great circles.These divide the sphere into eight spherical triangles, which are obviously con-gruent in antipodal pairs. View (b) shows the result of extending two sides, whichis a "slice" of the sphere with area proportional to the angle at its two ends.8.5.1 Letting the area of the triangle with angles α , β , γ be ∆αβ γ , and letting the areas of the other triangles be ∆α , ∆β , ∆γ as shown in view (a), prove that 2 ∆αβ γ + ∆α + ∆β + ∆γ = area of sphere, call it A. (1)8.5.2 Use view (b) to explain why α β γ ∆αβ γ + ∆α = A, ∆αβ γ + ∆β = A, ∆αβ γ + ∆γ = A. 2π 2π 2π8.5.3 Deduce from Exercise 8.5.2 that α +β +γ 3∆αβ γ + ∆α + ∆β + ∆γ = A (2) 2π α +β +γ −π8.5.4 Deduce from equations (1) and (2) that 4∆αβ γ = π A, and hence that ∆αβ γ = constant × (α + β + γ − π ).8.5.5 Using a formula for the area of the sphere, show that ∆αβ γ = α + β + γ − π on a sphere of radius 1.8.6 Non-Euclidean distance ¨So far we have found invariants of Mobius transformations by geometri-cally inspired guesses that can be confirmed by calculations with linearfractional functions. But still up our sleeve is the cross-ratio card, whichcarries the fundamental invariant of linear fractional transformations, andto find out what non-Euclidean distance is we finally have to play it. We know from Section 5.7 that the cross-ratio is invariant under thetransformations x → x + l and x → kx, and exactly the same calculationsapply to z → z + l and z → kz. It is invariant under z → −1/z, as can beshown by a calculation similar to, but shorter than, that given in Section5.7 for x → 1/x. However, it is not generally invariant under the M obius ¨transformation z → −z, because this replaces the cross-ratio by its complexconjugate. We can only say that Mobius transformations either leave the ¨cross-ratio invariant or change it to its complex conjugate.
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192 8 Non-Euclidean geometry Luckily, this does not matter, because we are interested in the cross-ratio only when the four points lie on a non-Euclidean line. It turns out thatthe cross-ratio of four points on a non-Euclidean line is real, and henceequal to its own complex conjugate. This is obvious when the points are pi, qi, ri, si on the upper y-axis,because, in this case, the cross-ratio equals the real number (r−p)(s−q) by (r−q)(s−p)cancellation of the i factors. It follows for any other non-Euclidean line Lby mapping the upper y-axis onto L by a Mobius transformation. ¨ • If L is another vertical line x = l, we map the upper y-axis to L by z → z + l. • If L is a semicircle with center on the x-axis, we first map the upper y-axis to x = 1 by z → z + 1, and then to the semicircle with ends at 0 and 1 by z → 1/z. Finally we map this semicircle to L by dilating it to the radius of L and then translating its center to the center of L. We know from the previous section that the transformations z → z + l,z → kz for k > 0, z → −z, and z → −1/z generate all Mobius transforma- ¨tions, so we have now proved that the cross-ratio of any four points on anon-Euclidean line is preserved by Mobius transformations. ¨ So far, so good, but distance is a function of two points, not four. If thecross-ratio is going to help us define distance, we need to specialize it to afunction of two variables. One of the beauties of a non-Euclidean line is that it lies between twoendpoints. The non-Euclidean line represented by the upper y-axis, forexample, consists of the points between 0 and ∞. The endpoints are notpoints of the line, but it is meaningful to include them in a cross-ratio, be-cause M¨ bius transformations apply to all complex numbers, and ∞. If we otake 0 and ∞ as the third and fourth members of the quadruple pi, qi, ri, sion the upper y-axis, then the cross-ratio of this quadruple simplifies as fol-lows: (r − p)(s − q) (r − p)(1 − q/s) = dividing top and bottom by s (r − q)(s − p) (r − q)(1 − p/s) r− p = because s = ∞ and 1/∞ = 0 r−q p = because r = 0. q
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8.6 Non-Euclidean distance 193Any M¨ bius transformation of the upper y-axis sends endpoints to end- opoints, as one can see from the generating transformations, but it is possiblefor 0 and ∞ to be exchanged. If r = ∞ and s = 0, we find that the cross- ¨ratio of pi, qi, ri, si is q/p, not p/q. Thus, q/p is not an invariant of M obiustransformations of the upper y-axis, but | log q | is, because log q = − log q p p pfor any p, q > 0. This prompts us to make the following definition.Distance on the upper y-axis. The non-Euclidean distance ndist(pi, qi)between points pi and qi on the upper y-axis is | log q |. p This definition of distance is appropriate for two reasons: • As already shown, non-Euclidean distance on the upper y-axis is in- variant under all M¨ bius transformations. o • Non-Euclidean distance is additive. That is, if pi, qi, ri lie on the upper y-axis in that order, then ndist(pi, ri) = ndist(pi, qi) + ndist(qi, ri). This is because r qr q r q r log = log = log + log = log + log p pq p q p q by the additive property of the logarithm function. It follows from this definition that the infinity of points 2n i, for integersn, are equally spaced along the upper y-axis, in the sense of non-Euclideandistance. The faces shown in Figure 8.9 are of equal size in this sense.The upper y-axis is not only infinite in the upward direction, but also in thedownward direction. There is infinite non-Euclidean distance between anyof its points and the x-axis. Thus, the upper y-axis satisfies Euclid's secondaxiom for "lines": Any segment of it can be "extended indefinitely." Having defined non-Euclidean distance on the upper y-axis, we canuse the axis as a "ruler" to measure the distance between two points in theupper half plane. Given any two points u and v, we find the unique non-Euclidean line L through u and v as described in Section 8.1, and thenmap L onto the upper y-axis by a Mobius transformation f as described ¨(in reverse) in the first part of this section. We take the non-Euclideandistance from u to v to be the non-Euclidean distance from f (u) to f (v),namely ndist( f (u), f (v)).
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194 8 Non-Euclidean geometry Figure 8.9: Faces of equal non-Euclidean size The quantity ndist( f (u), f (v)) does not depend on the Mobius trans- ¨formation f used to map L onto the upper y-axis. If g is another M obius ¨transformation mapping L onto the upper y-axis, then f g −1 is a M¨ bius otransformation that maps the upper y-axis onto itself and sends the pointsg(u) and g(v) to f (u) and f (v), respectively. Hence, ndist(g(u), g(v)) = ndist( f (u), f (v))by the invariance of non-Euclidean distance on the upper y-axis underM¨ bius transformations. oThe hidden geometry of the projective lineAs we mentioned in Section 7.1, Klein associated a "geometry" with eachgroup of transformations. We have set up the group of transformations ofthe half plane to be isomorphic to the group of transformations of RP 1 .Hence, the half plane and RP1 have isomorphic geometries in the senseof Klein, even though they seem very different. Indeed, we transferredgeometry from RP1 to the half plane mainly because of the difference:Geometry is much more visible in the half plane. Figure 8.7 is one illustration of this, and Figure 8.10 is another—aregular tiling of the half plane by fish that are congruent in the sense ofnon-Euclidean length. Figure 8.10 is essentially the picture Circle Limit I,by M. C. Escher, but mapped to the half plane by the transformation 1 − zi z→ . z−i
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8.6 Non-Euclidean distance 195 Figure 8.10: Half plane version of Escher's Circle Limit I By restricting M¨ bius transformations to the boundary of the half plane, ohalf plane geometry can be compressed into the geometry of RP 1 , eventhough RP1 has no concepts of length or angle. Conversely, length andangle emerge when RP1 is expanded to the half plane.ExercisesWe can now confirm the impression given by Figure 8.7, that each non-Euclideanline is infinite in both directions, as demanded by Euclid's second axiom.8.6.1 Show that the y-axis, and hence any non-Euclidean line, can be divided into infinitely many segments of equal non-Euclidean length.8.6.2 Find a M¨ bius transformation sending 0, ∞ to −1, 1, respectively, and hence o mapping the y-axis onto the unit semicircle.8.6.3 Using the transformation found in Exercise 8.6.2, find an infinite sequence of points on the unit semicircle that are equally spaced in the sense of non- Euclidean length.
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196 8 Non-Euclidean geometrySupposing that the equal faces shown in Figure 8.9 have non-Euclidean width ε ,which can be as small as we please, we can draw some interesting conclusionsabout the non-Euclidean distance between non-Euclidean lines.8.6.4 Show that the non-Euclidean distance between the lines x = 0 and x = 1 tends to zero as y tends to ∞.8.6.5 Show that the M¨ bius transformation z → 2/(1 − z) sends the unit circle o and the line x = 1 to the lines x = 1 and x = 0, respectively.8.6.6 Deduce from Exercise 8.6.5 that the non-Euclidean distance between the unit circle and the line x = 1 tends to zero as these non-Euclidean lines approach the x-axis.8.7 Non-Euclidean translations and rotationsLike the Euclidean plane, the half plane has isometries called translationsand rotations, which are products of two reflections. Their nature dependson whether the lines of reflection meet or have a common end. A translation is the product of reflections in non-Euclidean lines thatdo not meet and do not have a common end. A simple example is z → 2z,which √ the product of reflections in the circles with center 0 and radii is1 and 2. This translation maps each face in Figure 8.9 to the one aboveit. Any non-Euclidean translation maps a unique non-Euclidean line, calledthe translation axis, into itself. Also mapped into themselves are the curvesat constant non-Euclidean distance from the translation axis, which (fordistance > 0) are not non-Euclidean lines. For z → 2z, the translation axisis the y-axis and the equidistant curves are the Euclidean lines y = ax. Eachnon-Euclidean line perpendicular to the translation axis is mapped ontoanother such line. Figure 8.11 shows the translation axis, two equidistant curves (in gray),and some of their perpendiculars (on the left when the axis is vertical, andon the right when it is not). Notice that the equidistant curves in generalare Euclidean circles passing through the two ends of the translation axis.The translation moves each non-Euclidean perpendicular to the next. The product of reflections in two non-Euclidean lines that meet at apoint P is a non-Euclidean rotation about P. The point P remains fixedand points at non-Euclidean distance r from P remain at non-Euclideandistance r from P, since reflection is a non-Euclidean isometry. Hence,these points move on a non-Euclidean circle of radius r. It turns out thata non-Euclidean circle is a Euclidean circle, although its non-Euclidean
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8.7 Non-Euclidean translations and rotations 197 y 0 Figure 8.11: Non-Euclidean translationscenter (the point at constant non-Euclidean distance from all its points) isnot its Euclidean center. For example, if we take the product of the reflection z → −z in they-axis with the reflection z → −1/z in the unit circle, the result is a ro-tation through angle π about the point i where these two non-Euclideanlines meet. More generally, if we have two non-Euclidean lines throughP meeting at angle θ , then the product of reflections in these lines is a ro-tation about P through angle 2θ . Figure 8.12 shows four non-Euclideanlines through i and two non-Euclidean circles (in gray) with non-Euclideancenter at i. A rotation of π /4 about i moves each non-Euclidean line to thenext and maps each circle into itself. i 0 Figure 8.12: A non-Euclidean rotation about i
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198 8 Non-Euclidean geometry A limiting case of rotation is where the two lines of reflection do notmeet in the half plane, but have a common end P on the boundary R ∪ {∞}at infinity. Here P is a fixed point, each non-Euclidean line ending at P ismoved to another line ending at P, and each curve perpendicular to all theselines is mapped onto itself. This kind of isometry is called a limit rotation,and each curve mapped onto itself is called a limit circle or horocycle. The simplest example is the Euclidean horizontal translation z → z + 1,which is the product of reflections in the vertical lines x = 0 and x = 1/2.Each vertical line x = a is mapped to the line x = a + 1, and each horizontalline y = b is mapped onto itself. Thus, the horizontal lines y = b, which weknow are not non-Euclidean lines, are limit circles. 0 1 0 Figure 8.13: Limit rotations Like equidistant curves, limit circles can be Euclidean lines, but gen-erally they are Euclidean circles. Figure 8.13 shows the exceptional casez → z + 1, where the limit circles are the Euclidean horizontal lines (ingray), and the typical case z → z/(1 − z), where the limit circles are thegray circles tangential to the boundary at the fixed point z = 0. As in the previous pictures, the isometry moves each non-Euclideanline to the next, and maps each gray curve onto itself.Exercises8.7.1 Check that the product of reflections in the y-axis and the unit circle is z → −1/z, and that i is the fixed point of this map. √8.7.2 Show also that z → −1/z maps each circle of the form |z − ti| = t 2 − 1 onto itself.The limit rotation z → z/(1 − z) above is obtained by moving the limit rotationz → z + 1 about ∞ to a limit rotation about 0 with the help of the rotation z → −1/zthat exchanges 0 and ∞.
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8.8 Three reflections or two involutions 1998.7.3 If f (z) = z + 1 and g(z) = −1/z, show that g f g−1 (z) = z/(1 − z).8.7.4 Describe in words what g−1 , f , g in succession do to the half plane, and hence explain geometrically why g f g−1 has fixed point 0.8.8 Three reflections or two involutionsIt is possible to prove that each isometry of the half plane is the product ofthree reflections, following much the same approach as was used in Section3.7 to prove the three reflections theorem for the Euclidean plane. Thedetails of this approach are worked out in my book Geometry of Surfaces. However, our approach to isometries of the Euclidean plane began witha definition of Euclidean distance; we then had to find the transformationsthat leave it invariant. Here we know the isometries of the half plane—the M¨ bius transformations—so the only problem is to express them as o ¨products in some simple way. To do this, we can interpret Mobius trans-formations on RP1 , and exploit known theorems of projective geometry.Surprisingly, there is a theorem about RP1 that goes one better than thethree reflections theorem, namely the two involutions theorem from Veblenand Young's 1910 book Projective Geometry, p. 223. An involution is a transformation f such that f 2 is the identity. Thus,the involutions include the reflections, but some other transformations aswell, such as the function x → −1/x, which (when extended to the halfplane) represents a half turn about the point i. The name "involution" isone of many terms introduced into projective geometry by Desargues, andit is the only one that has stuck. To pave the way for the two involutions theorem (and the three reflec-tions theorem that follows from it), we first note three consequences of theresults in Section 5.8 about transformations of RP1 . • Any four points p, q, r, s ∈ RP1 can be mapped to q, p, s, r, respec- tively, by a linear fractional transformation. Notice that [p, q : r, s] = [q, p : s, r] because (r − p)(s − q) (s − q)(r − p) = . (r − q)(s − p) (s − p)(r − q) Hence, by the "criterion for four-point maps" in Section 5.8, there is a linear fractional f mapping p, q, r, s to q, p, s, r, respectively.
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200 8 Non-Euclidean geometry • If g is a linear fractional transformation that exchanges two points, then g is an involution. Suppose that p and q are two points with g(p) = q and g(q) = p. Let r be another point, not fixed by g, and suppose that g(r) = s. Because any linear fractional function is one-to-one, it follows that p, q, r, s are different. Hence, by the previous result, there is a linear fractional f mapping p, q, r, s to q, p, s, r, respectively. Because f agrees with g on the three points p, q, r, the functions f and g are identical by the "uniqueness of three-point maps" in Section 5.8. For any nonfixed point r of g, we therefore have g2 (r) = g(s) = f (s) = r, and if r is a fixed point, then g2 (r) = r obviously. Hence, g2 (x) = x for any x ∈ RP1 , and so g is an involution. • For any three points p, q, r, there is an involution that exchanges p, q and fixes r. By "existence of three-point maps" from Section 5.8, there is a linear fractional function g that sends p, q, r to q, p, r, respectively. Thus, g fixes r, and because it exchanges p and q, it is an involution by the previous result.Two involutions theorem. Any linear fractional transformation h of RP 1is the product of two involutions. If h = identity, then h = identity · identity, which is the product of twoinvolutions. If not, let p be a point not fixed by h, so h(p) = r = p,and let h(r) = q. Then q = r, because h−1 is also a linear fractional transfor-mation and hence one-to-one. If q = p, then h exchanges p and r. Hence,h is itself an involution by the second result above. We can therefore assume that p, q, r are three different points; in whichcase, the third result above gives a linear fractional involution f such that f (p) = q, f (q) = p, f (r) = r.Also, f h exchanges the two points p and r because f h(p) = f (r) = r, f h(r) = f (q) = p.
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8.8 Three reflections or two involutions 201Thus, f h is an involution too; call it g. Finally, applying f −1 to both sidesof f h = g, we get h = f −1 g.So h is the product of two involutions, f −1 = f and g, as required. We now consider M¨ bius transformations of the half plane, each of owhich is the unique extension of a linear fractional transformation of RP 1 .Such a function is determined by its values at three points on RP 1 , by"uniqueness of three-point maps." We use the same letter for a linear frac-tional transformation of RP1 and its extension to a M¨ bius transformation o ¨of the half plane, and we systematically use the fact that Mobius transfor-mations preserve non-Euclidean lines and angles.Three reflections theorem. Any M¨ bius transformation of the half plane ois the product of at most three reflections. The involution f in the proof above, which exchanges p, q and fixes r,necessarily maps the non-Euclidean line L from p to q into itself. Pointsof L near the end p are sent to points near the end q, and vice versa. Itfollows by continuity that some point u on L is fixed by f , and hence theunique non-Euclidean line M through u and ending at r is mapped intoitself by f . Also, because any M¨ bius transformation preserves angles, M omust be perpendicular to L (Figure 8.14). Thus, f has the same effect onp, q, r as reflection in the line M , so f is this reflection by "uniqueness ofthree-point maps." L u M p q r Figure 8.14: Lines involved in the involution f
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202 8 Non-Euclidean geometry Now consider the involution g, which is associated with a similar pairof lines L and M . Only the names of their ends are different, but thereader is invited to draw them to keep track. By the argument just used for f , the involution g is a reflection if ithas a fixed point on RP1 . In any case, g maps the line L with ends pand r into itself, exchanging the ends, so g has a fixed point u on L bythe argument just used for f . Also, because g preserves angles, g mapsthe non-Euclidean line M through u and perpendicular to L into itself.Thus, if g has no fixed point on RP1 , it necessarily exchanges the ends sand t of M . But then g has the same effect on the three points p, r, s as theproduct of reflections in L and M , so g is this product of reflections, by"uniqueness of three-point maps" again. Thus, f g, which is an arbitrary M¨ bius transformation by the theorem oabove, is the product of at most three reflections.ExercisesThe argument above appeals to "continuity" to show the existence of a fixed pointon a non-Euclidean line whose ends are exchanged by an involution. This ar-gument is valid, and it may be justified by the intermediate value theorem, wellknown from real analysis courses. However, some readers may prefer an actualcomputation of the fixed point. One way to do it is as follows. Suppose f (x) = ax+b and that f (p) = q, f (q) = p. cx+d8.8.1 Deduce that a = −d and b = cpq − a(p + q), so that f has the form a(x − p − q) + cpq k(x − p − q) + pq f (x) = = if c = 0. cx − a x−k8.8.2 Solve the equation k(x − p − q) + pq x= , x−k and hence show that the fixed points of f are u = k± (k − p)(k − q).8.8.3 Assuming that (k − p)(k − q) < 0, so one fixed point is in the upper half plane, show that its distance from the center (p + q)/2 of the semicircle with ends p and q is |(p − q)/2|.8.8.4 Deduce from Exercises 8.8.1–8.8.3 that f has a fixed point on the non- Euclidean line with ends p and q.
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8.9 Discussion 2038.9 DiscussionThe non-Euclidean parallel hypothesisIt has often been said that the germ of non-Euclidean geometry is in Eu-clid's own work, because Euclid recognized the exceptional character ofthe parallel axiom and used it only when it was unavoidable. Later geome-ters noted several plausible equivalents of the parallel axiom, such as • the equidistant curve of a line is a line, • the angle sum of a triangle is π , • similar figures of different sizes exist,but no outright proof of it from Euclid's other axioms was found. On thecontrary, attempts to derive a contradiction from the existence of manyparallels—what we will call the non-Euclidean parallel hypothesis—ledto a rich and apparently coherent geometry. This is the geometry we havebeen exploring in the half plane, now called hyperbolic geometry. Hyperbolic geometry diverges from Euclidean geometry in the oppo-site direction from spherical geometry—for example, the angle sum of atriangle is < π , not > π —but the divergence is less extreme. The "lines" ofspherical geometry violate all three of Euclid's axioms about lines, whereasthe "lines" of hyperbolic geometry violate only the parallel axiom. The first theorems of hyperbolic geometry were derived by the Ital-ian Jesuit Girolamo Saccheri in an attempt to prove the parallel axiom.In his 1733 book, Euclides ab omni naevo vindicatus (Euclid cleared ofevery flaw), Saccheri assumed the non-Euclidean parallel hypothesis, andsought a contradiction. What he found were asymptotic lines: lines thatdo not meet but approach each other arbitrarily closely. This discoverywas curious, and more curious at infinity, where Saccheri claimed that theasymptotic lines would meet and have a common perpendicular. Findingthis "repugnant to the nature of a straight line," he declared a victory forEuclid. But the common perpendicular at infinity is not a contradiction, andindeed (as we now know) it clearly holds in the half plane. There arenon-Euclidean lines that approach each other arbitrarily closely in non-Euclidean distance, such as the unit semicircle and the line x = 1, andthey have a common perpendicular at infinity—the x-axis. Saccheri hadunwittingly discovered not a bug, but a key feature of hyperbolic geometry.
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204 8 Non-Euclidean geometry The non-Euclidean geometry of the hyperbolic plane began to takeshape in the early 19th century. A small circle of mathematicians aroundCarl Friedrich Gauss (1777–1855) explored the consequences of the non-Euclidean parallel hypothesis, although Gauss did not publish on the sub-ject through fear of ridicule. Gauss was the greatest mathematician of histime, but he was unwilling to publish "unripe" work, and he evidently feltthat non-Euclidean geometry lacked a solid foundation. He knew of noconcrete interpretation, or model, of non-Euclidean geometry, and in fact,none was discovered in his lifetime. It is a great irony that some of hisown discoveries—in the geometry of curved surfaces and the geometry ofcomplex numbers—can provide such models. The first to publish comprehensive accounts of non-Euclidean geom-etry were Janos Bolyai in Hungary and Nikolai Lobachevsky in Russia.Around 1830 they discovered this geometry independently and became itsfirst "true believers." The richness and coherence of their results convincedthem that they had discovered a new geometric world, as real as the worldof mainstream geometry and not needing its support. In a sense, they wereright, but in their enthusiasm, they failed to notice another new geometrictheory that could have been a valuable ally. Gauss's Disquisitiones gen-erales circa superficies curvas (General investigations on curved surfaces)was published in 1827, but neither Bolyai, Lobachevsky, or Gauss noticedthat it gives models of non-Euclidean geometry, at least in small regions. The fundamental concept of Gauss's surface theory is the curvature,a quantity that is positive (and constant) for a sphere, zero for the planeand cylinder, and negative for surfaces that are "saddle-shaped" in theneighborhood of each point. In the Disquisitiones, Gauss investigatedthe relationship between the curvature of a surface and the behavior ofits geodesics, which are its curves of shortest length and hence its "lines."He found, for example, that a geodesic triangle has • angle sum > π on a surface of positive curvature, • angle sum π on a surface of zero curvature, • angle sum < π on a surface of negative curvature.Moreover, if the curvature is constant and nonzero, then, in any geodesictriangle, (angle sum −π ) is proportional to area. These results must havereminded Gauss of things he already knew in non-Euclidean geometry, soit is surprising that he failed to capitalize on them.
215.
8.9 Discussion 205Close encounters between the actual and the hypotheticalThe near agreement between geometry on surfaces of constant negativecurvature and non-Euclidean geometry was the first of several close en-counters over the next few decades. But usually the actual and hypotheticalgeometries passed each other like ships in a thick fog. For example, in the late 1830s, the German mathematician FerdinandMinding worked out the formulas of negative-curvature trigonometry. Hefound that they are like those of spherical trigonometry, but with hyper-bolic functions in place of circular functions. At about the same time (andin the same journal!), Lobachevsky showed that the same formulas holdfor triangles in his non-Euclidean plane. This would have been a nice timeto introduce the name "hyperbolic geometry" for the non-Euclidean ge-ometry of constant negative curvature, but apparently neither Minding norLobachevsky realized that they might have been talking about the samething. Perhaps they were aware of a difficulty with the known surfacesof negative curvature: They are incomplete in the sense that their "lines"cannot be extended indefinitely. Hence, they fail to satisfy Euclid's secondaxiom for lines. The simplest surface of constant negative curvature is called the pseu-dosphere (somewhat misleadingly, because constant curvature is about allit has in common with the sphere). It is more accurately known as thetractroid, because it is the surface of revolution of the curve known as thetractrix. The defining property of the tractrix is that its tangent has constantlength a between the curve and the x-axis (left half of Figure 8.15).a a Figure 8.15: The tractrix and the tractroid
216.
206 8 Non-Euclidean geometry It is an unavoidable consequence of this definition that the tractrix hasa singularity where the tangent becomes perpendicular to the x-axis. Thetractroid likewise has an edge (like the rim of a trumpet), beyond which itcannot be smoothly continued. Hence, geodesics on the tractroid cannotbe continued in both directions. In fact, the only geodesics on the tractroidthat are infinite in even one direction are the rotated copies of the originaltractrix. This problem is typical of what happens when one tries to con-struct a complete surface of constant negative curvature in ordinary space.The task was eventually shown to be impossible by Hilbert in 1901, but anobstacle to the construction of such surfaces was sensed much earlier. In 1854, Gauss's student Bernhard Riemann showed a way round theobstacle by proposing an abstract or intrinsic definition of curved spaces—one that does not require a "flat" space to contain the "curved" one. Thisidea made it possible to define a complete surface, or indeed a completen-dimensional space, of constant negative curvature. Riemann did exactlythis, but once again non-Euclidean geometry sailed by unnoticed, as far aswe know. (The elderly Gauss was very moved by Riemann's account ofhis discoveries. Whether he saw in them a vindication of non-Euclideangeometry, we will probably never know.) Another close encounter occurred in 1859, when Arthur Cayley devel-oped the concept of distance in projective geometry. He found that there isan invariant length for certain groups of projective transformations, such asthose that map the circle into itself. In effect, he had discovered a model ofthe non-Euclidean plane, but he did not notice that his invariant length hadthe same properties as non-Euclidean length. Despite the efforts of Bolyaiand Lobachevsky, non-Euclidean geometry remained an obscure subjectuntil the 1860s.Models of non-Euclidean geometryRiemann died in 1859, and his ideas first bore fruit in Italy, where he hadspent a lot of time in his final years. His most important successor wasEugenio Beltrami, who in 1868 finally brought non-Euclidean geometryand negative curvature together. Beltrami's first discovery, in 1865, established the special role of con-stant curvature in geometry: The surfaces of constant curvature are pre-cisely those that can be mapped to the plane in such a way that geodesicsgo to straight lines. The simplest example is the sphere, whose geodesicsare great circles, the intersections of the sphere with the planes through
217.
8.9 Discussion 207its center. Great circles can be mapped to straight lines by projecting thesphere onto the plane from its center (Figure 8.16). O Figure 8.16: Central projection of the sphere The geodesic-preserving map of the tractroid sends it to a wedge-shapedportion of the unit disk. The tractrix curves on the tractroid go to line seg-ments ending at the sharp end of the wedge (Figure 8.17). −→ Figure 8.17: Geodesic-preserving map of the tractroid Although this map preserves "lines," it certainly does not preservelength. Each tractrix curve has infinite length; yet it is mapped to a fi-nite line segment in the disk. The appropriate length function for the diskassigns a "pseudodistance" to each pair of points, equal to the geodesic dis-tance between the corresponding points on the tractroid. We do not needthe formula here; the important thing is that pseudodistance makes sense on
218.
208 8 Non-Euclidean geometrythe whole open disk, that is, for all points inside the boundary circle. Thecurve of shortest pseudodistance between any two points in the open diskis the straight line segment between them, and the pseudodistance betweenany point and the boundary is infinite. In 1868, Beltrami realized that this abstraction and extension of thetractroid is an interpretation of the non-Euclidean plane: a surface in whichthere is a unique "line" between any two points, "lines" are infinite, andthe non-Euclidean parallel hypothesis is satisfied. Figure 8.18 shows why:Many "lines" through the point P do not meet the "line" L . P L Figure 8.18: Why the non-Euclidean parallel hypothesis holds Beltrami wrote two epic papers on models of non-Euclidean geometryin 1868, and English translations of them may be found in my book Sourcesof Hyperbolic Geometry. The first paper arrives at the non-Euclidean planeas an extension of the tractroid through the idea of "unwinding" infinitelythin sheets wrapped around it. (The dotted paths in the right half of Fig-ure 8.17, all converging to the endpoint of the wedge, are the limit circlestraced by circular sections of the tractroid as they unwind.) Beltrami was atpains to be as concrete as possible, because Riemann's ideas were not wellunderstood or accepted in 1868. However, at the end of the paper, Beltramiforeshadows the more abstract and general approach he intends to take inhis second paper: where the most general principles of non-Euclidean geometry are considered independently of their possible relations with ordinary geometric entities. In the present work we have been
219.
8.9 Discussion 209 interested mainly in offering a concrete counterpart of abstract geometry; however, we do not wish to omit a declaration that the validity of the new order of concepts does not depend on the possibility of such a counterpart. In the second paper, Beltrami vindicates this ringing endorsement ofRiemann's ideas with whole families of models of non-Euclidean geometryin any number of dimensions. Among them is the half-plane model usedin this chapter, and its generalization to three dimensions, the "half-spacemodel." The half-space model has • "points" that are the points (x, y, z) ∈ R3 with z > 0, • "lines" that are the vertical Euclidean half lines in R3 and the vertical semicircles with centers on the plane z = 0, • "planes" that are the vertical Euclidean half planes in R3 and hemi- spheres with centers on z = 0.It turns out that non-Euclidean distance on a plane z = a is a constant mul-tiple of Euclidean distance. This surprising result gives probably the sim-plest proof of a result first discovered by Friedrich Wachter, a member ofGauss's circle, in 1816: Three-dimensional non-Euclidean geometry con-tains a model of the Euclidean plane. Another model of the hyperbolic plane, discovered by Beltrami, is theconformal disk model. It is like the half plane in being angle-preserving,but unlike it in being finite. Its "points" are the interior points of the unitdisk (the points z with |z| < 1, if we work in the plane of complex numbers),and its "lines" are circular arcs perpendicular to the unit circle. Figure 8.19,which is the original M. C. Escher picture Circle Limit I, can be viewed asa picture of the conformal disk model. The fish are arranged along "lines,"and they are all of the same hyperbolic length. As mentioned in connectionwith Figure 8.10, the transformed Circle Limit I, the function 1 − zi z→ z−imaps the conformal disk model onto the half-plane model. It should be stressed that all models of non-Euclidean geometry, in agiven dimension, are isomorphic to the half-space model. For example,models of the non-Euclidean plane satisfy Hilbert's axioms (Section 2.9)
220.
210 8 Non-Euclidean geometry Figure 8.19: The conformal disk modelwith the parallel axiom replaced by the non-Euclidean parallel hypothesis.And Hilbert in his Grundlagen showed that the "lines" satisfying theseaxioms have "ends" that behave like the points of RP1 . Thus, any non-Euclidean plane is essentially the same as the half plane discussed in thischapter, so we can call it the non-Euclidean plane or the hyperbolic plane.Non-Euclidean realityIn Beltrami's original model, the open disk in which "lines" are line seg-ments ending on the unit circle, isometries map Euclidean lines to Eu-clidean lines, and so they are projective maps. For this reason, the modelis often called the projective disk. It can also be constructed by methods ofprojective geometry, and indeed this is essentially what Cayley did in 1859.The first to connect all the dots between projective and non-Euclidean ge-ometry was Klein in 1871. An English translation of his paper may befound in Sources of Hyperbolic Geometry. Although Klein had only tofill a few technical gaps, it was he who first made the important concep-tual point that a model of non-Euclidean geometry ensures that the non-Euclidean parallel hypothesis is not contradictory. Hence, Euclid's paral-lel axiom does not follow from his other axioms.
221.
8.9 Discussion 211 In 1872, Klein also made the great advance of linking geometries togroups of transformations. This link gives a deeper reason for the presenceof non-Euclidean geometry in projective geometry: The real projective lineand the non-Euclidean plane have isomorphic groups of transformations. The group of the non-Euclidean plane was first described explicitly bythe French mathematician Henri Poincar´ in 1882, along with its interpre- etation as the group of M¨ bius transformations of the half plane.The relevant oparts of his work may also be found in Sources of Hyperbolic Geometry.Poincar´ became interested in non-Euclidean geometry when he noticed ethat some functions of a complex variable have non-Euclidean periodicity. An ordinary periodic function, such as cos x, has Euclidean periodicityin the sense that its values repeat when x undergoes the Euclidean transla-tion x → x + 2π . A complex function can have non-Euclidean periodicity,and one example is the modular function j(z). Its definition is too long toexplain here, but its periodicity is simple: The values of j(z) repeat underthe M¨ bius transformations z → z + 1 and z → −1/z. As we know, these oare isometries of the half plane. If one applies them over and over, to thelines x = 0, x = 1, and the unit semicircle, they produce the non-Euclideanregular tessellation shown in Figure 8.20. −1 0 1 Figure 8.20: The modular tessellation The modular function and its periodicity were already part of mathe-matical reality, having been known to Gauss and others since early in the19th century. But Poincar´ was the first to see its non-Euclidean symmetry. eHe used non-Euclidean geometry to study large classes of functions whosebehavior had until then seemed intractable. Poincar´ was also the first to eview the half plane as an extension of the real projective line, as we have
222.
212 8 Non-Euclidean geometrydone in this chapter. In fact, he went much further, noticing that the half-space model of non-Euclidean space is a natural extension of the complexprojective line CP1 = C ∪ {∞}. Just as the real projective line R ∪ {∞} comes with the linear fractionaltransformations ax + b x→ , where a, b, c, d ∈ R and ad − bc = 0, cx + dthe complex projective line C ∪ {∞} comes with the linear fractional trans-formations az + b z→ , where a, b, c, d ∈ C and ad − bc = 0. cz + dAnd just as the linear fractional transformations of R ∪ {∞} extend toM¨ bius transformations of the half plane, the linear fractional transfor- omations of C ∪ {∞} extend to M¨ bius transformations of the half space, ofor which there is likewise an invariant non-Euclidean distance, and thenon-Euclidean "lines" and "planes" mentioned above. It is a great advantage to have a concept of distance, even if the dis-tance is non-Euclidean and one needs an extra dimension to acquire it.By passing to the third dimension, Poincar´ could understand transforma- etions of C whose behavior is almost incomprehensible when viewed in theplane. Understanding comes by viewing these transformations as com-pressed versions of isometries of non-Euclidean space, which behave quitesimply (like isometries of the half plane). Thus, expanding from a pro-jective line to a non-Euclidean space is not just an interesting theoreticalpossibility—it is sometimes the best way to understand the mysteries ofprojection.
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Project MathWORKS! reaches out to high school math and science instructors, introduces them to operations research methods that help solve real-world problems, and excites them into teaching math in ways that help improve student learning. Instructors and undergraduates at the United States Military Academy conduct workshops that relate mathematical ideas to familiar context - such as reducing waiting time in a fast food restaurant, maximizing profit for a manufacturing company, or choosing of the most appropriate college upon graduating from high school - so that high school instructors can then incorporate the material into their classrooms. MathWORKS! seeks to introduce practical methods and tools that help promote the love of learning, dispel the notion that mathematics is incomprehensible, and empower teachers and students alike with the capacity and imagination to leverage math in the solution of everyday problems. In short, MathWORKS! aims to show that math does actually work
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Course Description: Intended primarily for students
who need only one mathematics foundation course, this course includes topics
selected from (but not limited to) graph theory, geometry, number theory,
algebra, combinatorics, and statistics. Students
address topical applications from management sciences, social sciences,
environmental sciences, information technologies, and the arts, with an
emphasis on quantitative reasoning.
Text:Excursions in Modern Mathematics, 7th
edition, by Tannenbaum/Arnold, Prentice Hall,
with MathXL.
Required supplies: You will need a
calculator, but it does not need to
be a graphing calculator.Any calculator
that can handle arithmetic operations and exponents, with parentheses to set
order of operations, will be fine.
Prerequisites: Grade of C or better in MA
097 (Intermediate Algebra) or equivalent, or appropriate score on the Math
Assessment Test, or approval of the professor.NOTE:If you enrolled and do not
have the prerequisites met, you may be dropped from the class.
Course layout:We will be covering 3 of the 4 parts in the
textbook - "Social Choice", "Management Science", and "Shape, Growth, and Form".Each part consists of 4 loosely related
chapters. We may speed up or slow down
at various times, but we will typically spend approximately 1 week on each
chapter.
Grades:You grade for the course will be based on 5
components:
ØQuizzes, which will follow each chapter.I will drop the 1 lowest quiz, and average the
others.
ØHomework, which will be due weekly.Most
answers will be submitted on MathXL, but I will also
collect work in which you show step-by-step how you arrived at your solutions,
along with explanations in the form of neatly drawn pictures and/or complete
English sentences.
Ø3 unit tests.
I
will drop the lowest of these 5 grades, and average the other 4 equally.If your average is then 90 or higher, you
will earn an 'A' for the course; 80-89 will earn a 'B'; 70 - 79 a 'C'; 60-69 a
'D'; and 0-59 an 'F'.
Make-up Policy:If you miss a class, it is your
responsibility to contact me as soon as possible so that you can be prepared
for the next class session – do not wait until the next class meeting
to ask for missed work!I
will often be able to e-mail you copies of the missed activities so that you
don't fall even further behind.
Late
homework will not be accepted.If you
know in advance that you will miss a quiz or test due to a college sponsored
activity or religious holiday, I will try to make arrangements for you to take
an alternate test before the test day. No make-ups will be given after the fact
for any reason.
Important dates:September 17:last day to drop the class (no 'W')
November 12:last
day to drop the class (receiving a grade of 'W')
December 11:final exam:10:15-12:15a.m.
E-mail communications:Communications for this class, when necessary, will be made via college
e-mail.Be sure to check yours regularly!
Inclement weather and other emergencies
If
inclement weather forces the College or any campus or College facility to
suspend classes or close, public service announcements will be provided to
local radio and television stations as early as possible. You may also call MC
at 240-567-5000 or check the college website
to verify MC school closings.Any exams
planned on days classes are suspended will be administered at the first class
meeting once classes resume.Note that
the Montgomery County Public Schools (MCPS) and Montgomery College do not
follow the same school closing procedures.
In the event class is unexpectedly cancelled, please
check your college e-mail for information and assignments.I will do my best to keep class going even if
we can't meet in person.
Math Science Center:Located in 02 Macklin Tower,
this is where you can go to borrow a math book, work in a group study area,
work in a quiet study area, use a computer for a math or science class, borrow
a calculator, or, best of all, get free tutoring.The phone number there is 240-567-5200, and
the hours are: Mon. – Thurs. 8am – 8pm, Fri. 8am – 4pm, Sat.
10am – 3pm.
Accommodations for Students with Disabilities:Disability
Support Services(240-567-5058)
Any
student who may need an accommodation due to a disability, please make an
appointment to see me during my office hour. A letter from Disability Support
Services (CB122) authorizing your accommodations will be needed. Any student
who may need assistance in the event of an emergency evacuation must identify
to the Disability Support Services Office; guidelines for emergency evacuations
for individuals with disabilities are found at:
This
course fulfills a General Education Program Math foundationrequirement. Montgomery
College's General Education Program is designed to ensure that students have
the skills, knowledge and attitudes to carry them successfully through their
work and personal lives. This course provides multiple opportunities to develop
the following competencies: written and oral communication, scientific and
quantitative reasoning, critical analysis and reasoning, and technological
competency. For more information, please see
I would love nothing better
than to have everyone earn a passing grade, or better yet an 'A'.However, the grades that I give will be those
that are earned, as described above.If you "absolutely, positively, must
pass this class"to graduate / for
your job / to keep your full-time status/ so your dog doesn't run away / or for
any other reason, the time to think about that is now, not after you have dug yourself into too deep a hole.
I expect that when you come
to class, you are doing so with the intention of learning.I will do my part to make the atmosphere as
conducive to learning as possible, and I ask you to do the same.Feel free to ask questions, or to answer
questions for other students.But please
keep unnecessary distractions down, and,
| 677.169 | 1 |
Elementary Geometry for College Students
9781439047903
ISBN:
1439047901
Edition: 5 Pub Date: 2010 Publisher: Brooks Cole
Summary: If you want to rent Elementary Geometry for College Students online, we can help you. This text book, written by Daniel C Alexander and Geralyn M Koeberlein, was published by Brooks Cole in 2010. Now you can get cheap Elementary Geometry for College Students here in its 5th edition for an affordable price. We specialize in providing great deals that are heavily discounted for previously owned copies. You can buy Elem...entary Geometry for College Students online here for a price far lower than you might think, and sell back later on too. We provide the whole deal for every college student.
Alexander, Daniel C. is the author of Elementary Geometry for College Students, published 2010 under ISBN 9781439047903 and 1439047901. Four hundred seventy Elementary Geometry for College Students textbooks are available for sale on ValoreBooks.com, seventy six used from the cheapest price of $36.88, or buy new starting at $91.79The primary subject of this book was getting across the basics of geometry. I found this book very effective because it gave you review and test questions/answers which was very helpful when preparing for a test. Most of the time teachers will use problems from here for quizzes or test so practicing these problems is crucial. The examples in each chapter are very helpful as well because they give a break down of each problem.
If I could change one thing about this book it would be to provide all the answers for every other problem. Sometimes in certain chapters, answers to the odd problems would be missing. But other than that this book was very helpful in helping me pass with an A this semester!
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Abstract: Although we no longer doubt the legitimacy of split domains as a function, students still have difficulty perceiving a graph having a shape like that of Euler and d'Alembert's string as the graph of a single function rather than of several joined functions (Vinner 1989). Given the centrality of the function concept in the school curriculum (NCTM 1989; Amit and Koren 1993), one cannot underestimate the importance of exposing students to this type of function. A method of introducing students to this concept while furnishing the teacher with material that has its own intrinsic interest (Satianov 1995).
The National Council of Teachers of Mathematics is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research.
| 677.169 | 1 |
Why Head First Algebra?
Let's just be honest here - Head First Algebra is a whole new direction - it's a long way from Java and Software Development, right? The truth is, Algebra is one of those subjects that just confuses people. You may be taking it for the first time in school, learning it on your own, or trying to help your own kid - it's hard for everyone.
There are three big reasons that Algebra needs to be taken on Head First:
Algebra is important.
It may not seem that way when the most boring teacher in the world is teaching it - but Algebra is everywhere. And knowing it helps you in the real world every day - buying cars, figuring out sale prices on clothes, and making any kind of financial decision.
Algebra is hard.
Let's face it - this stuff is confusing. When did a letter work like a number, anyway? It all seems pretty abstract and if you can't get grounded, it's useless.
Algebra is the tipping point!
When you start working with Algebra, you move into a whole new arena of math. It turns out that all of the adding, multiplying, and fractions you did before is just what you need to really get going. Algebra is where math goes from crunching numbers to being about logic, problem solving, and thinking. Algebra problems also become significantly more complex which means that they model the REAL WORLD!
If you can't get past Algebra, you are cut off from all of higher math: Geometry, Calculus, Algebra 2, Trigonometry and beyond.
What does that get you?
You have no interest in Geometry you say? What about putting carpet on the floor in your house or painting a wall? To figure out how much carpet or paint to buy you need to figure out area.
Who cares about Statistics? Every medicine you take tells you the statistics on the odds of your arm falling off if you take it. After Stats, you'll know that if there's 1 in 2 chance of that you should change medicine.
Calculus? That's pretty high level, maybe you won't use it. Unless you'd like a career in any of the following areas:
But why does that mean it should be a Head First book? There are lots of Algebra books out there.
That's true. But did you ever notice that with most textbooks you skip the part that explains the concept and just do the exercises? The teacher explains the ideas and the book is just for practice, right?
| 677.169 | 1 |
9780201726343
ISBN:
0201726343
Edition: 5 Pub Date: 2003 Publisher: Pearson
Summary: This text is organised into 4 main parts - discrete mathematics, graph theory, modern algebra and combinatorics (flexible modular structuring). It includes a large variety of elementary problems allowing students to establish skills as they practice.
Ralph P. Grimaldi is the author of Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, published 2003 under ISBN 9780201726343 and 0...201726343. Six hundred forty five Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition textbooks are available for sale on ValoreBooks.com, sixty six used from the cheapest price of $102.03, or buy new starting at $169.62201726343-4-0-3 Orders ship the same or next business day. Expedited shipping within U.S. [more]
May include moderately worn cover, writing, markings or slight discoloration. SKU:9780201726
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Elementary Statistics-Text - 8th edition
Summary: ELEMENTARY STATISTICS: A STEP BY STEP APPROACH is for introductory statistics courses with a basic algebra prerequisite. The book is non-theoretical, explaining concepts intuitively and teaching problem solving through worked examples and step-by-step instructions. In recent editions, Al Bluman has placed more emphasis on conceptual understanding and understanding results, along with increased focus on Excel, MINITAB, and the TI-83 Plus and TI-84 Plus graphing calculators; computing ...show moretechnologies commonly used in such Good condition. May or may not contain highlighting. Expedited shipping available.
$15.94 +$3.99 s/h
Good
HPB-Ohio Columbus, OH
Hardcover Good Item may show signs of shelf wear. Pages may include limited notes and highlighting. Includes supplemental or companion materials if applicable. Access codes may or may not work. Con...show morenecting readers since 1972. Customer service is our top priority. ...show less
$15.9798 +$3.99 s/h
Good
Back Alley Textbooks WI Sun Prairie, WI
2011 Hardcover Used book / Very good condition / may have minimal highlighting, underlining or writing / mild cover wear. Quick
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Designed for advanced undergraduate and beginning graduate students in linear or abstract algebra, Advanced Linear Algebra covers theoretical aspects of the subject, along with examples, computations, and proofs. It explores a variety of advanced topics in linear algebra that highlight the rich...
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Algebra and Trigonometry - 7th edition
Summary: For undergraduate courses in Algebra and Trigonometry with optional Graphing Calculator usage.
The Seventh Edition of this dependable text retains its best features -- accuracy, precision, depth, strong student support, and abundant exercises -- while substantially updating content and pedagogy. After completing the book, students will be prepared to handle the algebra found in subsequent courses such as finite mathematics, business mathematics, and engine...show moreering calculus. ...show less
Angles and Their Measure. Right Triangle Trigonometry. Computing the Values of Trigonometric Functions of Given Angles. Trigonometric Functions of General Angles. Unit Circle Approach; Properties of the Trigonometric Functions. Graphs of the Sine and Cosine Functions. Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions. Phase Shift; Sinusoidal Curve Fitting Slightly wear and some writings Used-Acceptable Slightly wear and some writings.53 +$3.99 s/h
Good
Wonder Book Frederick, MD
Prentice Hall, 01/15/2004, Hardcover, Good condition. 7th edition.
$3.573.5757 +$3.99 s/h
Good
Goodwill Industries Miami, FL
Good Used-Good
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A Level
Edexcel AS and A Level Modular Mathematics
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Publishing for all modules, to provide the best possible match to the 2008 Edexcel GCE maths specification, Edexcel AS and A Level Modular Mathematics motivates students by making maths more accessible.
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More About
This Textbook
Overview
With the continued advance of computing power and accessibility, the view that "real mathematicians don't compute" no longer has any traction for a newer generation of mathematicians. The goal in this book is to present a coherent variety of accessible examples of modern mathematics where intelligent computing plays a significant role and in so doing to highlight some of the key algorithms and to teach some of the key experimental approaches.
What People Are SayingHerbert S. Wilf
--(Prof. Herbert S. Wilf, author of generatingfunctionology)
Richard E. Crandall--(Richard E. Crandall, Apple Distinguished Scientist, Apple, Inc.)
Editorial ReviewsMAA Reviews
Computers will change the face of mathematics . . . There certainly is a new generation rising using computers adroitly to expand the boundaries of pure mathematics.
Science News
Based on a short course taught by the authors, this book describes the shift in the way in which mathematics has been practiced over the past 20 years - a shift that has moved from theory to computation
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More About
This Textbook
Overview
The wide-ranging debate brought about by the calculus reform movement has had a significant impact on calculus textbooks. In response to many of the questions and concerns surrounding this debate, the authors have written a modern calculus textbook, intended for students majoring in mathematics, physics, chemistry, engineering and related fields. The text is written for the average student — one who does not already know the subject, whose background is somewhat weak in spots, and who requires a significant motivation to study calculus.
The authors follow a relatively standard order of presentation, while integrating technology and thought-provoking exercises throughout the text. Some minor changes have been made in the order of topics to reflect shifts in the importance of certain applications in engineering and science. This text also gives an early introduction to logarithms, exponentials and the trigonometric functions. Wherever practical, concepts are developed from graphical, numerical, and algebraic perspectives (the "Rule of Three") to give students a full understanding of calculus. This text places a significant emphasis on problem solving and presents realistic applications, as well as open-endedGreat On Line Resources
This textbook hasn't been too difficult to follow, and I could reccommend it. I like the on line resources that come with this book. It includes a bind-in OLC (on line Learning Center) card in it, so that you can register for the OLC from McGraw Hill. The OLC has a solutions manual, and great topics to help demistify Calculus. This book inlcludes proofs to most theorems, but a few theorems do not have proofs that follow them; the authors have left those as an exercise. This may be a negative for some, but a positive for others.
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Making Mathematics Practical: An Approach to Problem Solving
Publisher:
World Scientific
Number of Pages:
155
Price:
42.00
ISBN:
978-981-4355-00-1
When in the area of problem solving there are the "how to solve it" steps and, for educators, there are the "how to teach how to solve it" steps. The first set of steps was definitively established by George Pólya in his classic work How to Solve It, a book that should be required reading in all mathematics curricula, especially those with the goal of teaching teachers.
The second set of steps is a bit harder because learning is such an individual thing. There are cases in which what is clear to half of a class might as well, for the rest, have been stated in ancient Mayan. This is especially the case in mathematics classes, where some people have to overcome significant insecurities before they can even begin to understand and execute a problem solving strategy.
This book is a textbook designed to teach the educator how to pedagogically present the solving of problems. A small set of well known problems are presented and then thoroughly sliced, diced and dissected into minute segments. Some of the problems are:
Key heuristics, plans of attack, solutions and possible student responses or errors are given for each problem.
The materials in this book have been used in the Singaporean secondary schools. Given that Singapore traditionally ranks first or second in the world and has held that position for almost 20 years, this book should be taken very seriously. Sometimes there is no substitute for grinding away at a concept. That is what often must be done in mathematical problem solving. Repeating the basic tactics of problem solving for different problems is a necessity and this book contains all the necessary details
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This textbook on linear algebra includes the key topics of the subject that most advanced undergraduates need to learn before entering graduate school. All the usual topics, such as complex vector spaces, complex inner products, the Spectral theorem for normal operators, dual spaces, the minimal polynomial, the Jordan canonical form, and the rational... more...
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This book introduces algebraic, combinatorial, and geometric methods for hidden tree models. It presents a focused introduction to some concepts in algebraic statistics and shows how the methods can be applied in statistics. The book also gives a broad overview of the current research on hidden tree models. Readers will gain a complete geometric and... more...
Elementary Number Theory takes an accessible approach to teaching students about the role of number theory in pure mathematics and its important applications to cryptography and other areas.
The first chapter of the book explains how to do proofs and includes a brief discussion of lemmas, propositions, theorems, and corollaries. The core ofThis unique volume presents a fruitful and beautiful mathematical world hidden in Caianiello's neuronic equations, which describe the instantaneous behavior of a model of a brain or thinking machine. The detailed analysis from a viewpoint of "dynamical systems", even in a single neuron case, enables us to obtain amazingly good... more...
One of the most exciting new subjects in Algebraic Number Theory and Arithmetic Algebraic Geometry is the theory of Euler systems. Euler systems are special collections of cohomology classes attached to p-adic Galois representations. Introduced by Victor Kolyvagin in the late 1980s in order to bound Selmer groups attached to p-adic representations,... more...
This book, part of the series?Contributions in Mathematical and Computational Sciences,?reviews recent developments in the theory of vertex operator algebras (VOAs) and their applications to mathematics and physics.The mathematical theory of VOAs originated from the famous monstrous moonshine conjectures of J.H. Conway and S.P. Norton, which predicted... more...
An exciting approach to the history and mathematics of number theory ". . . the author's style is totally lucid and very easy to read . . .the result is indeed a wonderful story." —Mathematical Reviews Written in a unique and accessible style for readers of varied mathematical backgrounds, the Second Edition of Primes... more...
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Overview
This book is an introduction to classical knot theory. Topics covered include: different constructions of knots, knot diagrams, knot groups, fibred knots, characterisation of torus knots, prime decomposition of knots, cyclic coverings and Alexander polynomials and modules together with the free differential calculus, braids, branched coverings and knots, Montesinos links, representations of knot groups, surgery of 3-manifolds and knots. Knot theory has expanded enormously since the first edition of this book published in 1985. A special feature of this second completely revised and extended edition is the introduction to two new constructions of knot invariants, namely the Jones and homfly polynomials and the Vassiliev invariants. The book contains many figures and some tables of invariants of knots. This comprehensive account is an indispensable reference source for anyone interested in both classical and modern knot theory. Most of the topics considered in the book are developed in detail; only the main properties of fundamental groups and some basic results of combinatorial group theory are assumed to be known. The text is accessible to advanced undergraduate and graduate students in
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״This text provides the students with simple "cookbook" recipes for solving problems they might face in their studies of...
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״This text provides the students with simple "cookbook" recipes for solving problems they might face in their studies of economics. Since the target audience was supposed to have some mathematical background (admittance to the program requires at least BA level mathematics), the main goal was to refresh students' knowledge of mathematics rather than teach them math "from scratch״.״
This is a free, online textbook that can be downloaded as a pdf file. According to the site, "This is a textbook for the...
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This is a free, online textbook that can be downloaded as a pdf file. According to the site, "This is a textbook for the course of multidimensional geometry and linear algebra. This course is a part of the basic mathematical education. Therefore, it is taught at Physical and Mathematical Departments in all Universities of Russia during one or two semesters.״
According to OER Commons, "Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus has two...
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According to OER Commons, "Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus has two major goals: to improve mathematics education at two-year colleges and at the lower division of four-year colleges and universities and to encourage more students to study mathematics. The document presents standards that are intended to revitalize the mathematics curriculum preceding calculus and to stimulate changes in instructional methods so that students will be engaged as active learners in worthwhile mathematical tasks. Preparation of these standards has been guided by the principle that faculty must help their students think critically, learn how to learn, and find motivation for the study of mathematics in appreciation of its power and usefulness' (direct from website). Users can access all chapters of the book as well as the Illinois Mathematics Association of Community Colleges.״
This is the site of a textbook that is published by Cambridge University, as part of Cambridge Series in Statistical and...
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This is the site of a textbook that is published by Cambridge University, as part of Cambridge Series in Statistical and Probabilistic Mathematics.. Each of the chapters, however, in draft form can be downloaded as a pdf file for free. ״This book should be on the shelf of every practising statistician who designs experiments. Good design considers units and treatments first, and then allocates treatments to units. It does not choose from a menu of named designs. This approach requires a notation for units that does not depend on the treatments applied. Most structure on the set of observational units, or on the set of treatments, can be defined by factors. This book develops a coherent framework for thinking about factors and their relationships, including the use of Hasse diagrams. These are used to elucidate structure, calculate degrees of freedom and allocate treatment subspaces to appropriate strata. Based on a one-term course the author has taught since 1989, the book is ideal for advanced undergraduate and beginning graduate courses. Examples, exercises and discussion questions are drawn from a wide range of real applications: from drug development, to agriculture, to manufacturing.״
This is a free, online textbook offered by Bookboon.com. According to the author, "This is the fifth book of examples from...
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This is a free, online textbook offered by Bookboon.com. According to the author, "This is the fifthAccording to the author, "The book is intended as a help for students and researchers in the biomedical fields, enabling them...
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According to the author, "The book is intended as a help for students and researchers in the biomedical fields, enabling them to choose the best model for their dose-response data obtained from wet-lab experiments. The focus is on how to interpret and handle dose-response data, with a recommendation to down-play analytical methods developed before the era of personal computers, such as the Lineweaver–Burk and Scatchard data conversion, the null-methods by Gaddum and Schild, or the use of meaningless mathematical manipulations, as for instance the implementation of a Hill-exponentiation. Instead, when fishing for system constants, the readers learn how to get access to the free-way of analytical tools that offer forward formulated physical functions to be fixed with non-linear fitting procedures. The approach I have taken is different from that of many other textbooks on the analysis of equilibrium dose-responses, which follow in the tradition of data-linearization developed more than 70 years ago. A book in point is Segel's 'Enzyme kinetics' (1975), reissued as a non-revised edition in 1993 and still considered a standard textbook on enzyme kinetics: it lacks almost completely in analysis of the so-called two-state models.״
This is a free, online textbook that is available as a pdf. "This book is intended to serve as the textbook for a rst-year...
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This is a free, online textbook that is available as a pdf. "This book is intended to serve as the textbook for a rst-year graduate course in econometrics. It can be used as a stand-alone text, or be used as a supplement to another text. Students are assumed to have an understanding of multivariate calculus, probability theory, linear algebra, and mathematical statistics. A prior course in undergraduate econometrics would be helpful, but not required.״This is a free, online textbook that is offered through MIT's OpenCourseWare. "this book is a useful resource for educators...
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This is a free, online textbook that is offered through MIT's OpenCourseWare. "this topics covered in the textbook. The The short form of these videos contains the demonstrations only. The long form also presents theory, diagrams, and calculations in support of the demonstrations.״
According to the website, "Published in 1989 by Prentice-Hall, this book is a useful resource for educators and self-learners...
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According to the website, "PublishedThese video demonstrations convey electromagnetism concepts. The demonstrations are related to topics covered in the textbook. They were prepared by Markus Zahn, James R. Melcher, and Manuel L. Silva and were produced by the Department of Electrical Engineering and Computer Science at the Massachusetts Institute of Technology.TheThe short form of these videos contains the demonstrations only. The long form also presents theory, diagrams, and calculations in support of the demonstrations.״
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Mathematics I
Mathematics is rarely thought of in the context of real world problems. This course teaches you how to think of Mathematics not as a subject in itself, but rather as a collection of tools which are necessary for any rigorous and complete analysis of economic problems.
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From the makers of MATLAB, a technical computing environment for high-performance numeric computation and visualization, and Simulink, an interactive environment for modeling, analyzing, and simulating a wide variety of dynamic systems, including discrete, analog, and mixed signal systems (MATLAB is required to use Simulink). MATLAB integrates numerical analysis, matrix computation, signal processing, and graphics in an easy-to-use environment. See MATLAB in Education for demos, basics, information on the student edition, and resources by topic - select mathematics to browse teaching examples, products, books, and third-party products. The File Exchange offers a wealth of utilities, tips, graphics, and other MATLAB files contributed by users; subscribe to a list to receive alerts of new additions. MathWorks sponsors a programming contest semi-annually, previous topics of which include Matlab golf, folding protein, arranging molecules, playing Mastermind, surveying Mars, and gerrymandering. Download by ftp miscellaneous M-file contributions from MathWorks developers and technical support engineers.
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"Linear and Nonlinear Integral Equations: Methods and Applications" is a self-contained book divided into two parts. Part I offers a comprehensive and systematic treatment of linear integral equations of the first and second kinds. The text brings together newly developed methods to reinforce and complement the existing procedures for solving... more...
Imagine that you've finally found a parking space after a long and harrowing search, but are now encountering some difficulty in trying to enter this space. Wouldn't it be great if you knew a formula that allowed you to enter the space without difficulty? Are you annoyed because your soda can doesn't remain upright during a picnic? Would...The best way to master math is to practice, practice, practice?and 1,001 Math Problems offers ?mathophobes? and others who just need a little math tutoring the practice they need to succeed. Whether students need help calculating a tip or facing a standardized math test that could determine their future, the 1,001 math questions in this useful... more...
There are some mathematical problems whose significance goes beyond the ordinary - like Fermat's Last Theorem or Goldbach's Conjecture - they are the enigmas which define mathematics.
The Great Mathematical Problems explains why these problems exist, why they matter, what drives mathematicians to incredible lengths to solve them and where... more...
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GCSE Maths Revision Guide (with Online Edition) - Higher (Paperback)
This updated and refreshed version of CGP's bestselling Revision Guide is the ideal companion to Higher Level GCSE Maths - it even includes a free online edition that can be used wherever you have internet access. Every topic is explained in a concise, friendly style, with a sprinkling of CGP humour to keep things interesting. Grade information is included to show the difficulty level of each topic, and there are summary questions at the bottom of each page to test you on the important skills. And finally, a unique code is printed in the book that gives you access to the free online digital version (which also includes fully worked answers to all the test questions in the book).
Book details
Published 07/06/2006
Publisher Coordination Group Publications Ltd (CGP)
ISBN 9781841465364
GCSE Maths Revision Guide (with Online Edition) - Higher
4
5
1
1
Really useful
I bought this for my daughter's Gcse revision and to help with her homework.
She's found it really useful and said that she would recommend this guide to maths boffins everywhere.
08 October 2010
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College Algebra : Real Mathematics - 6th edition
Summary: Ideal for courses that require the use of a graphing calculator, COLLEGE ALGEBRA: REAL MATHEMATICS, REAL PEOPLE, 6th Edition, features quality exercises, interesting applications, and innovative resources to help you succeed. Retaining the book's emphasis on student support, selected examples include notations directing students to previous sections where they can review concepts and skills needed to master the material at hand. The book also achieves accessibility through careful wr...show moreiting and design--including examples with detailed solutions that begin and end on the same page, which maximizes readability. Similarly, side-by-side solutions show algebraic, graphical, and numerical representations of the mathematics and support a variety of learning styles. Reflecting its new subtitle, this significant revision focuses more than ever on showing readers the relevance of mathematics in their lives and future careers70.57 +$3.99 s/h
Good
Recycle-A-Textbook Lexington, KY
111157510
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Trigonometry-Text Only - 3rd edition
Summary: Dugopolski's Trigonometry gives students the essential strategies to help them develop the comprehension and confidence they need to be successful in this course. Students will find enough carefully placed learning aids and review tools to help them do the math without getting distracted from their objectives. Regardless of their goals beyond the course, all students will benefit from Dugopolski's emphasis on problem solving and critical thinking, which is enhanced b...show morey the addition of nearly 1,000 exercises in this edition. Instructors will also find this book a pleasure to use, with the support of an Annotated Instructor's Edition which maps each group of exercises back to each example within the section; pop quizzes for every section; and answers on the page for most exercises plus a complete answer section at the back of the text. An Insider's Guide provides further strategies for successful teaching with Dugopolski
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08665146 Tessellations
This clear introduction to tessellations and other intriguing geometric designs help students explore polygons, regular polygons and combinations of regular polygons, Escher-type tessellations, Islamic art designs, and tessellating letters. Step-by-step directions for creating tessellations support discussions of the symmetries and transformations involved. The companion book of overhead masters contains more than 270 patterns students can use to create and analyze polygonal, regular, demi-regular, and demi-regular tessellations; star polygons, tessellating curves; and polyominoes. Reproducible. Grades 6-12
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Description
This course is a rapid coverage of high school-level algebra. Topics include: integer and rational exponents, operations with polynomials and factoring, operations with rational and radical expressions, solving quadratic and rational equations, graphs of lines and parabolas, systems of equations, complex numbers, functions, and applications of all of the aforementioned. Prerequisite: 2.0 or better in MATH 097 or COMPASS test placement. (Previously MTH 095)
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