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Books to prepare for maths common entrance
Does anyone know of a book which explains the concepts and gives questions (with answers in case I can't do it!). My son already uses the Galore Park books and doesn't really get on with them. Does anyone know of anything else?
A maths teacher friend of mine - so I can't say I am unbiased! - has just published a revision workbook which has notes on all the topics. It is called Common Entrance Mathematic 13+ Revision Workbook. You can find it on Amazon.
I don't really think GCSE revision books are as good as those specifically prepared for Common Entrance. Although the majority of the topics are the same, the type of questions often isn't. Much better to stick to the books which use CE style practice questions.
The ISEB revision guide and associated practice exercise textbook is the only resource I recommend to parents in my school, as it is the only resource I have found that exactly mirrors the style of questions in CE, which as we all know are highly formulaic!
Thanks for all the help. I have now received my copy of "Common Entrance 13+ Revision and Practice" from Amazon. It looks to be exactly what I wanted for my son's revision. It has notes, lots of worked examples and plenty of practice questions ( with answers in the back ). All I have to do now is persuade him to use it!
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Wk 2 Math 116 What Mathematical Concepts You Learned That May Help You Understand Appendix C Reading Essays and Term Papers
, explain how? Whatconcepts investigated in this course apply to your personal and professional life? In what ways did you use MyMathLab® for extra support?
• "I teach on-site and online math courses, and I must say that students who take online math courses have more available resources, more help...
students need to set aside the arguments about responsibility for this breakdown and get down to the basics of teach students to become literate in Math. To understandwhat it means to be literate in Math one must first realize that Math is a language unto itself. Students who are fluent in readingmay...
E3.1: Attitudinal Pre-Survey r r
14. How many COLLEGE math courses have you taken prior to this course? Please count the total number of semesters or quarters.
0 1 2 3 4 5 6 7 or more
15. What grade do you expect to receive in this course?
A AB+ B BC+ C CD F
Your academic background
16. What...
|
| | | | |
| |Has the content in this course allowed you to think of math as a useful tool? If so, explain | | |
| |how. Whatconcepts investigated in this course can apply to your personal and professional...
.
Course Assignments
1. Final Exam
• Complete the Week Nine Final Test in MyMathLab®. This test assesses content learned in Weeks One through Eight.
2. Capstone Discussion Question
• Has the content in this course allowed you to think of math as a useful tool? If so, explain how. What...
between mathematics and music?
(2) What are the mathematicalconcepts used develop musical theories?
(3) What are the proofs that music helps enhance one's mathematical skills?
B. Significance of the Study
This research will be able to aid those students who need help with their studies...
whatyoulearned high school. The pedagogy enables you to progress at your own pace and adapt the book to your own needs. FEATURES 1066 fully solved problems Clear, concise explanations of all mathematicalconcepts for business and economics Covers all course fundamentals
Professional References...
Comprehension in English and Math. Action research project, Saint Xavier University, Chicago, IL.
APPENDIX
An example of a graphic organizer used to help high school algebra students understandmath word problems.
(Kuzniewski et al., 1998)...
you to see that the same concept or relationship may be expressed by many different verbal forms and grammatical constructions, and to exclude cases where the form is right but the meaning in context is not. Each set of threads in a category was given a holistic reading where I noted the salient...
I don't know how to deal
with the problems. I need to find a way to help
me remember the concepts. Maybe I could use
word associations or cue cards to help me
remember the concepts.
What makes it difficult for you to study?
I don't understand any new words. I need to find
out all the...
can you split a bag of M&Ms and what portion is each student going to receive? This activity supports constructivist teaching theory by providing an opportunity for students to build upon their existing schema and use what they learned to solve a real life problem. To construct and understand...
three activities that helped youunderstand the concept of area? How did these activities helpyouunderstand the concept? Might the same activities help children understand the concept? Explain.
Week Nine Discussion Questions
What two mathematicalconcepts that you have learned in this course...
failure of the course). Passing off the work of other as your own can be either plagiarism or collusion. Both are scholastic offences that the instructors will not tolerate. The definitions and descriptions in appendix A below will helpyou know what acts constitute either violation. If you have any...
coverage.
The good news is that this help is now available to you. Think of the guidelines below as ideas on whatyou can expect during science high school entrance tests and some tips on how you can do well on the exam.
Half the Battle
Mathmay be a very broad field in itself, but certain sub...
Appendix B) was redesigned using the actual templates from the UbD workbook. Appendix D shows the "Concept Map" that was the skeleton of the Unit shown in Appendix B. It shows in a nutshell what is happening in the curriculum and is helpful in summarizing the template. The assessments that were put...
in mathematical literacy is to learn one of the smallest parts of literacy learning: vocabulary. Instruction in the language specific to mathematics is vital for students to understand the concepts within the math curriculum. Through the use of the vocabulary strategies within my handbook, Effective...
important concepts and to helpyou explore
2
Chapter 1: Introduction
these in more depth by guided reading of the set text and selected internet
resources.
Youmay also need to research particular topics in more depth as part
of the coursework. When doing so, you should carefully evaluate...
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Introduction to Number Theory - 05 edition
Summary: An Introduction to Number Theory provides an introduction to the main streams of number theory. Starting with the unique factorization property of the integers, the theme of factorization is revisited several times throughout the book to illustrate how the ideas handed down from Euclid continue to reverberate through the subject. In particular, the book shows how the Fundamental Theorem of Arithmetic, handed down from antiquity, informs much of the teaching of modern number theory. T...show morehe result is that number theory will be understood, not as a collection of tricks and isolated results, but as a coherent and interconnected theory. A number of different approaches to number theory are presented, and the different streams in the book are brought together in a chapter that describes the class number formula for quadratic fields and the famous conjectures of Birch and Swinnerton-Dyer. The final chapter introduces some of the main ideas behind modern computational number theory and its applications in cryptography. Written for graduate and advanced undergraduate students of mathematics, this text will also appeal to students in cognate subjects who wish to learn some of the big ideas in number theory. ...show less
Edition/Copyright:05 Cover: Hardcover Publisher:Springer-Verlag New York Published: 06/15/2005 International: No
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Additional Sellers for Introduction to Number TheoryLondon, England 2007 Hard cover Good. The book has been read but remains in clean condition. All pages are intact and the cover is intact. Some minor wear to the spine. Glued binding. Paper over boa...show morerds. 294 p. Contains: Illustrations, black & white. Graduate Texts in Mathematics, 23279 +$3.99 s/h
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This volume is the first to offer a comprehensive, research-based, multi-faceted look at issues in early algebra. In recent years, the National Council for Teachers of Mathematics has recommended that algebra become a strand flowing throughout the K-12 curriculum, and the 2003 RAND Mathematics...
From Children's Ideas To Classroom Practice
Bringing Out the Algebraic Character of Arithmetic contributes to a growing body of research relevant to efforts to make algebra an integral part of early mathematics instruction, an area of studies that has come to be known as Early Algebra. It provides both a rationale for promoting algebraic...
Published August 17
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How does reading a good book help you in real life?
How does playing a game of golf help you in real life?
How does listening to Mozart help you in real life ?
How does doing a cross-word puzzle help you in real life ?
How does listening to your mp3 players help you in real life ?
How does solving a tough math problem help you in real life ?
etc.
Actually, it doesn't necessarily depend what your so-called real life is like.
Real life is real life, believe it or not. Oh my god. So confusing. Not.
Sorry to be so blunt.
So back to the question. It could help you solve problems that could actually occur to you, like complex interest equations can help you save money. It's like having an inside look at your life. You can know exactly what to do in certain situations.
MATH - cos(x)=2 and sin(x)=5 Thanks for your previous answers mathmate. No. It's...
MATH - ok so I'm learning to solve problems like cos(x)=2 and sin(x)=5 I've ...
english - A life skill is something you need to get along in the world. ...
Tech Ed - How can simulations be used in education? A.They allow people to touch...
9th Algebra - Im in 9th grade algebra. and we are learning adding and ...
Math - I need help with scales and converting them to real-life measurements. So...
Communications - I am supposed to survey people because I am doing a research ...
Algebra - Can you please give me some real life examples of using variables and ...
2nd Math - My daughter is learning about comparison problems. I would like to ...
Introducing Literature - The term bibliotheraphy means a. using books to give ...
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If you are a student that started at GVSU BEFORE FALL 2011:
1. You must meet MTH 110 Proficiency, even if you fulfill GE Foundation - Mathematical Science (F-MS) or MACRAO elsewhere OR
2. You must adopt a new catalog
If you are a New Transfer Student:
1. If you fulfilled MACRAO then you have fulfilled F-MS
2. You can transfer in any course in F-MS. If so, you have then fulfilled F-MS OR
If you need F-MS, you must show MTH 110 Proficiency
If you are a New Freshman:
1. MTH110 course, or proficiency (see list above), is a pre-requisite for F-MS
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Discrete Mathematics and Its designed for the sophomore/junior level introduction to discrete mathematics taken by students preparing for future coursework in areas such as math, computer science and engineering. Rosen has become a bestseller largely due to how effectively it addresses the main portion of the discrete market, which is typically characterized as the mid to upper level in rigor. The strength of Rosen's approach has been the effective balance of theory with relevant applications, as well as the overall comprehensive nature of the topic coverage.
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For many students interested in pursuing - or required to pursue - the study of mathematics, a critical gap exists between the level of their secondary school education and the background needed to understand, appreciate, and succeed in mathematics at the university level. A Concise Introduction to Pure Mathematics provides a robust bridge over this gap. In nineteen succinct chapters, it covers the range of topics needed to build a strong foundation for the study of the higher mathematics.
Written in a relaxed, readable style, A Concise Introduction to Pure Mathematics leads students gently but firmly into the world of higher mathematics. It demystifies some of the perceived abstractions, intrigues its readers, and entices them to continue their exploration on to analysis, number theory, and beyond.
Editorial Reviews
Review
"Liebeck's book, A Concise Introduction to Pure Mathematics, is one of the best I have seena gentle but fascinating introduction into the culture of mathematicsmathematics will [no longer] be viewed as some abstract black box" - From the Foreword by Robert Guralnick, University of Southern California, Los Angeles
Most Helpful Customer Reviews
I'm not sure how many of us there are out there, but I am one of a breed of consumers of applied mathematics who learned some pretty sophisticated mathematical technology without the rigor of pure mathematics. Although this book is aimed at freshmen entering mathematics programs who need to be inculturated into the world of pure mathematics, I found it to be the crucial link I needed to advance my own applied mathematical training. I reached a point where I was ready to move from applied texts to the more cryptic world of math texts written for graduate mathematicians. Unfortunately, I was not properly trained to decypher their special language and way of doing things, particularly that of the formal proof. I found reading introductory texts in analysis to be like trying to learn Japanese from books which were themselves written in Japanese. Then I found this wonderful little text. It made things much more accessible to me and helped me crack enough of the code where I could find my way around those analysis texts, which in turn allowed me to move on to the graduate math texts containing the methods I am studying now. The book is a bit high priced for its size, but for me it was well worth it.
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Summary: Focusing on the important ideas of geometry, this book shows how to investigate two- and three-dimensional shapes with very young students. It introduces methods to describe location and position, explores simple transformations, and addresses visualization, spatial reasoning, and the building and drawing of constructions. Activities in each chapter pose questions that stimulate students to think more deeply about mathematical ideas. The CD-ROM features fourteen arti...show morecles from NCTM publications. The supplemental CD-ROM also features interactive electronic activities, master copies of activity pages for students, and additional readings for teachers. ...show less
Edition/Copyright: 01 Cover: Paperback Publisher: National Council of Teachers of Mathematics Published: 01/28/2001 International: No
087353511178 +$3.99 s/h
VeryGood
No Particular Books Sun City West, AZ
2001 Very good in very good dust jacket. 98 p. Principles and Standards for School Mathematics Navigations. Audience: General/trade. Excellent condition, not in plastic. Includes CD. Great price. Sh...show moreips fast
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This is a Stand Alone Instructional Resource project, for the subject of Algebra 1. This StAIR is done through PowerPoint...
see more
This is a Stand Alone Instructional Resource project, for the subject of Algebra 1. This StAIR is done through PowerPoint 2010 and covers the topic of Probability…mostly probability of compound events. The point of this stair is that students can work their way through it on their own. The goal of this module is for students to master the concept on how to find the probability of compound events and review their knowledge on experimental and theoretical probability. Students review examples, vocabulary, watch videos, take mini quizzes, etc.
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A solid foundation of basic math skills is essential for early success in math. Children who can connect their understanding of math to the world around them will be ready for the challenges of mathematics as they advance to more complex topics. The games and puzzles in this workbook are designedA Tour of the Calculus
Written by David Berlinski
Format: Trade Paperback
ISBN: 9780679747888
Our Price: $16.95
Were it not for the calculus, mathematicians would have no way to describe the acceleration of a motorcycle or the effect of gravity on thrown balls and distant planets, or to prove that a man could cross a room and eventually touch the opposite wall. Just how calculus makes these thingsThese simple math secrets and tricks will forever change how you look at the world of numbers.
Secrets of Mental Math will have you thinking like a math genius in no time. Get ready to amaze your friends—and yourself—with incredible calculations you never thought you could master, as renowned "mathemagician" Arthur Benjamin...
Read more >
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This video, which is suitable for high school students, starts off with a black screen because the narrator uses it as a 'chalkboard'. More intuition of what a derivative is. Determining the derivatives of simple polynomials. (09:50)
This video, which is suitable for high school students, starts off with a black screen because the narrator uses it as a 'chalkboard'. The product rule. Examples using the Product and Chain rules. (08:48)
This video, which is suitable for high school students, starts off with a black screen because the narrator uses it as a 'chalkboard'. Why the quotient rule is the same thing as the product rule. Introduction to the derivative of e^x, ln x, sin x, co...s x, and tan x. (09:31)[more]
This video, which is suitable for high school students, starts off with a black screen because the narrator uses it as a 'chalkboard'. In this segment, the instructor offers more 'intuition' of what a derivative is using the derivative to find the sl...ope at any point along f(x)=x^2. (09:31)[more]
This video starts off with a black screen because the narrator uses it as a 'chalkboard'. The instructor demonstrates how to find the slope of a tangent line to a curve (the derivative). Introduction to Calculus. (09:24
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Geometry Seeing, Doing, Understanding
9780716743613
ISBN:
0716743612
Edition: 3 Pub Date: 2003 Publisher: W H Freeman & Co
Summary: Jacobs innovative discussions, anecdotes, examples, and exercises to capture and hold students' interest. Although predominantly proof-based, more discovery based and informal material has been added to the text to help develop geometric intuition.
Jacobs, Harold R. is the author of Geometry Seeing, Doing, Understanding, published 2003 under ISBN 9780716743613 and 0716743612. One hundred forty five Geometry ...Seeing, Doing, Understanding textbooks are available for sale on ValoreBooks.com, seventeen used from the cheapest price of $99.00, or buy new starting at $165.88.[read more]
Seller Rating:(0)
Ships From:Tempe, AZShipping:StandardComments:WE HAVE NUMEROUS COPIES -HARDCOVER, Mild shelf wear to cover, edges, and corners, a ding to the ... [more]WE HAVE NUMEROUS COPIES -HARDCOVER, Mild shelf wear to cover, edges, and corners, a ding to the top and bottom of spine, otherwise book is NEW
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Students shoot a ball and earn higher scores the closer they are to hitting a target. They can improve their scores by interpreting data from the linear model that relates the distance of the target t... More: lessons, discussions, ratings, reviews,...
Students find the optimal price for an insurance company premium in this game by interpreting data and applying their understanding of linear and quadratic models. [Access requires setting up a (free)... More: lessons, discussions, ratings, reviews,...
This application is a basic tool for learning algebra. Users try to solve a system of linear equations with two equations and two unknown variables. It is based on the same problem used in the free ap... More: lessons, discussions, ratings, reviews,...
Use your TI-Nspire to consider this scenario and question: Sam and Teri have bank accounts. Sam always withdraws money; Teri always saves it. When will they have equal balances -- or will they ever?
... More: lessons, discussions, ratings, reviews,...
A Java applet that illustrates the use of the derivative in finding a linear approximation to a function, with questions to answer using the applet. Includes a re-scalable presentation version with la... More: lessons, discussions, ratings, reviews,...
Make an open-top box from a sheet of paper by cutting squares out of the corners. See how the volume of the box changes as the sides of the squares change. The link to the tool itself downloads the ... More: lessons, discussions, ratings, reviews,...
Students explore the sum of certain distances in an equilateral triangle, and develop a proof to explain their observations. The link to the activity itself is to a zip file that contains both the... More: lessons, discussions, ratings, reviews,...
Students investigate and prove a property of kites, and use the proof to make a discovery about other quadrilaterals. The link to the activity itself is to a zip file that contains both the activi... More: lessons, discussions, ratings, reviews,...
Students form a conjecture about the ratio of two areas in a quadrilateral, and use Sketchpad to search for a proof. In the process they verify or disprove their results for two different kinds of
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These simple math secrets and tricks will forever change how you look at the world of numbers. Secrets of Mental Math will have you thinking like a math genius in no time. Get ready to amaze your friends?and yourself?with incredible calculations you never thought you could master, as renowned ?mathemagician? Arthur Benjamin shares his techniques... more...
Please, Mr Murdstone! Don't beat me! I've tried to learn my lessons, really I have, sir!' sobs David.Although... more...
A concise revision guide for Cambridge IGCSE Economics written by one of the authors of the popular Complete Economics for Cambridge IGCSE. Suitable for students of all ability levels, it provides not only revision material but a huge bank of practice tests and advice on preparing for examinations. It also contains vocabulary-based exercises to support... more...
The ideal review for your basic circuit analysis course
More than 40 million students have trusted Schaum?s Outlines for their expert knowledge and helpful solved problems. Written by renowned experts in their respective fields, Schaum?s Outlines cover everything from math to science, nursing to language. The main feature for all these books is... more...
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Photos and Videos tagged with #10212
Book #10212 Using Math On A Space Mission Perma-Bound (from Original Paperback. Using Math on a Space Mission (Mathworks!) by - Powell's Books The End of Your Life Book Club Sale;. Using Math on a Space Mission (Mathworks!): Anne Brumfitt, Steve. Using Math on a Space Mission - Hilary Koll, Steve Mills, Anne. Using Math on a Space Mission (Mathworks!) [Library Binding] Using Math on a Space Mission (Mathworks!) [Anne Brumfitt, Steve Mills] on Amazon.com. Social Mission; Contact Us ;. *FREE* super saver shipping on qualifying offers. Using Math on a Space Mission (Mathworks!) [Anne Brumfitt, Steve Mills] on Amazon.com. Mathworks! (Gareth Stevens) - Perma-Bound Books Mathworks! (Gareth Stevens). Using Math on a Space Mission - Google Books Our best-selling math series has six new titles! Correlated to NCTM grade 3 curriculum standards, Mathworks! motivates students by relating math concepts and skills. Library of Books, Links & More: ELEMENTARY AND MIDDLE SCHOOL MATH. Using Math on a Space Mission (Mathworks!) by Anne Brumfitt; MathWorks - MATLAB and Simulink for Technical Computing MathWorks is the world's leading developer of technical computing software for engineers and scientists in industry, government,. Vote for Your Favorite Book of 2012! The 2012 Holiday Gift Guide ; Announcing Indiespensable #37: The Death of Bees;. *FREE* super saver shipping on qualifying offers. Using Math on a Space Mission (Mathworks!) Using math on a space mission (Open Library) Using math on a space mission by Hilary Koll,. Using Math on a Space Mission (Mathworks!) Comment on this title and you could win free books! Synopses & Reviews Using Math on a Space Mission (Mathworks!) by Hilary Koll - Powell
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College Algebra Essentials -With CD - 3rd edition
Summary: Chapter P. Prerequisites: Fundamental Concepts of Algebra.P.1 Algebraic Expressions, Mathematical Models, and Real Numbers1. Evaluate algebraic expressions.2. Use mathematical models.3. Find the intersection of two sets.4. Find the union of two sets.5. Recognize subsets of the real numbers.6. Use inequality symbols.7. Evaluate absolute value.8. Use absolute value to express distance.9. Identify properties of the real numbers.10. Simplify algebraic expressions.P.2 Exponents and Scient...show moreific Notation1. Use the product rule.2. Use the quotient rule.3. Use the zero-exponent rule.4. Use the negative-exponent rule.5. Use the power rule.6. Find the power of a product.7. Find the power of a quotient.8. Simplify exponential expressions.9. Use scientific notation.P.3 Radicals and Rational Exponents1. Evaluate square roots.2. Simplify expressions of the form ?a23. Use the product rule to simplify square roots.4. Use the quotient rule to simplify square roots.5. Add and subtract square roots.6. Rationalize denominators.7. Evaluate and perform operations with higher roots.8. Understand and use rational exponents.P.4 Polynomials1. Understand the vocabulary of polynomials.2. Add and subtract polynomials.3. Multiply polynomials.4. Use FOIL in polynomial multiplication.5. Use special products in polynomial multiplication.6. Perform operations with polynomials in several variables.Mid-Chapter Check PointP.5 Factoring Polynomials1. Factor out the greatest common factor of a polynomial.2. Factor by grouping.3. Factor trinomials.4. Factor the difference of squares.5. Factor perfect square trinomials.6. Factor the sum or difference of two cubes.7. Use a general strategy for factoring polynomials.8. Factor algebraic expressions containing fractional and negative exponents.P.6 Rational Expressions1. Specify numbers that must be excluded from the domain of rational expressions.2. Simplify rational expressions.3. Multiply rational expressions.4. Divide rational expressions.5. Add and subtract ...show less
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Advanced Mathematical Concepts lessons develop mathematics using numerous examples, real-world applications, and an engaging narrative. Graphs, diagrams, and illustrations are used throughout to help students visualize concepts. Directions clearly indicate which problems may require the use of a graphing calculator
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318
Spring
Offering
6 subjects
including precalculus precalculus precalculus precalculus
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Algebra II For Dummies is the fun and easy way to get a handle on this subject and solve even the trickiest algebra problems. This friendly guide shows you how to get up to speed on exponential functions, laws of logarithms, conic sections, matrices, and other advanced algebra concepts. In no time you'll have the tools you need to:
Interpret quadratic functions
Find the roots of a polynomial
Reason with rational functions
Expose exponential and logarithmic functions
Cut up conic sections
Solve linear and non linear systems of equations
Equate inequalities
Simplifyy complex numbers
Make moves with matrices
Sort out sequences and sets
This straightforward guide offers plenty of multiplication tricks that only math teachers know. It also profiles special types of numbers, making it easy for you to categorize them and solve any problems without breaking a sweat. When it comes to understanding and working out algebraic equations, Algebra II For Dummies is all you need to succeed!
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...
Show More because the authors provide the necessary background from graph theory, number theory, group theory and representation theory. Thus the text can be used as a brief introduction to these subjects and their synthesis in modern mathematics
| 677.169 | 1 |
Elementary Algebra
9780321577290
ISBN:
0321577299
Pub Date: 2010 Publisher: Pearson Education
Summary: Carson, Tom is the author of Elementary Algebra, published 2010 under ISBN 9780321577290 and 0321577299. One hundred forty two Elementary Algebra textbooks are available for sale on ValoreBooks.com, one hundred ten used from the cheapest price of $8.38, or buy new starting at $62.25.Comments: ALTERNATE EDITION: no cd or access cards-ANNOTATED TEACHER EDITION-same content as the student ed- with teaching tips and answers- May havehilighting-writing-Has Used Book stickers on the cover.
There wasn't a part of the book that was least helpful to me because all the steps to the equation or story problem were explained well and I did not need to go and waste extra time to try and figure it out on my own.
The material that was needed for the class I had to take and it had all the answers I needed to learn the material for that class. It gave great examples for problems in each lesson of the chapters!!
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hematics Review/Preview
5th grade Pre-Algebra • Pre-Algebra • Algebra • Geometry • Algebra 2
Prepare your child for next year or review last year's mathematics. Strengthen your child's foundation of algebra; it's the base for all future mathematics classes. We review next year's textbook, chapter by chapter, and introduce many new concepts every day. This way when they are in class, they are already familiar with the concept. Private collaborative Groups available.
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feel that this will better prepare them for future math courses. I was always taught to just "follow the formula", and I found that this method can be detrimental when entering a theoretical math course where the student is challenged to think deeper about the problem and ask why do we get the
| 677.169 | 1 |
From inside the book
Don't listen to the imbisol who rated this as a 1, this is why I rated it a 5 otherwise it would have been a 4. This book is a thoroughly invaluable book when it comes toto knowing the foundations of mathematics. There's plenty of examples/work/problems to challenge yourself with. There's practically all the information you need to go from 7th grade algebra to aceing a college precalculus course and more. The front has all sorts of common factoring identities to reference, while the back has all the important geometry functions. Also, it explains how to factor in the back. For the record though I haven't used this book in years and am not that good at math, although I do have a good grasp on the visual aspect of it. Math isn't just numbers for the sake of numbers. If that's your attitude, you'll more than likely hate this book - but for god's sakes don't give it a 1 star it's a great book.
Review: Precalculus, Seventh Edition
User Review - David Glad - Goodreads
This book is great. I am sure everyone has had some bad mathematical experiences, but this one was good. It did a decent job explaining concepts and instead of just giving you the (final) answers in ...Read full review
About the author (2003)
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How to Count: An Introduction to Combinatorics, Second Edition (Discrete Mathematics and Its Applications)
9781420082609
ISBN:
1420082604
Edition: 2 Pub Date: 2010 Publisher: C R C Press LLC
Summary: Allenby, Regnaud B. J. T. is the author of How to Count: An Introduction to Combinatorics, Second Edition (Discrete Mathematics and Its Applications), published 2010 under ISBN 9781420082609 and 1420082604. Four hundred eighty three How to Count: An Introduction to Combinatorics, Second Edition (Discrete Mathematics and Its Applications) textbooks are available for sale on ValoreBooks.com, one hundred four used from ...the cheapest price of $54.65, or buy new starting at $64.0420082609
ISBN:1420082604
Edition:2nd
Pub Date:2010 Publisher:C R C Press LLC
ValoreBooks.com is the top book store for cheap How to Count: An Introduction to Combinatorics, Second Edition (Discrete Mathematics and Its Applications) rentals, or new and used condition books for purchase.
| 677.169 | 1 |
Resena: Passing grades in two years of algebra courses are required for high school graduation. Algebra II Essentials For Dummies covers key ideas from typical second-year Algebra coursework to help s...
Resena: When did the first civilizations arise? How many human languages exist? The answers are found in anthropology - and this friendly guide explains its concepts in clear detail. You'll see how an...
Resena: Calculus II is a prerequisite for many popular college majors, including pre-med, engineering, and physics. Calculus II For Dummies offers expert instruction, advice, and tips to help second ...
Resena: Good manners matter – and not just during formal events. Etiquette isn't just about which fork to use or how to hold a teacup. Good manners and etiquette are as important in the supermarket ch...
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Find a La Porte Geometry...Thomas and received an A in the course. Linear Algebra is the study of matrices and their properties. The applications for linear algebra are far reaching whether you want to continue studying advanced algebra or computer science
| 677.169 | 1 |
Enter 3 parts of a triangle and then enter the number of solutions and then your solution(s) accurate to the hundredths...
see more
Enter 3 parts of a triangle and then enter the number of solutions and then your solution(s) accurate to the hundredths place. This program checks your answer without giving away the answers.trig, trigonometry, triangle, triangles, precalculus, solving triangles, math, mathematics
This website is called SMILE- Science and Math Initiative for Learning Enhancement. It's kind of like Merlot, but a much...
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This website is called SMILE- Science and Math Initiative for Learning Enhancement. It's kind of like Merlot, but a much smaller data source. It is a contributed collection website of single-concept lesson plans from teachers all over the U.S. It is free and not only do they offer the subject of chemistry, but if you click on the homepage link at the bottom of this page, you'll find subjects on Biology, Physics, and Math as well.
Intended for the advanced undergraduate or graduate student, Electromagnetic Field Theory is a textbook on the theory of...
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Intended for the advanced undergraduate or graduate student, Electromagnetic Field Theory is a textbook on the theory of electrodynamics, at roughly the same level as the well-known textbooks by Jackson and Panofsky&Phillips. The book is written mainly from a classical field theoretical point of view, emphasising fundamental and subtle properties of the EM field and includes a comprehensive appendix on the mathematical methods used.Sample exercises and exam problems are also available for download.The overarching purpose of the present study, commissioned by UNESCO's Institute for Statistics, is to compile, analyze...
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The overarching purpose of the present study, commissioned by UNESCO's Institute for Statistics, is to compile, analyze and describe commonalities and differences in the intended primary curriculum in reading and mathematics across a diverse set of developing countries. The specific activities to be carried out include the following: Compile materials on the intended reading and mathematics curriculum in the final grades of primary education from a significant number of developing countries (around 25-30), and ensure adequate coverage by region and language (i.e., at least in English, Arabic, Spanish and French); Develop and validate a coding scheme to systematically record, retrieve and compare the intended reading and mathematics curriculum in different primary education systems; Discuss an interim set of products emerging from the aforementioned project activities, including the coding scheme, with the IWG through electronic means and IWG meetings; Complete all compilation activities and cross-national analyses of the intended reading and mathematics curriculum and submit a draft report for review by the IWG and UNESCO colleagues. Submit a final report with the study's main findings
This activity guide for Grades 5-8 explores the potential for global climate change to increase the frequency and intensity...
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This activity guide for Grades 5-8 explores the potential for global climate change to increase the frequency and intensity of hurricanes and storm surges, and the impacts that could result. Designed to teach through scientific inquiry, the activity seeks to stimulate thought about the long-term impact of a warmer planet. The activity responds to national education standards in the English language arts, geography, social studies, mathematics, and science. The submitter notes that the mirror site will take you right to the .pdf curriculum.
Provides both theoretical and practical coverage of all data mining topics. Includes extensive number of integrated examples...
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Provides both theoretical and practical coverage of all data mining topics. Includes extensive number of integrated examples and figures.Offers instructor resources including solutions for exercises and complete set of lecture slides. Assumes only a modest statistics or mathematics background, and no database knowledge is needed.Topics covered include; predictive modeling, association analysis, clustering, anomaly detection, visualization.
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Essential Mathematics with Applications
As in previous editions, the focus in ESSENTIAL MATHEMATICS with APPLICATIONS remains on the Aufmann Interactive Method (AIM). Students are ...Show synopsisAs in previous editions, the focus in ESSENTIAL MATHEMATICS with APPLICATIONS remains on the Aufmann Interactive Method (AIM). Students are encouraged to be active participants in the classroom and in their own studies as they work through the How To examples and the paired Examples and You Try It problems. The role of "active participant" is crucial to success. Presenting students with worked examples, and then providing them with the opportunity to immediately work similar problems, helps them build their confidence and eventually master the concepts. To this point, simplicity plays a key factor in the organization of this edition, as in all other editions. All lessons, exercise sets, tests, and supplements are organized around a carefully-constructed hierarchy of objectives. This "objective-based" approach not only serves the needs of students, in terms of helping them to clearly organize their thoughts around the content, but instructors as well, as they work to design syllabi, lesson plans, and other administrative documents. The Eighth Edition features a new design, enhancing the Aufmann Interactive Method and the organization of the text around objectives, making the pages easier for both students and instructors to follow
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Numerical Mathematics and Computing
by Ward Cheney Publisher Comments
Authors Ward Cheney and David Kincaid show students of science and engineering the potential computers have for solving numerical problems and give them ample opportunities to hone their skills in programming and problem solving. NUMERICAL MATHEMATICS... (read more)
Calculus I with Precalculus
by Ron Larson Publisher Comments
Carefully developed for one-year courses that combine and integrate material from Precalculus through Calculus I, this text is ideal for instructors who wish to successfully bring students up to speed algebraically within precalculus and transition them... (read more)
Schaum's Outline of Precalculus (Schaum's Outlines)
by Fred Safier Publisher Comments... (read more)
Pre-Calculus Demystified
by Rhonda Huettenmueller Publisher Comments
HERE'S AN ABSOLUTE VALUE FOR ANYONE WISHING TO MASTER PRE-CALCULUS! Calculus is a cinch with pre-calculus under your belt -- and calculus is a must for any science, math, or computer science major. Pre-calculus by itself deepens your understanding of... (read more)
Super Reviews Calculus (Super Reviews; All You Need to Know)
by Rea Publisher Comments
CALCULUS SUPER REVIEW Need help with calculus? Want a quick review or refresher for class? This is the book for you! CONCISE SUBJECT REVIEW Covers the material students typically learn in an introductory calculus course. Clear, easy-to-understand format... (read more)
Calculus 3RD Edition
by James Stewart Book News Annotation
A textbook that introduces topics with an intuitive geometrical or
physical description and ties mathematical concepts to the students'
experience. It can be used with or without technology, and special
symbols indicate when a particular type of... (read more)
Multivariable Calculus 3RD Edition
by William G Mccallum Publisher Comments
The new edition exhibits the same strengths from earlier editions including the Rule of Four, an emphasis on modeling, exposition that students can read and understand and a flexible approach to technology. * The conceptual and modeling problems, praised... (read more)
Basic Real Analysis
by Houshang H. Sohrab Publisher Comments
Basic Real Analysis demonstrates the richness of real analysis, giving students an introduction both to mathematical rigor and to the deep theorems and counter examples that arise from such rigor. In this modern and systematic text, all the... (read more)
Concepts in Calculus, II
by Miklos Bona Publisher Comments
From the University of Florida Department of Mathematics, this is the second volume in a three volume presentation of calculus from a concepts perspective. The emphasis is on learning the concepts behind the theories, not the rote completion
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provides algorithmically generated practice exercises that correlate at the objective level to the content of the text. Every exercise is accompanied by an example and a guided solution, and selected exercises also include a video clip. The software provides helpful feedback and can generate printed summaries of studentsâ progress.
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Elementary Algebra
The Sullivan/Struve/Mazzarella Algebra program is designed to motivate students to "do the math"- at home or in the lab-and supports a variety of ...Show synopsisThe Sullivan/Struve/Mazzarella Algebra program is designed to motivate students to "do the math"- at home or in the lab-and supports a variety of learning environments. The text is known for its two-column example format that provides annotations to the left of the algebra. These annotations explain what the authors are about to do in each step (instead of what was just done), just as an instructor would do.Hide synopsis
Description:New. ***Please Read***Annotated Instructor's Edition. Still in...New. ***Please Read***Annotated Instructor's Edition. Still in Brand new, hardcover, 2nd Edition, NO ACCESS CODE. It has the same content as student edition, except it has ANNOTATED INSTRUCTOR'S EDITION printed on the COVER of the BOOK. The text and pages are otherwise the EXACT SAME as the student edition, and may include notes/answers in margins or back of book. From California.
Description:New. 0131467662 Purchased as new and in great condition. We...New. 0131467662Description:****Instructor's Edition / Same as Student Version but may...****Instructor's Edition / Same as Student Version but may contain additional notes or answers. May have tape on cover. ***New Book / Never Used*** May not include supplements such as CDs, infotrac or other web access codes. Quick shipping.
Reviews of Elementary Algebra
Received the book in a timely manner. It was in good condition. I don't think this will happen every time, so don't expect this, but it turned out to be an instructor's manual, which I didn't ask for, but I'm glad I got - the answers are right next to the problems; no flipping to the back to check my work anymore.
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Intermediate Algebra Textbooks
Intermediate algebra textbooks are the next step from introductory algebra. Intermediate algebra textbooks expand on and introduce new concepts in addition to what students have already learned and an intermediate algebra textbook can be a secondary or post secondary required text. Intermediate algebra textbooks also make great resources for intermediate algebra help. Textbooks.com offers a wide variety of titles and editions for intermediate algebra textbooks available in new and used so you can buy cheap textbooks online and save.
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-to-understand, ESSENTIALS OF TRIGONOMETRY starts with the right-angle definition, and applications involving the solution of right triangles to help you investigate and understand the trigonometric functions, their graphs, their relationships to one another, and ways in which they can be used in a variety of real-world applications. The accompanying CD-ROM and online tutorials give you the practice you need to improve your grade in the course.
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9780828401 Calculus of Variations in the 18th Century
Shortly after the invention of differential and integral calculus, the calculus of variations was developed. The new calculus looks for functions that minimize or maximize some quantity, such as the brachistochrone problem, which was solved by Johann Bernoulli, Leibniz, Newton, Jacob Bernoulli and l'H pital and is sometimes considered as the starting point of the calculus of variations. In Woodhouse's book, first published in 1810, he has interwoven the historical progress with the scientific development of the subject.
The reader will have the opportunity to see how calculus, during its first one hundred years, developed by seemingly tiny increments to become the highly polished subject that we know today. Here, Woodhouse's interweaving of history and science gives his special point of view on the mathematics. As he states in his preface: "Indeed the authors who write near the beginnings of science are, in general, the most instructive; they take the reader more along with them, show him the real difficulties and, which is the main point, teach him the subject, the way they themselves learned it
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Complete Book of Algebra and Geometry Grades 5-6
9780769643304
ISBN:
0769643302
Publisher: Carson-Dellosa Publishing, LLC
Summary: The Complete Book of Algebra and Geometry offers children in grades 5-6 easy-to-understand lessons in higher math concepts, skills, and strategies. This best-selling, 352 page workbook teaches children how to understand algebraic and geometric languages and operations. Children complete a variety of activities that help them develop skills and then complete lessons that apply these skills and concepts to everyday sit...uations. Including a complete answer key this workbook features a user friendly format perfect for browsing, research, and review. Basic Skills Include: -Order of Operations -Numbers -Variables -Expressions -Integers -Powers -Exponents -Points -Lines -Rays -Angles -Area Over 4 million in print! The best-selling "Complete Book series" offers a full complement of instruction, activities, and information about a single topic or subject area. Containing over 30 titles and encompassing preschool to grade 8 this series helps children succeed in every subject area!
Carson-Dellosa Publishing Staff is the author of Complete Book of Algebra and Geometry Grades 5-6, published under ISBN 9780769643304 and 0769643302. Fifty one Complete Book of Algebra and Geometry Grades 5-6 textbooks are available for sale on ValoreBooks.com, ten used from the cheapest price of $8.47, or buy new starting at $46.84
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Elementary and Intermediate Algebra Worksheets for Classroom or Lab Practice
Mathxl 12Mo Stu Cpn Business Mathematics
MathXL Tutorials on CD for Elementary and Intermediate Algebra
Pass the Test (Standalone) for Elementary and Intermediate Algebra
Video Lectures on CD for Elementary and Intermediate Algebra
Summary
This study skills workbook, written by Alan Bass, expands upon George Woodburys study skills feature in the text, Building Your Study Strategy, and introduces new topics to help students be more successful in developmental math. Topics include: time management, note-taking, homework, and test preparation skills, overcoming math anxiety, among other topics. This no-nonsense approach to developing better math study skills provides students with the basic skills needed to be successful in developmental math.
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ISBN: 1429210737 / ISBN-13: 9781429210737
Calculus: Early Transcendentals
What's the ideal balance? How can you make sure students get both the computational skills they need and a deep understanding of the significance of ...Show synopsisWhat's the ideal balance? How can you make sure students get both the computational skills they need and a deep understanding of the significance of what they are learning? With your teaching--supported by Rogawski's "Calculus Second Edition"--the most successful new calculus text in 25 years! Widely adopted in its first edition, Rogawski's "Calculus" "Calculus" success continues in a meticulously updated new edition. Revised in response to user feedback and classroom experiences, the new edition provides an even smoother teaching and learning experience.Hide synopsis
Hide Calculus: Early Transcendentals
This is a great book and it is very easy to understand. But the only downfall of it is I wish the answers to the selected answers showed each step to finding the solution because even though the problems are based on the same concept, some questions are harder. But it's an overall good back and totally worth it :)
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9780792357 of Braids (Mathematics and Its Applications (closed))
This book provides a comprehensive exposition of the theory of braids, beginning with the basic mathematical definitions and structures. Among the many topics explained in detail are: the braid group for various surfaces; the solution of the word problem for the braid group; braids in the context of knots and links (Alexander's theorem); Markov's theorem and its use in obtaining braid invariants; the connection between the Platonic solids (regular polyhedra) and braids; the use of braids in the solution of algebraic equations. Dirac's problem and special types of braids termed Mexican plaits are also discussed. Audience: Since the book relies on concepts and techniques from algebra and topology, the authors also provide a couple of appendices that cover the necessary material from these two branches of mathematics. Hence, the book is accessible not only to mathematicians but also to anybody who might have an interest in the theory of braids. In particular, as more and more applications of braid theory are found outside the realm of mathematics, this book is ideal for any physicist, chemist or biologist who would like to understand the mathematics of braids. With its use of numerous figures to explain clearly the mathematics, and exercises to solidify the understanding, this book may also be used as a textbook for a course on knots and braids, or as a supplementary textbook for a course on topology or algebra
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Pearson Debuts Interactive NovaNET Geometry
Pearson has launched a new online geometry course for its NovaNET 15.0 service targeted toward students in grades 6 through 12 and adult education.
Person's NovaNET is an online, standards-based courseware system designed for middle- and high-school students. Aligned to the 2007 Prentice Hall Geometry textbook, the new NovaNET Geometry course includes 77 multimedia lessons and includes instructional strategies for each. Additional features include:
Interactive practices;
Feedback and remediation;
Ongoing, formative and summative assessments for each lesson; and
Support for special needs students, including struggling readers.
According to Pearson, the previous geometry course remains available, but the new version is designed for split-semester geometry schedules divided into Geometry A and B
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fifth edition continues to improve on the features that have made it the market leader. The text offers a flexible organization, enabling instructors to adapt the book to their particular courses. The book is both complete and careful, and it continues to maintain its emphasis on algorithms and applications. Excellent exercise sets allow students to perfect skills as they practice. This new edition continues to feature numerous computer science applications-making this the ideal text for preparing students for advanced study.
I will once again be teaching discrete mathematics this summer, so I am searching through the mathematical publishing pathways looking for a suitable textbook. Therefore, that is the context within which I examined this book.
It certainly is the largest discrete book that I have encountered; including the appendices and problem solutions, there are over one thousand pages. Grimaldi has tried to include every topic that falls under the discrete mathematics tent. Therefore, this is a book that could be used for a two semester sequence in discrete mathematics.
When examining discrete books for possible adoption I start with the simple premise that logic, set theory and functions and relations must be covered very early. In my ideal world, they are the first three chapters. Set theory and relations are so fundamental a part of other areas that I am surprised when authors don't cover them first. The first chapter in this book covers basic counting principles. While this doesn't break too much from my ideal sequence, I see no overpowering reason why fundamental counting should be before set theory. Given that the rules of counting for sums and products can easily be related to sets, there is a strong justification for putting set theory first.Read more ›
This is a bad book if you are not already familiar with the basic concepts of the material. The author was more interested in showing worked examples than explaining concepts, and the more difficult problems in the exercise sections do not have solutions in the back of the book, so even 'self-learning' is extremely hard.
Unless you have a very good teacher, you will not benefit from the way the material is presented inside this book. 'Solutions' and 'examples' are presented 'as is' without explanations. One of my friends into math did mention it's not a bad reference guide for proofs, but he was as unimpressed with this book as a learning tool as I was. The level of rigor is very high, but the simple explanations to go with it are not present. I advise finding a good source on the subject instead of this unfriendly text, which has a target audience of math professionals.
I bought this book as a supplement to a summer course in Discrete Math, and since this was my first ever exposure to mathematical proof and dialog, I first thought this book mostly alien, with occaisional sections of brevity; it did help me fill in some gaps left behind in Rosen's book, especially on some basic proofs dealing with integers and with combinatorial reasoning--something this book is REALLY good at... I'm in my first course of Combinatorics with a teacher that assumes we know alot more calculus than we do. We use Tucker's Applied combinatorics 5th, and I was cruising along just fine until we hit Generating Functions. Brick wall. Rosen's book didn't cover it (well; there's a great page of known identities, but not an intro-level version), neither did Epp, so I dusted this tome off my shelf and cracked it open... section 9.1 presents Generating functions on such an easy to use language and analytic explanation that I went from getting every problem wrong in Tucker's book to getting them all right; all due to the clarity of exposition.
I've also found that as my 'mathematical maturity' has grown in the last year, so has the comprehensibility of this text. It may be too deep for a beginner--I would agree that it would be too much for all but your brightest minus an excellent teacher--but this book teaches 'real math' and does so *very* well.
In conclusion, if you have the available student loan $$ and want a very good supplementary book that you really can take with you to higher classes, put this at the top of your list.
I also own Epp and Rosen's discrete math texts, and have to say that for me ultimately I needed all three as a beginner; plus a few extra books from the library for special topics. But what I learned stayed with me and all three have their positives and negatives, but if I were to choose only one to stay on my shelf, THIS would be the one.
It is common to feel you need someone to explain what you are reading while studying from a book and even more if the subject is mathematics. That is what surprises readers while starting to explore this interesting book.
At the beginning it is hard to believe how simple it becomes to understand the different topics. That is a consequence of the easy way readers assimilate what is learnt by analyzing general and particular examples. That is the way in which the book presents the different units: the usual incomprehensible explanations are replaced by a lot of short examples which are easily understandable. Students not only feel they understand what they read but also enjoy and are attracted by a subject that is nice when comprehended.
Even if it seems to be too long, its more than eight-hundred pages do not reflect the period of time which takes to learn each unit. They are considerably short and are also divided in sections that reduce the difficulty of continuous reading, especially after having stopped for a wile, leaving aside the need to go over the last pages.
I consider this is a recommendable book for those students who are studying all the mathematic points which are analyzed in the volume. I believe it is the best complement for daily classes or a good option to study on your own.
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in the series of highly respected Swokowski/Cole mathematics texts retains the elements that have made it so popular with instructors and students alike: its exposition is clear, the time-tested exercise sets feature a variety of applications, its uncluttered layout is appealing, and the difficulty level of problems is appropriate and consistent. The goal of this text is to prepare students for further courses in mathematics.This book is set apart from the competition in a number of ways: it is mathematically sound, it focuses on preparing students for further courses in mathematics, and it has excellent problem sets. This edition has been improved in many respects. All of the chapters include numerous technology inserts with specific keystrokes for the TI-83 Plus and the TI-86, ideal for students who are working with a calculator for the first time. The new design of the text makes the technology inserts easily identifiable, so if a professor prefers to skip these sections it is simple to do so.
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Algebra 1
Description
Students learn how algebra relates to the physical world with an outstanding textbook presenting mathematics as a study of absolutes. Concepts are developed and mastered through an abundance of worked examples and student exercises. Designed to be used in grades 8 or 9 and is 374 pages.
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Find a South Houston help students develop the ability to see computational problems from a mathematical perspective. Discrete math is normally divided into six areas: sets, functions, and relations; basic logic; proof techniques; counting basics; graphs and trees; and discrete probability. I show students how these topics are interwoven with computer science applications.
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Created for the Connected Curriculum Project, the purpose of this module is to learn the basics of Mathematica for use in a linear algebra course. This is one resource within a larger set of learning modules hosted by...
Created by Lang Moore, David Smith and Jim Tomberg for the Connected Curriculum Project, the purpose of this module is to earn the basics of Maple for use in a linear algebra course. This is a portion of a larger set of...
Created by William Barker and David Smith for the Connected Curriculum Project, the purposes of this module are to develop a mathematical model for decay of radioactive substances, and to develop a technique for...
Created by Lawrence Moore and David Smith for the Connected Curriculum Project, the purpose of this module is to carry out an exploration of functions defined by data; to learn about data entry and plotting operations. ...
Created by David Smith for the Connected Curriculum Project, the purpose of this module is to apply linear algebra concepts to study the properties of sequences defined by difference equations. This is one within a...
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Pirnot believes that conceptual understanding is the key to a student's success in learning mathematics. He focuses on explaining the thinking behind the subject matter, so that students are able to truly understand the material and apply it to their lives. This textbook maintains a conversational tone throughout and focuses on motivating students and the mathematics through current applications. Ultimately, students who use this book will become more educated consumers of the vast amount of technical and mathematical information that they encounter daily, transforming them into mathematically aware citizens.
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What is Mathematics?: An Elementary Approach to Ideas and Methods
The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ...Show synopsisThe teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but not real understanding or greater intellectual independence. The new edition of this classic work seeks to address this problem. Its goal is to put the meaning back into mathematics. "Lucid . . . easily understandable".--Albert Einstein. 301 linecuts 1979-Paperback-Used-Acceptable--Shows substantial...Acceptable. 1979 study copy, shows heavy wear, text has markings, a...Fair. Good study copy, shows heavy wear, text has markings, a good study copy. We take great pride in accurately describing the condition of our books, ship within 48 hours and offer a 100% money back guarantee.
Description:Fair. The book has some limited-writing in pencil, as well as...Fair. The book has some limited-writing in pencil, as well as general-coverwear (edges, corners, scuffs/scratches, and possibly creases What is Mathematics?: An Elementary Approach to Ideas and Methods
For more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning ofmathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but does not lead to real understanding or to greater intellectual independence. This new edition of Richard Courant's and Herbert Robbins's classic work seeks to address thisproblem. Its goal is to put the meaning back into mathematics. Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics?, Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from naturalnumbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving.With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts. Brought up to date with a new chapter by Ian Stewart, What is Mathematics?, Second Edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat's LastTheorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved. Formal mathematics is like spelling and grammar--a matter of the correct application of local rules. Meaningful mathematics is like journalism--it tells an interesting story. But unlike some journalism, the story has to be true. The best mathematics is like literature--it brings a story to lifebefore your eyes and involves you in it, intellectually and emotionally. What is Mathematics is like a fine piece of literature--it opens a window onto the world of mathematics for anyone interested to view
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Beginning Algebra With Applications
9780618803590
ISBN:
0618803599
Pub Date: 2007 Publisher: Houghton Mifflin
Summary: Intended for developmental math courses in beginning imm...ediate feedback, reinforcing the concept, identifying problem areas, and, overall, promoting student success."New!" "Interactive Exercises" appear at the beginning of an objective's exercise set (when appropriate), and provide students with guided practice on some of the objective's underlying principles."New!" "Think About It" Exercises are conceptual in nature and appear near the end of an objective's exercise set. They ask the students to think about the objective's concepts, make generalizations, and apply them to more abstract problems. The focus is on mental mathematics, not calculation or computation, and help students synthesize concepts."New!" "Important Points" have been highlighted to capture students' attention. With these signposts, students are able to recognize what is most important and to study more efficiently."New!" A Concepts of Geometry section has been added to Chapter 1."New!" Coverage of operations on fractions has been changed in Section 1.3 so that multiplication and division of rational numbers are presented first, followed by addition and subtraction"New!" A Complex Numbers section has been added to Chapter 11, "Quadratic Equations.""New Media!" Two key components have been added to the technology package: HM Testing (powered by Diploma) and, as part of the Eduspace course management tool, HM Assess, an online diagnostic assessment tool.
Aufmann, Richard N. is the author of Beginning Algebra With Applications, published 2007 under ISBN 9780618803590 and 0618803599. Two hundred ninety five Beginning Algebra With Applications textbooks are available for sale on ValoreBooks.com, one hundred twenty three used from the cheapest price of $4.59, or buy new starting at $45.29.[read more]
Ships From:Jackosnville, FLShipping:StandardComments:Book is in acceptable condition; cover shows signs of wear. Pages include markings from pencil/ p... [more]Book is in acceptable condition; cover shows signs of wear. Pages include markings from pencil/ pen/highlighter, but text is not obscured. Used stickers on binding and back cover. [less]
Ships From:Castleton, NYShipping:StandardComments: 0618803599 AtAGlance Books--Orders ship next business day, with tracking numbers, from our wareh... [more] 0618803599 AtAGlance Books--Orders ship next business day, with tracking numbers, from our warehouse in upstate NY. This book is in brand new condition
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Complex Analysis
Complex Analysis
This is a free online course offered by the Saylor Foundation.'...
More
This is a free online course offered by the Saylor Foundation.
' inherently geometric flavor of complex analysis, this course will feel quite different from Real Analysis, although many of the same concepts, such as open sets, metrics, and limits will reappear. Simply put, you will be working with lines and sets and very specific functions on the complex plane—drawing pictures of them and teasing out all of their idiosyncrasies. You will again find yourself calculating line integrals, just as in multivariable calculus. However, the techniques you learn in this course will help you get past many of the seeming dead-ends you ran up against in calculus. Indeed, most of the definite integrals you will learn to evaluate in Unit 7 come directly from problems in physics and cannot be solved except through techniques from complex variables.
We will begin by studying the minimal algebraically closed extension of real numbers: the complex numbers. The Fundamental Theorem of Algebra states that any non-constant polynomial with complex coefficients has a zero in the complex numbers. This makes life in the complex plane very interesting. We will also review a bit of the geometry of the complex plane and relevant topological concepts, such as connectedness.
In Unit 2, we will study differential calculus in the complex domain. The concept of analytic or holomorphic function will be introduced as complex differentiability in an open subset of the complex numbers. The Cauchy-Riemann equations will establish a connection between analytic functions and differentiable functions depending on two real variables. In Unit 3, we will review power series, which will be the link between holomorphic and analytic functions. In Unit 4, we will introduce certain special functions, including exponentials and trigonometric and logarithmic functions. We will consider the Möbius Transformation in some detail.
In Units 5, 6, and 7 we will study Cauchy Theory, as well as its most important applications, including the Residue Theorem. We will compute Laurent series, and we will use the Residue Theorem to evaluate certain integrals on the real line which cannot be dealt with through methods from real variables alone. Our final unit, Unit 8, will discuss harmonic functions of two real variables, which are functions with continuous second partial derivatives that satisfy the Laplace equation, conformal mappings, and the Open Mapping Theorem.'
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College Algebra: Enhanced with Graphing Utilities
Michael Sullivan??? s time-tested approach focuses students on the fundamental skills they need for the course: preparing for class, practicing ...Show synopsisMichael Sullivan??? s time-tested approach focuses students on the fundamental skills they need for the course: preparing for class, practicing with homework, and reviewing the concepts. The Enhanced with Graphing Utilities Series has evolved to meet today??? s course needs by integrating the usage of graphing calculator, active-learning, and technology in new ways to help students be successful in their course, as well as in their future endeavors95649321795649-5-1Good. Hardcover. Missing components. May include moderately...Good. Hardcover. Missing components. May include moderately worn cover, writing, markings or slight discoloration. SKU: 978032183211532115The book is not that bad but the way it is organized leaves a lillte to be desired. I have found that the index is not correct when looking for particular items as they are off by a few pages and this has happened several times.
Still, this is an OK book for my college
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Trigonometry provides you with all you need to know to understand the basic concepts of trigonometry — whether you need a supplement to your textbook and classes or an at-a-glance reference. Trigonometry isn't just measuring angles; it has many applications in the real world, such as in navigation, surveying, construction, and many other branches of science, including mathematics and physics. As you work your way through this review, you'll be ready to tackle such concepts as
Trigonometric functions, such as sines and cosines
Graphs and trigonometric identities
Vectors, polar coordinates, and complex numbers
Inverse functions and equations
You can use CliffsQuickReview Trigonometry in any way that fits your personal style for study and review — you decide what works best with your needs. You can read the book from cover to cover or just look for the information you want and put it back on the shelf for later. Here are just a few ways you can search for topics:
With titles available for all the most popular high school and college courses, CliffsQuickReview guides are a comprehensive resource that can help you get the best possible grades.
Editorial Reviews
From the Back Cover
Leading educators help you succeed When it comes to pinpointing the stuff you really need to know, nobody does it better than CliffsNotes. This fast, effective tutorial helps you master core trigonometer concepts — from trigonometric functions and trigonometric identities to vectors, polar coordinates, and complex numbers — and get the best possible grade.
At CliffsNotes, we're dedicated to helping you do your best, no matter how challenging the subject. Our authors are veteran teachers and talented writers who know how to cut to the chase—and zero in on the essential information you need to succeed.
Master the basics—fast
Complete coverage of core concepts
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About the Author
DAVE KAY is a writer, engineer, and aspiring naturalist and artist. He has written or cowritten more than a dozen computer books.
Most Helpful Customer Reviews
This book is good for one thing, a Quick Review. I suppose I can't blame CliffsNotes, since that is given in the title of the book. I bought this book when I was beginning to learn Trigonometry, along with another textbook. I found that a lot of the things the textbook went over was not even mentioned in this Quick Review book. If you have taken Trigonometry in the past and want to refresh some of the basic trigonometry concepts, then this book is for you. However, if you know nothing about Trigonometry and want to learn about it, this book will do nothing but CONFUSE you.
Most people that will look into these books are people who want to learn the subject for their first time, so take my advice: instead of buying this book, save that money towards a better, complete basic trigonometry textbook.
I'm a (returning :P) university Freshman preparing for the College Board CLEP tests. I was already familiar with the material covered in this book, but needed to refresh my memory. This review turned out to be *exactly* what I needed. The author's ability to explain the material to the student are just shy of enlightening. The discussions & theorem proofs are written in a very concise, clear style. I'm a big advocate of the Cliff's QuickReview series. Intended as a course supplement, these books are also *GREAT* for students wanting to refine their skills. Most of them are also very accessible to students with less familiarity on the subject; trying to learn it for the first time. After reading this, I bought the Calculus & Differential Equations QuickReviews & I'm looking forward to reading them!
After several years in a corporate engineering job, I started moonlighting as a math tutor. The Cliff's Quick Review Guides are wonderful to have in my "back pocket" when I need to quickly look something up that is covered in dust in the "archives of my brain."
Trig is something you always have to practice if you want to remain competent. When practicing basic trig problems (identities, equations, vectors, graphs, angles, cmplx.#'s....) this book gives me just enough explanation with the info I need. But like others said, you should already have some trig under your belt before purchasing it.
| 677.169 | 1 |
Mathematics
Finite Mathematics
1 Semester / 1 Credit(s)
This course expands students' mathematical reasoning and problem-solving skills as they cover topics such as mathematics of voting, weighted voting systems, fair division, apportionment, Euler circuits, Hamilton circuits, mathematics of networks, and game theory. The course will encourage students to make mathematical connections from the classroom to the world after high school, while learning the importance of mathematics in everyday life. This course is offered as an addition to Pre-Calculus/Trigonometry, not a replacement. A SCIENTIFIC CALCULATOR IS REQUIRED.
Probability & Statistics
1 Semester / 1 Credit(s)
This course introduces and examines the statistical topics that are applied during the decision-making process. Topics include: descriptive statistics, probability, and statistical inference. Techniques investigated include: data collection through experiments or surveys, data organization, sampling theory and making inferences from samples. Computers are used for data analysis and data presentation. This course should not be taken as a replacement for Pre-Calculus/Trigonometry in a college preparatory course of study. A SCIENTIFIC CALCULATOR IS REQUIRED.
Honors Probability & Statistics
1 Semester / 1 Credit(s)
This course introduces students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. This course is required for students taking MAT 230 in the spring. Students complete a rigorous study of the following concepts: describing patterns and departures from patterns, planning and conducting a statistical study, exploring random phenomena using probability and simulation, and estimating population parameters and testing hypotheses. A GRAPHING CALCULATOR IS REQUIRED.
AP Calculus
2 Semesters / 2 Credit(s)
This course is intended for students who have a thorough knowledge of college preparatory mathematics. It covers both the theoretical basis for and applications of differentiation and integration. Concepts and problems are approached graphically, numerically, analytically and verbally. All students enrolled in this course will take the AP Calculus (AB) Exam. A GRAPHING CALCULATOR IS REQUIRED.
Honors Pre-Calculus/Trigonometry
2 Semesters / 2 Credit(s)
This course covers the same topics as Pre-Calculus/Trigonometry listed above. Greater emphasis is placed on applications and developing the depth of understanding and skills necessary for success in AP Calculus. This course is required for students who plan to take AP Calculus. A GRAPHING CALCULATOR IS REQUIRED.
This course expands and develops the topics learned in Honors Algebra I. Content areas include the topics listed for Algebra II with greater emphasis on preparation for upper level mathematics content. The course is required for students who plan to take AP Calculus, and it is recommended that this course be taken at the same time as Honors Geometry unless Honors Geometry was taken as a freshman. A GRAPHING CALCULATOR IS REQUIRED.
Algebra II
2 Semesters / 2 Credit(s)
This course further develops the topics learned in Algebra I with extensive work on learning to graph equations and inequalities in the Cartesian coordinate system. Topics include: relations and functions, systems of equations and inequalities, conic sections, polynomials, algebraic fractions, logarithmic and exponential functions, sequences and series, and counting principles and probability. A GRAPHING CALCULATOR IS REQUIRED.
Honors Geometry
2 Semesters / 2 Credit(s)
This course covers the same topics as Geometry, but with greater emphasis on complex direct deductive proof and indirect proof and on utilization of more advanced algebraic techniques. Content is extended to include topics such as analytic geometry and the interrelationships of inscribed polyhedra. A GRAPHING CALCULATOR IS REQUIRED.
Geometry (1&2)
1 Semester / 2 Credit(s)
This Geometry course is designed for those students who did not receive a C- average or better in Geometry. This course will meet everyday.
Geometry
2 Semesters / 2 Credit(s)
The purpose of Geometry is to use logical thought processes to develop spatial skills. Students work with figures in one, two- and three-dimensional Euclidean space. The interrelationships of the properties of figures are studied through visualization, using computer drawing programs and constructions, as well as through formal proof and algebraic applications. A GRAPHING CALCULATOR IS REQUIRED.
Honors Algebra I
2 Semesters / 2 Credit(s)
The same topics as in Algebra I are covered with more emphasis on problem solving and critical thinking skills in order to challenge the mathematically talented student. Projects are incorporated into the lessons for the purpose of applying the mathematical concepts. A GRAPHING CALCULATOR IS REQUIRED.
Algebra I (1&2)
1 Semester / 2 Credit(s)
This course is designed for those students who did not receive a B- average or better in Algebra I or did not pass the ISTEP+ Algebra I Graduation Exam. This course will meet everyday.
Algebra Enrichment is a mathematics support course for Algebra I. The course provides students with additional time to build the foundations necessary for high school math courses, while concurrently having access to rigorous, grade-level appropriate courses. The five critical areas of Algebra Enrichment align with the critical areas of Algebra I: Relationships between Quantities and Reasoning with Equations; Linear and Exponential Relationships; Descriptive Statistics; Expressions and Equations; and Quadratic Functions and Modeling. However, whereas Algebra I contains exclusively grade-level content, Algebra Enrichment combines standards from high school courses with foundational standards from the middle grades. This course counts as a Mathematics Course for the General Diploma only or as an Elective for the Core 40, Core 40 with Academic Honors and Core 40 with Technical Honors diplomas. Algebra Enrichment is designed as a support course for Algebra I. As such, a student taking Algebra Enrichment should also be enrolled in Algebra I during the same academic year.
A GRAPHING CALCULATOR IS REQUIRED.
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| 677.169 | 1 |
Buy Article:
Abstract:
Designing an optimal Norman window is a standard calculus exercise. How much more difficult (or interesting) is its generalization to deploying multiple semicircles along the head (or along head and sill, or head and jambs)? What if we use shapes beside semi-circles? As the number of
copies of the shape increases and the optimal Norman windows approach a rectangular shape, what proportions arise? How does the perimeter of the limiting rectangle compare to the limit of the perimeters? These questions provide challenging optimization problems for students and the graphical
depiction of the geometry of these window sequences illustrates more vividly than sequences of numbers, the concept of limit.
The College Mathematics Journal is designed to enhance classroom learning and stimulate thinking regarding undergraduate mathematics. CMJ publishes articles, short Classroom Capsules, problems, solutions, media reviews and other pieces. All are aimed at the college mathematics curriculum with emphasis on topics taught in the first two years.
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to view
additional materials from
7
00:00:13 --> 00:00:16
hundreds of MIT courses, visit
MIT OpenCourseWare
8
00:00:16 --> 00:00:22
at ocw.mit.edu.
9
00:00:22 --> 00:00:28
PROFESSOR STRANG: Just to give
an overview in three lines: the
10
00:00:28 --> 00:00:32
text is the book of that name,
Computational Science
11
00:00:32 --> 00:00:33
and Engineering.
12
00:00:33 --> 00:00:38
That was completed just last
year, so it really ties
13
00:00:38 --> 00:00:41
pretty well with the course.
14
00:00:41 --> 00:00:43
I don't cover everything in
the book, by all means.
15
00:00:43 --> 00:00:47
And I don't, certainly, don't
stand here and read the book.
16
00:00:47 --> 00:00:50
That would be no good.
17
00:00:50 --> 00:00:55
But you'll be able, if you
miss a class -- well,
18
00:00:55 --> 00:00:56
don't miss a class.
19
00:00:56 --> 00:01:01
But if you miss a class,
you'll be able, probably,
20
00:01:01 --> 00:01:05
to see roughly what we did.
21
00:01:05 --> 00:01:08
OK, so the first part of
the semester is applied
22
00:01:08 --> 00:01:10
linear algebra.
23
00:01:10 --> 00:01:13
And I don't know how many of
you have had a linear algebra
24
00:01:13 --> 00:01:16
course, and that's why I
thought I would start
25
00:01:16 --> 00:01:19
with a quick review.
26
00:01:19 --> 00:01:23
And you'll catch on.
27
00:01:23 --> 00:01:26
I want matrices to come
to life, actually.
28
00:01:26 --> 00:01:31
You know, instead of just being
a four by four array of
29
00:01:31 --> 00:01:34
numbers, there are four by
four, or n by n or m by n
30
00:01:34 --> 00:01:36
array of special numbers.
31
00:01:36 --> 00:01:38
They have a meaning.
32
00:01:38 --> 00:01:41
When they multiply a
vector, they do something.
33
00:01:41 --> 00:01:47
And so it's just part of this
first step is just, like,
34
00:01:47 --> 00:01:50
getting to recognize,
what's that matrix doing?
35
00:01:50 --> 00:01:52
Where does it come from?
36
00:01:52 --> 00:01:53
What are its properties?
37
00:01:53 --> 00:01:57
So that's a theme at the start.
38
00:01:57 --> 00:02:04
Then differential equations,
like Laplace's equation,
39
00:02:04 --> 00:02:06
are beautiful examples.
40
00:02:06 --> 00:02:11
So here we get, especially,
to numerical methods;
41
00:02:11 --> 00:02:14
finite differences, finite
elements, above all.
42
00:02:14 --> 00:02:17
So I think in this class you'll
really see how finite elements
43
00:02:17 --> 00:02:20
work, and other ideas.
44
00:02:20 --> 00:02:21
All sorts of ideas.
45
00:02:21 --> 00:02:25
And then the last part of the
course is about Fourier.
46
00:02:25 --> 00:02:29
That's Fourier series, that
you may have seen, and
47
00:02:29 --> 00:02:30
Fourier integrals.
48
00:02:30 --> 00:02:34
But also, highly important,
Discrete Fourier
49
00:02:34 --> 00:02:36
Transform, DFT.
50
00:02:36 --> 00:02:40
That's a fundamental step
for understanding what
51
00:02:40 --> 00:02:42
a signal contains.
52
00:02:42 --> 00:02:46
Yeah, so that's great
stuff, Fourier.
53
00:02:46 --> 00:02:52
OK, what else should I
say before I start?
54
00:02:52 --> 00:02:56
I said this was my favorite
course, and maybe I
55
00:02:56 --> 00:03:01
elaborate a little.
56
00:03:01 --> 00:03:06
Well, I think what I want to
say is that I really feel my
57
00:03:06 --> 00:03:12
life is here to teach you
and not to grade you.
58
00:03:12 --> 00:03:15
I'm not going to spend this
semester worrying about
59
00:03:15 --> 00:03:18
grades, and please don't.
60
00:03:18 --> 00:03:19
They come out fine.
61
00:03:19 --> 00:03:22
We've got lots to learn.
62
00:03:22 --> 00:03:26
And I'll do my very best
to explain it clearly.
63
00:03:26 --> 00:03:30
And I know you'll do your best.
64
00:03:30 --> 00:03:31
I know from experience.
65
00:03:31 --> 00:03:36
This class goes for it
and does it right.
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00:03:36 --> 00:03:40
So that's what
makes it so good.
67
00:03:40 --> 00:03:41
OK.
68
00:03:41 --> 00:03:46
Homeworks, by the way, well,
the first homework will simply
69
00:03:46 --> 00:03:50
be a way to get a grade list,
a list of everybody
70
00:03:50 --> 00:03:52
taking the course.
71
00:03:52 --> 00:03:55
They won't be graded
in great detail.
72
00:03:55 --> 00:03:59
Too large a class.
73
00:03:59 --> 00:04:03
And you're allowed to talk to
each other about homework.
74
00:04:03 --> 00:04:05
So homework is not
an exam at all.
75
00:04:05 --> 00:04:09
So let me just leave any
discussion of exams and
76
00:04:09 --> 00:04:12
grades for the future.
77
00:04:12 --> 00:04:14
I'll tell you, you'll
see how informally the
78
00:04:14 --> 00:04:18
first homework will be.
79
00:04:18 --> 00:04:21
And I hope it'll go
up on the website.
80
00:04:21 --> 00:04:23
The first homework
will be for Monday.
81
00:04:23 --> 00:04:29
So it's a bit early, but
it's pretty open-ended.
82
00:04:29 --> 00:04:33
If you could take three
problems from 1.1, the first
83
00:04:33 --> 00:04:38
section of the book, any three,
and any three problems from
84
00:04:38 --> 00:04:45
1.2, and print your name on the
homework -- because we're going
85
00:04:45 --> 00:04:48
to use that to create
the grade list -- I'll
86
00:04:48 --> 00:04:50
be completely happy.
87
00:04:50 --> 00:04:52
Well, especially if you get
them right and do them
88
00:04:52 --> 00:04:53
neatly and so on.
89
00:04:53 --> 00:04:59
But actually we won't know.
90
00:04:59 --> 00:05:02
So that's for Monday.
91
00:05:02 --> 00:05:03
OK.
92
00:05:03 --> 00:05:05
And we'll talk more about it.
93
00:05:05 --> 00:05:11
I'll announce the TA on the
website and the TA hours, the
94
00:05:11 --> 00:05:12
office hours, and everything.
95
00:05:12 --> 00:05:17
There'll be a Friday afternoon
office hour, because homeworks
96
00:05:17 --> 00:05:20
will typically come Monday.
97
00:05:20 --> 00:05:20
OK.
98
00:05:20 --> 00:05:27
Questions about the course
before I just start?
99
00:05:27 --> 00:05:30
OK.
100
00:05:30 --> 00:05:31
Another time for
questions, too.
101
00:05:31 --> 00:05:41
OK, so can we just start
with that matrix?
102
00:05:41 --> 00:05:45
So I said about matrices, I'm
interested in their properties.
103
00:05:45 --> 00:05:47
Like, I'm going to
ask you about that.
104
00:05:47 --> 00:05:51
And then, I'm interested
in their meaning.
105
00:05:51 --> 00:05:53
Where do they come from?
106
00:05:53 --> 00:05:56
You know, why that matrix
instead of some other?
107
00:05:56 --> 00:06:01
And then, the numerical part
is how do we deal with them?
108
00:06:01 --> 00:06:05
How do we solve a linear system
with that coefficient matrix?
109
00:06:05 --> 00:06:07
What can we say
about the solution?
110
00:06:07 --> 00:06:09
So the purpose.
111
00:06:09 --> 00:06:10
Right.
112
00:06:10 --> 00:06:15
OK, now help me out.
113
00:06:15 --> 00:06:18
So I guess my plan with the
video taping is, whatever
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00:06:18 --> 00:06:20
you say, I'll repeat.
115
00:06:20 --> 00:06:27
So say it as clearly as
possible, and it's fantastic
116
00:06:27 --> 00:06:30
to have discussion,
conversation here.
117
00:06:30 --> 00:06:34
So I'll just repeat it so that
it safely gets on the tape.
118
00:06:34 --> 00:06:35
So tell me its properties.
119
00:06:35 --> 00:06:41
Tell me the first property that
you notice about that matrix.
120
00:06:41 --> 00:06:41
Symmetric.
121
00:06:41 --> 00:06:42
Symmetric.
122
00:06:42 --> 00:06:44
Right.
123
00:06:44 --> 00:06:46
I could have slowed down a
little and everybody probably
124
00:06:46 --> 00:06:48
would have said that at once.
125
00:06:48 --> 00:06:51
So that's a symmetric matrix.
126
00:06:51 --> 00:06:54
Now we might as well pick
up some matrix notation.
127
00:06:54 --> 00:06:58
How do I express the fact that
this a symmetric matrix?
128
00:06:58 --> 00:07:03
In simple matrix notation,
I would say that K is
129
00:07:03 --> 00:07:07
the same as K transpose.
130
00:07:07 --> 00:07:11
The transpose, everybody knows,
it comes from -- oh, I
131
00:07:11 --> 00:07:14
shouldn't say this -- flipping
it across the diagonal.
132
00:07:14 --> 00:07:17
That's not a very
"math" thing to do.
133
00:07:17 --> 00:07:21
But that's the way
to visualize it.
134
00:07:21 --> 00:07:27
And let me use a capital
T for transpose.
135
00:07:27 --> 00:07:30
So it's symmetric.
136
00:07:30 --> 00:07:31
Very important.
137
00:07:31 --> 00:07:32
Very, very important.
138
00:07:32 --> 00:07:34
That's the most important
class of matrices,
139
00:07:34 --> 00:07:35
symmetric matrices.
140
00:07:35 --> 00:07:39
We'll see them all the time,
because they come from
141
00:07:39 --> 00:07:41
equilibrium problems.
142
00:07:41 --> 00:07:44
They come from all sorts of
-- they come everywhere
143
00:07:44 --> 00:07:47
in applications.
144
00:07:47 --> 00:07:49
And we will be doing
applications.
145
00:07:49 --> 00:07:55
The first week or week and a
half, you'll see pretty much
146
00:07:55 --> 00:08:00
discussion of matrices and the
reasons, what their meaning is.
147
00:08:00 --> 00:08:03
And then we'll get to
physical applications;
148
00:08:03 --> 00:08:05
mechanics and more.
149
00:08:05 --> 00:08:06
OK.
150
00:08:06 --> 00:08:08
All right.
151
00:08:08 --> 00:08:11
Now I'm looking for
properties, other
152
00:08:11 --> 00:08:13
properties, of that matrix.
153
00:08:13 --> 00:08:18
Let me write "2" here so that
you got a spot to put it.
154
00:08:18 --> 00:08:21
What are you going to tell
me next about that matrix?
155
00:08:21 --> 00:08:22
Periodic.
156
00:08:22 --> 00:08:23
Well, okay.
157
00:08:23 --> 00:08:25
Actually, that's
a good question.
158
00:08:25 --> 00:08:31
Let me write
periodic down here.
159
00:08:31 --> 00:08:36
You're using that word, because
somehow that pattern is
160
00:08:36 --> 00:08:37
suggesting something.
161
00:08:37 --> 00:08:43
But you'll see I have a little
more to add before I would
162
00:08:43 --> 00:08:44
use the word periodic.
163
00:08:44 --> 00:08:47
So that's great to
see that here.
164
00:08:47 --> 00:08:47
What else?
165
00:08:47 --> 00:08:50
Somebody else was going
to say something.
166
00:08:50 --> 00:08:51
Please.
167
00:08:51 --> 00:08:52
Sparse!
168
00:08:52 --> 00:08:53
Oh, very good.
169
00:08:53 --> 00:08:54
Sparse.
170
00:08:54 --> 00:08:59
That's also an obvious
property that you see from
171
00:08:59 --> 00:09:01
looking at the matrix.
172
00:09:01 --> 00:09:03
What does sparse mean?
173
00:09:03 --> 00:09:05
Mostly zeros.
174
00:09:05 --> 00:09:07
Well that isn't mostly
zeros, I guess.
175
00:09:07 --> 00:09:11
I mean, that's got what,
out of sixteen entries,
176
00:09:11 --> 00:09:13
it's got six zeros.
177
00:09:13 --> 00:09:14
That doesn't sound like sparse.
178
00:09:14 --> 00:09:19
But when I grow the matrix
-- because this is
179
00:09:19 --> 00:09:21
just a four by four.
180
00:09:21 --> 00:09:24
I would even call this one K_4.
181
00:09:24 --> 00:09:30
When the matrix grows to 100
by 100, then you really
182
00:09:30 --> 00:09:31
see it as sparse.
183
00:09:31 --> 00:09:35
So if that matrix was
100 by 100, how many
184
00:09:35 --> 00:09:37
non-zeros would it have?
185
00:09:37 --> 00:09:44
So if n is 100, then the number
of non-zeros -- wow, that's the
186
00:09:44 --> 00:09:46
first MATLAB command
I've written.
187
00:09:46 --> 00:09:51
A number of non-zeros of
K would be -- anybody
188
00:09:51 --> 00:09:53
know what it would be?
189
00:09:53 --> 00:10:00
I'm just asking to go
up to five by five.
190
00:10:00 --> 00:10:03
I'm asking you to keep
that pattern alive.
191
00:10:03 --> 00:10:08
Twos on the diagonal, minus
ones above and below.
192
00:10:08 --> 00:10:13
So yeah, so 298, would it be?
193
00:10:13 --> 00:10:20
A hundred diagonal entries,
99 and 99, maybe 298?
194
00:10:20 --> 00:10:27
298 out of 100 by
100 would be what?
195
00:10:27 --> 00:10:29
It's been a long summer.
196
00:10:29 --> 00:10:32
Yeah, a lot of zeros.
197
00:10:32 --> 00:10:32
A lot.
198
00:10:32 --> 00:10:33
Right.
199
00:10:33 --> 00:10:37
Because the matrix has got what
100 x 100, 10,000 entries.
200
00:10:37 --> 00:10:39
Out of 10,000.
201
00:10:39 --> 00:10:42
So that's sparse.
202
00:10:42 --> 00:10:46
But we see those all the
time, and fortunately we do.
203
00:10:46 --> 00:10:48
Because, of course, this
matrix, or even 100
204
00:10:48 --> 00:10:53
by 100, we could deal
with if it was dense.
205
00:10:53 --> 00:10:58
But 10,000, 100,000, or 1
million, which happens
206
00:10:58 --> 00:11:02
all the time now in
scientific computation.
207
00:11:02 --> 00:11:05
A million by million dense
matrix is not a nice
208
00:11:05 --> 00:11:07
thing to think about.
209
00:11:07 --> 00:11:13
A million by million matrix
like this is a cinch.
210
00:11:13 --> 00:11:14
OK.
211
00:11:14 --> 00:11:15
So sparse.
212
00:11:15 --> 00:11:18
What else do you want to say?
213
00:11:18 --> 00:11:19
Toeplitz.
214
00:11:19 --> 00:11:22
Holy Moses.
215
00:11:22 --> 00:11:23
Exactly right.
216
00:11:23 --> 00:11:29
But I want to say, before I
use that word, so that'll be
217
00:11:29 --> 00:11:30
my second MATLAB command.
218
00:11:30 --> 00:11:31
Thanks.
219
00:11:31 --> 00:11:33
Toeplitz.
220
00:11:33 --> 00:11:35
What's that mean?
221
00:11:35 --> 00:11:40
So this matrix has a
property that we see
222
00:11:40 --> 00:11:46
right away, which is?
223
00:11:46 --> 00:11:51
I want to stay with Toeplitz
but everybody tell me something
224
00:11:51 --> 00:11:54
more about properties
of that matrix.
225
00:11:54 --> 00:11:56
Tridiagonal.
226
00:11:56 --> 00:12:02
Tridiagonal, so that's almost
a special subcase of sparse.
227
00:12:02 --> 00:12:05
It has just three diagonals.
228
00:12:05 --> 00:12:08
Tridiagonal matrices
are truly important.
229
00:12:08 --> 00:12:11
They come in all the time,
we'll see that they come from
230
00:12:11 --> 00:12:14
second order differential
equations, which are, thanks
231
00:12:14 --> 00:12:17
to Newton, the big ones.
232
00:12:17 --> 00:12:23
Ok, now it's more than
tridiagonal and what more?
233
00:12:23 --> 00:12:26
So what further, we're
getting deeper now.
234
00:12:26 --> 00:12:32
What patterns do you see beyond
just tridiagonal, because
235
00:12:32 --> 00:12:35
tridiagonal would allow any
numbers there but those are
236
00:12:35 --> 00:12:39
not, there's more of a
pattern than just three
237
00:12:39 --> 00:12:42
diagonals, what is it?
238
00:12:42 --> 00:12:45
Those diagonals are constant.
239
00:12:45 --> 00:12:48
If I run down each of
those three diagonals,
240
00:12:48 --> 00:12:50
I see the same number.
241
00:12:50 --> 00:12:53
Twos, minus ones, minus
ones, and that's what
242
00:12:53 --> 00:12:55
the word Toeplitz means.
243
00:12:55 --> 00:13:05
Toeplitz is constant diagonal.
244
00:13:05 --> 00:13:05
Ok.
245
00:13:05 --> 00:13:09
And that kind of matrix
is so important.
246
00:13:09 --> 00:13:18
It corresponds, yeah, if we
were in EE, I would use the
247
00:13:18 --> 00:13:23
words time invariant filter,
linear time invariant.
248
00:13:23 --> 00:13:27
So it's linear because we're
dealing with a matrix.
249
00:13:27 --> 00:13:31
And it's time invariant,
shift invariant.
250
00:13:31 --> 00:13:36
I just use all these equivalent
words to mean that we're seeing
251
00:13:36 --> 00:13:41
the same thing row by row,
except of course, at shall
252
00:13:41 --> 00:13:44
I call that the boundary?
253
00:13:44 --> 00:13:46
That's like, the end of the
system and this is like the
254
00:13:46 --> 00:13:51
other end and there
it's chopped off.
255
00:13:51 --> 00:13:56
But if it was ten by ten I
would see that row eight times.
256
00:13:56 --> 00:13:58
100 by 100 I'd see it 98 times.
257
00:13:58 --> 00:14:05
So it's constant diagonals and
the guy who first studied
258
00:14:05 --> 00:14:08
that was Toeplitz.
259
00:14:08 --> 00:14:14
And we wouldn't need that great
historical information except
260
00:14:14 --> 00:14:18
that MATLAB created a command
to create that matrix.
261
00:14:18 --> 00:14:25
K, MATLAB is all set to
create Toeplitz matrices.
262
00:14:25 --> 00:14:30
Yeah, so I'll have to put
what MATLAB would put.
263
00:14:30 --> 00:14:37
I realize I'm already
using the word MATLAB.
264
00:14:37 --> 00:14:42
I think that MATLAB language
is really convenient to
265
00:14:42 --> 00:14:44
talk about linear algebra.
266
00:14:44 --> 00:14:46
And how many know MATLAB
or have used it?
267
00:14:46 --> 00:14:49
Yeah.
268
00:14:49 --> 00:14:51
You know it better than I.
269
00:14:51 --> 00:14:56
I talk a good line with MATLAB
but I, the code never runs.
270
00:14:56 --> 00:14:58
Never!
271
00:14:58 --> 00:15:02
I always forget some
stupid semicolon.
272
00:15:02 --> 00:15:04
You may have had
that experience.
273
00:15:04 --> 00:15:11
And I just want to say it now
that there are other languages,
274
00:15:11 --> 00:15:14
and if you want to do homeworks
and want to do your own work
275
00:15:14 --> 00:15:18
in other languages,
that makes sense.
276
00:15:18 --> 00:15:23
So the older established
alternatives were Mathematica
277
00:15:23 --> 00:15:29
and Maple and those two have
symbolic, they can deal with
278
00:15:29 --> 00:15:33
algebra as well as numbers.
279
00:15:33 --> 00:15:34
But there are newer languages.
280
00:15:34 --> 00:15:37
I don't know if you know them.
281
00:15:37 --> 00:15:41
I just know my friends say,
Yes they're terrific.
282
00:15:41 --> 00:15:46
Python is one.
283
00:15:46 --> 00:15:47
And R.
284
00:15:47 --> 00:15:51
I've just had a email saying,
Tell your class about R.
285
00:15:51 --> 00:15:53
And others.
286
00:15:53 --> 00:15:59
Ok, so but we'll use MATLAB
language because that's really
287
00:15:59 --> 00:16:01
a good common language.
288
00:16:01 --> 00:16:03
Ok, so what is a
Toeplitz matrix?
289
00:16:03 --> 00:16:06
A Toeplitz matrix is one
with constant diagonals.
290
00:16:06 --> 00:16:09
You could use the word time
invariant, linear time
291
00:16:09 --> 00:16:11
invariant filter.
292
00:16:11 --> 00:16:16
And to create K, this
is an 18.085 command.
293
00:16:16 --> 00:16:19
It's just set up for us.
294
00:16:19 --> 00:16:26
I can create K by telling
the system the first row.
295
00:16:26 --> 00:16:32
Two, minus one, zero, zero.
296
00:16:32 --> 00:16:37
That would, then if it wasn't
symmetric I would have to
297
00:16:37 --> 00:16:40
give the first column also.
298
00:16:40 --> 00:16:42
Toeplitz would be constant
diagonal, it doesn't
299
00:16:42 --> 00:16:44
have to be symmetric.
300
00:16:44 --> 00:16:47
But if it's symmetric, then the
first row and first column are
301
00:16:47 --> 00:16:50
the same vector, so I just
have to give that vector.
302
00:16:50 --> 00:16:55
Okay, so that's the
quickest way to create K.
303
00:16:55 --> 00:17:00
And of course, if it was bigger
then I would, rather than
304
00:17:00 --> 00:17:08
writing 100 zeros, I could
put zeros of 98 and one.
305
00:17:08 --> 00:17:09
Wouldn't I have to say that?
306
00:17:09 --> 00:17:11
Or is it one and 98?
307
00:17:11 --> 00:17:15
You see why it doesn't run.
308
00:17:15 --> 00:17:18
Well I guess I'm thinking
of that as a row.
309
00:17:18 --> 00:17:19
I don't know.
310
00:17:19 --> 00:17:24
Anyway.
311
00:17:24 --> 00:17:27
I realize getting this
videotaped means I'm supposed
312
00:17:27 --> 00:17:28
to get things right!
313
00:17:28 --> 00:17:31
Usually it's like, we'll
get it right later.
314
00:17:31 --> 00:17:36
But anyway, that might work.
315
00:17:36 --> 00:17:37
Okay.
316
00:17:37 --> 00:17:39
So there's a command
that you know.
317
00:17:39 --> 00:17:44
Zeros that creates a matrix
of this size with all zeros.
318
00:17:44 --> 00:17:45
Okay.
319
00:17:45 --> 00:17:48
That would create
the 100 by 100.
320
00:17:48 --> 00:17:48
Good.
321
00:17:48 --> 00:17:50
Ok.
322
00:17:50 --> 00:17:52
Oh, by the way, as long
as we're speaking about
323
00:17:52 --> 00:17:55
computation I've gotta
say something more.
324
00:17:55 --> 00:17:59
We said that the
matrix is sparse.
325
00:17:59 --> 00:18:02
And this 100 by 100 matrix
is certainly sparse.
326
00:18:02 --> 00:18:07
But if I create it this way,
I've created all those zeros
327
00:18:07 --> 00:18:13
and if I ask MATLAB to work
with that matrix, to square it
328
00:18:13 --> 00:18:19
or whatever, it would carry
all those zeros and do all
329
00:18:19 --> 00:18:21
those zero computations.
330
00:18:21 --> 00:18:25
In other words, it would treat
K like a dense matrix and it
331
00:18:25 --> 00:18:27
would just, it wouldn't know
the zeros were there
332
00:18:27 --> 00:18:29
until it looked.
333
00:18:29 --> 00:18:33
So I just want to say that if
you have really big systems
334
00:18:33 --> 00:18:38
Sparse MATLAB is the way to go.
335
00:18:38 --> 00:18:42
Because Sparse MATLAB keeps
track only of the non-zeros.
336
00:18:42 --> 00:18:45
So it knows-- and their
locations, of course.
337
00:18:45 --> 00:18:47
What the numbers are
and their location.
338
00:18:47 --> 00:18:50
So I could create a sparse
matrix out of that,
339
00:18:50 --> 00:18:53
like KS for K sparse.
340
00:18:53 --> 00:18:59
I think if I just did
sparse(K) that would
341
00:18:59 --> 00:19:01
create a sparse matrix.
342
00:19:01 --> 00:19:07
And then if I do stuff to it,
MATLAB would automatically know
343
00:19:07 --> 00:19:11
those zeros were there and not
spend it's time multiplying by
344
00:19:11 --> 00:19:15
zero But of course, this isn't
perfect because I've created
345
00:19:15 --> 00:19:17
the big matrix before
sparsifying it.
346
00:19:17 --> 00:19:20
And better to have created
it in the first place
347
00:19:20 --> 00:19:22
as a sparse matrix.
348
00:19:22 --> 00:19:27
Ok.
349
00:19:27 --> 00:19:32
So those were properties
that you could see.
350
00:19:32 --> 00:19:36
Now I'm looking for
little deeper.
351
00:19:36 --> 00:19:39
What's the first question I
would ask about a matrix if I
352
00:19:39 --> 00:19:43
have to solve a system of
equations, say KU=F
353
00:19:43 --> 00:19:46
or something.
354
00:19:46 --> 00:19:53
I got a 4 by 4 matrix, four
equations, four unknowns.
355
00:19:53 --> 00:19:57
What would I want to know next?
356
00:19:57 --> 00:19:59
Is it invertible?
357
00:19:59 --> 00:20:04
Is the matrix invertible?
358
00:20:04 --> 00:20:07
And that's an important
question and how do you
359
00:20:07 --> 00:20:10
recognize an invertible matrix?
360
00:20:10 --> 00:20:12
This one is invertible.
361
00:20:12 --> 00:20:15
So let me say K is invertible.
362
00:20:15 --> 00:20:17
And what does that mean?
363
00:20:17 --> 00:20:21
That means that there's another
matrix, K inverse such that
364
00:20:21 --> 00:20:27
K times K inverse is
the identity matrix.
365
00:20:27 --> 00:20:33
The identity matrix in MATLAB
would be eye(n) and it's
366
00:20:33 --> 00:20:35
the diagonal matrix of one.
367
00:20:35 --> 00:20:39
It's the unit matrix is the
matrix that doesn't do
368
00:20:39 --> 00:20:43
anything to a vector.
369
00:20:43 --> 00:20:48
So this K has an inverse.
370
00:20:48 --> 00:20:49
But how do you know?
371
00:20:49 --> 00:20:53
How can you recognize that
a matrix is invertible?
372
00:20:53 --> 00:20:56
Because obviously that's a
critical question and many,
373
00:20:56 --> 00:21:00
many-- since our matrices are
not-- a random matrix would be
374
00:21:00 --> 00:21:06
invertible, for sure, but our
matrices have patterns, they're
375
00:21:06 --> 00:21:11
created out of a problem and
the question of whether that
376
00:21:11 --> 00:21:13
matrix is invertible
is fundamental.
377
00:21:13 --> 00:21:18
I mean finite elements has
these, zero energy modes that
378
00:21:18 --> 00:21:24
you have to watch out for
because, what are they?
379
00:21:24 --> 00:21:28
They produce non-invertible
stiffness matrix.
380
00:21:28 --> 00:21:28
Ok.
381
00:21:28 --> 00:21:31
So how did we know, or
how could we know that
382
00:21:31 --> 00:21:34
this K is invertible?
383
00:21:34 --> 00:21:37
Somebody said invertible
and I wrote it down.
384
00:21:37 --> 00:21:39
Yeah?
385
00:21:39 --> 00:21:41
Well ok.
386
00:21:41 --> 00:21:45
Now I get to make a speech
about determinants.
387
00:21:45 --> 00:21:46
Don't deal with them!
388
00:21:46 --> 00:21:49
Don't touch determinants.
389
00:21:49 --> 00:21:54
I mean this particular four
by four happens to have
390
00:21:54 --> 00:21:56
a nice determinant.
391
00:21:56 --> 00:21:58
I think it's five.
392
00:21:58 --> 00:22:04
But if it was a 100 by 100
how would we show that the
393
00:22:04 --> 00:22:06
matrix was invertible?
394
00:22:06 --> 00:22:10
And what I mean by this is the
whole family is invertible.
395
00:22:10 --> 00:22:13
All sizes are invertible.
396
00:22:13 --> 00:22:17
K_ n is invertible for every n,
not just this particular guy,
397
00:22:17 --> 00:22:20
whose determinant
we could take.
398
00:22:20 --> 00:22:24
But as five by five, six by
six, we would be up in the--
399
00:22:24 --> 00:22:28
but you're completely right.
400
00:22:28 --> 00:22:33
The determinant is a test.
401
00:22:33 --> 00:22:35
Alright.
402
00:22:35 --> 00:22:40
But I guess I'm saying
that it's not the test
403
00:22:40 --> 00:22:45
that I would use.
404
00:22:45 --> 00:22:49
So what I do?
405
00:22:49 --> 00:22:52
I would row reduce.
406
00:22:52 --> 00:22:58
That's the default option
in linear algebra.
407
00:22:58 --> 00:23:01
If you don't know what to do
with a matrix, if you want to
408
00:23:01 --> 00:23:03
see what's going
on, row reduce.
409
00:23:03 --> 00:23:04
What does that mean?
410
00:23:04 --> 00:23:09
That means, shall I try it?
411
00:23:09 --> 00:23:21
So let me just start it just
so I'm not using a word
412
00:23:21 --> 00:23:24
that we don't need.
413
00:23:24 --> 00:23:25
Ok.
414
00:23:25 --> 00:23:29
And actually, maybe the third
lecture, maybe next Monday
415
00:23:29 --> 00:23:33
we'll come back to row reduce.
416
00:23:33 --> 00:23:38
So I won't make heavy weather
of that, certainly not now.
417
00:23:38 --> 00:23:43
So what is row reduce,
just so you know.
418
00:23:43 --> 00:23:46
I want to get that minus
one to be a zero.
419
00:23:46 --> 00:23:50
I'm aiming for a
triangular matrix.
420
00:23:50 --> 00:23:55
I want to clean out below the
diagonal because if my matrix
421
00:23:55 --> 00:23:59
is triangular then I can see
immediately everything.
422
00:23:59 --> 00:24:01
Right?
423
00:24:01 --> 00:24:06
Ultimately I'll reach a matrix
U that'll be upper triangular
424
00:24:06 --> 00:24:11
and that first row won't change
but the second row will change.
425
00:24:11 --> 00:24:13
And what does it change to?
426
00:24:13 --> 00:24:17
How do I clean out, get a
zero in that where the
427
00:24:17 --> 00:24:21
minus one is right now?
428
00:24:21 --> 00:24:29
Well I want to use the first
row, the first equation.
429
00:24:29 --> 00:24:32
I want to add some
multiple of the first
430
00:24:32 --> 00:24:36
row to the second row.
431
00:24:36 --> 00:24:38
And what should
that multiple be?
432
00:24:38 --> 00:24:41
I want to multiply that
row by something.
433
00:24:41 --> 00:24:43
And I'll say "add" today.
434
00:24:43 --> 00:24:47
Later I'll say "subtract."
But what shall I do?
435
00:24:47 --> 00:24:50
Just tell me what
the heck to do.
436
00:24:50 --> 00:24:53
I've got that row and I want
to use it, I want to take a
437
00:24:53 --> 00:24:55
combination of these two rows.
438
00:24:55 --> 00:24:59
This row and some multiple
of this one that'll
439
00:24:59 --> 00:25:00
produce a zero.
440
00:25:00 --> 00:25:02
This is called the pivot.
441
00:25:02 --> 00:25:04
That's the first
pivot P-I-V-O-T.
442
00:25:04 --> 00:25:07
Pivot.
443
00:25:07 --> 00:25:11
And then that's the pivot row.
444
00:25:11 --> 00:25:14
And what do I do?
445
00:25:14 --> 00:25:15
Tell me what to do.
446
00:25:15 --> 00:25:18
Add half this row to this one.
447
00:25:18 --> 00:25:21
When I add half of that row
to that one, what do I get?
448
00:25:21 --> 00:25:22
I get that zero.
449
00:25:22 --> 00:25:26
What do I get here for
the second pivot?
450
00:25:26 --> 00:25:27
What is it?
451
00:25:27 --> 00:25:30
1.5, 3/2.
452
00:25:30 --> 00:25:32
Because half of
that is, so 3/2.
453
00:25:32 --> 00:25:39
And the rest won't change.
454
00:25:39 --> 00:25:43
So I'm happy with that zero.
455
00:25:43 --> 00:25:48
Now I've got a couple more
entries below that first pivot,
456
00:25:48 --> 00:25:49
but they're already zero.
457
00:25:49 --> 00:25:52
That's where the
sparseness pays off.
458
00:25:52 --> 00:25:54
The tridiagonal
really pays off.
459
00:25:54 --> 00:25:59
So those zeros say the
first column is finished.
460
00:25:59 --> 00:26:02
So I'm ready to go on
to the second column.
461
00:26:02 --> 00:26:08
It's like I got to this smaller
problem with the 3/2 here.
462
00:26:08 --> 00:26:12
And a zero there.
463
00:26:12 --> 00:26:13
What do I do now?
464
00:26:13 --> 00:26:16
There is the second pivot, 3/2.
465
00:26:16 --> 00:26:17
Below it is a non-zero.
466
00:26:17 --> 00:26:20
I gotta get rid of it.
467
00:26:20 --> 00:26:23
What do I multiply by now?
468
00:26:23 --> 00:26:24
2/3.
469
00:26:24 --> 00:26:28
2/3 of that new, second row
added to the third row will
470
00:26:28 --> 00:26:30
clean out the third row.
471
00:26:30 --> 00:26:32
This was already cleaned out.
472
00:26:32 --> 00:26:34
This is already a zero.
473
00:26:34 --> 00:26:38
But I want to have 2/3 of this
row added to this one so
474
00:26:38 --> 00:26:41
what's my new third row?
475
00:26:41 --> 00:26:43
Starts with zero and what's
the third pivot now?
476
00:26:43 --> 00:26:46
You see the pivots appearing?
477
00:26:46 --> 00:26:51
The third pivot will be 4/3
because I've got 2/3 this
478
00:26:51 --> 00:26:56
minus one and two is
6/3 so I have 6/3.
479
00:26:56 --> 00:27:00
I'm taking 2/3 away, I
get 4/3 and that minus
480
00:27:00 --> 00:27:01
one is still there.
481
00:27:01 --> 00:27:07
So you see that
I'm-- this is fast.
482
00:27:07 --> 00:27:08
This is really fast.
483
00:27:08 --> 00:27:11
And the next step, maybe
you can see the beautiful
484
00:27:11 --> 00:27:13
patterns that are coming.
485
00:27:13 --> 00:27:16
Do you want to just
guess the fourth pivot?
486
00:27:16 --> 00:27:21
5/4, good guess, right.
487
00:27:21 --> 00:27:24
5/4.
488
00:27:24 --> 00:27:29
Now this is actually how MATLAB
would find the determinant.
489
00:27:29 --> 00:27:32
It would do elimination.
490
00:27:32 --> 00:27:36
I call that elimination because
it eliminated all those numbers
491
00:27:36 --> 00:27:39
below the diagonal
and got zeros.
492
00:27:39 --> 00:27:42
Now what's the determinant?
493
00:27:42 --> 00:27:45
If I asked you for the
determinant, and I will very
494
00:27:45 --> 00:27:51
rarely use the word
determinant, but I guess I'm
495
00:27:51 --> 00:27:55
into it now, so tell
me the determinant.
496
00:27:55 --> 00:27:58
Five.
497
00:27:58 --> 00:27:59
Why's that?
498
00:27:59 --> 00:28:01
I guess I did say five earlier.
499
00:28:01 --> 00:28:06
But how do you know it's five?
500
00:28:06 --> 00:28:10
Whatever the determinant of
that matrix is, why is it five?
501
00:28:10 --> 00:28:12
Because it's a
triangular matrix.
502
00:28:12 --> 00:28:16
Triangular matrices, you've
got all these zeros.
503
00:28:16 --> 00:28:18
You can see what's happening.
504
00:28:18 --> 00:28:21
And the determinant of a
triangular matrix is just the
505
00:28:21 --> 00:28:24
product down the diagonal.
506
00:28:24 --> 00:28:25
The product of these pivots.
507
00:28:25 --> 00:28:29
The determinant is the
product of the pivots.
508
00:28:29 --> 00:28:32
And that's how MATLAB would
compute a determinant.
509
00:28:32 --> 00:28:36
And it would take two times 3/2
times 4/3 times 5/4 and it
510
00:28:36 --> 00:28:40
would give answer five.
511
00:28:40 --> 00:28:45
My friend Alan Edelman told
me something yesterday.
512
00:28:45 --> 00:28:54
MATLAB computes in
floating point.
513
00:28:54 --> 00:29:02
So 4/3, that's 1.3333, etc.
514
00:29:02 --> 00:29:06
So MATLAB would not, when it
does that multiplication,
515
00:29:06 --> 00:29:08
get a whole number.
516
00:29:08 --> 00:29:09
Right?
517
00:29:09 --> 00:29:14
Because in MATLAB that would be
1.333 and probably it would
518
00:29:14 --> 00:29:18
make that last pivot a
decimal, a long decimal.
519
00:29:18 --> 00:29:22
And then when it multiplies
that it gets whatever it gets.
520
00:29:22 --> 00:29:25
But it's not exactly
five I think.
521
00:29:25 --> 00:29:30
Nevertheless MATLAB will
print the answer five.
522
00:29:30 --> 00:29:31
It's cheated actually.
523
00:29:31 --> 00:29:36
It's done that calculation and
I don't know if it takes the
524
00:29:36 --> 00:29:42
nearest integer when it knows
that the-- I shouldn't tell
525
00:29:42 --> 00:29:46
you this, this isn't
even interesting.
526
00:29:46 --> 00:29:50
If the determinant of an
integer matrix, whole number is
527
00:29:50 --> 00:29:54
a whole number, so MATLAB says,
Better get a whole number.
528
00:29:54 --> 00:29:58
And somehow it gets one.
529
00:29:58 --> 00:30:01
Actually, it doesn't
always get the right one.
530
00:30:01 --> 00:30:09
So maybe later I'll know the
matrix whose determinant
531
00:30:09 --> 00:30:11
might not come out right.
532
00:30:11 --> 00:30:15
But ours is right, five.
533
00:30:15 --> 00:30:19
Now where was this going?
534
00:30:19 --> 00:30:23
It got thrown off track
by the determinant.
535
00:30:23 --> 00:30:25
What's the real test?
536
00:30:25 --> 00:30:28
Well so I said there are
two ways to see that a
537
00:30:28 --> 00:30:30
matrix is invertible.
538
00:30:30 --> 00:30:32
Or not invertible.
539
00:30:32 --> 00:30:34
Here we're talking
about the first way.
540
00:30:34 --> 00:30:38
How do I know that this
matrix-- I've got an
541
00:30:38 --> 00:30:39
upper triangular matrix.
542
00:30:39 --> 00:30:41
When is it invertible?
543
00:30:41 --> 00:30:47
When is an upper triangular
matrix invertible?
544
00:30:47 --> 00:30:48
Upper triangular is great.
545
00:30:48 --> 00:30:50
When you've got it in
that form you should
546
00:30:50 --> 00:30:51
be able to see stuff.
547
00:30:51 --> 00:30:58
So this key question of
invertible, which is not
548
00:30:58 --> 00:31:04
obvious for a typical
matrix is obvious for
549
00:31:04 --> 00:31:06
a triangular matrix.
550
00:31:06 --> 00:31:06
And why?
551
00:31:06 --> 00:31:10
What's the test?
552
00:31:10 --> 00:31:12
Well, we could do the
determinant but we can say it
553
00:31:12 --> 00:31:15
without using that long word.
554
00:31:15 --> 00:31:18
The diagonal is non-zero.
555
00:31:18 --> 00:31:22
K as invertible because the
diagonal-- no, it's got
556
00:31:22 --> 00:31:24
a full set of pivots.
557
00:31:24 --> 00:31:26
It's got four non-zero pivots.
558
00:31:26 --> 00:31:28
That's what it takes.
559
00:31:28 --> 00:31:31
That's what it's going to
take to solve systems.
560
00:31:31 --> 00:31:33
So this is the first step
in solving this system.
561
00:31:33 --> 00:31:38
In other words, to decide if
a matrix is invertible, you
562
00:31:38 --> 00:31:41
just go ahead and use it.
563
00:31:41 --> 00:31:45
You don't stop first
necessarily to check
564
00:31:45 --> 00:31:46
invertibility.
565
00:31:46 --> 00:31:49
You go forward, you get to this
point and you see non-zeros
566
00:31:49 --> 00:31:53
there and then you're
practically got to
567
00:31:53 --> 00:31:55
the answer here.
568
00:31:55 --> 00:32:00
I'll leave for another day the
final back to going back
569
00:32:00 --> 00:32:03
upwards that gives
you the answer.
570
00:32:03 --> 00:32:05
So K is invertible.
571
00:32:05 --> 00:32:15
That means full set of
pivots. n non-zero pivots.
572
00:32:15 --> 00:32:20
And here they are, two,
3/2, 4/3 and 5/4.
573
00:32:20 --> 00:32:23
Worth knowing because this
matrix K is so important.
574
00:32:23 --> 00:32:24
We'll see it over
and over again.
575
00:32:24 --> 00:32:33
Part of my purpose today is to
give some matrices a name
576
00:32:33 --> 00:32:35
because we'll see them again
and you'll know them and
577
00:32:35 --> 00:32:38
you'll recognize them.
578
00:32:38 --> 00:32:44
While I'm on this invertible
or not invertible business I
579
00:32:44 --> 00:32:48
want to ask you to change K.
580
00:32:48 --> 00:32:52
To make it not invertible.
581
00:32:52 --> 00:32:54
Change that matrix.
582
00:32:54 --> 00:32:56
How could I change that matrix?
583
00:32:56 --> 00:32:58
Well, of course, many ways.
584
00:32:58 --> 00:33:01
But I'm interested in another
matrix and this'll be
585
00:33:01 --> 00:33:04
among my special matrices.
586
00:33:04 --> 00:33:07
And it will start out the same.
587
00:33:07 --> 00:33:14
It'll have these
same diagonals.
588
00:33:14 --> 00:33:16
It'll be Toeplitz.
589
00:33:16 --> 00:33:23
I'm going to call it C and I
want to say the reason I'm
590
00:33:23 --> 00:33:25
talking about it now is that
it's not going to
591
00:33:25 --> 00:33:29
be invertible.
592
00:33:29 --> 00:33:38
And I'm going to tell you a C
and see if you can tell me
593
00:33:38 --> 00:33:40
why it is not invertible.
594
00:33:40 --> 00:33:42
So here's the difference;
I'm going to put minus
595
00:33:42 --> 00:33:45
one in the corners.
596
00:33:45 --> 00:33:49
Still zeros there.
597
00:33:49 --> 00:33:56
So that matrix C still
has that pattern.
598
00:33:56 --> 00:33:58
It's still a Toeplitz
matrix, actually.
599
00:33:58 --> 00:34:04
That would still be the matrix
Toeplitz of two, minus
600
00:34:04 --> 00:34:04
one, zero, minus one.
601
00:34:04 --> 00:34:07
602
00:34:07 --> 00:34:14
I claim that matrix is not
invertible and I claim that we
603
00:34:14 --> 00:34:19
can see that without computing
determinants, we can see it
604
00:34:19 --> 00:34:22
without doing elimination, too.
605
00:34:22 --> 00:34:24
MATLAB would see it by
doing elimination.
606
00:34:24 --> 00:34:30
We can see it by just
human intelligence.
607
00:34:30 --> 00:34:33
Now why?
608
00:34:33 --> 00:34:39
How do I recognize a matrix
that's not invertible?
609
00:34:39 --> 00:34:44
And then, by converse, how a
matrix that is invertible.
610
00:34:44 --> 00:34:49
I claim-- and let may
say first, let me say
611
00:34:49 --> 00:34:51
why that letter C.
612
00:34:51 --> 00:35:00
That letter C stands for
circulant. it's because this
613
00:35:00 --> 00:35:03
word circulant, why circulant,
it's because that diagonal
614
00:35:03 --> 00:35:09
which only had three guys
circled around to the fourth.
615
00:35:09 --> 00:35:12
This diagonal that only had
three entries circled around
616
00:35:12 --> 00:35:14
to the fourth entry.
617
00:35:14 --> 00:35:16
This diagonal with two
zeros circled around to
618
00:35:16 --> 00:35:17
the other two zeros.
619
00:35:17 --> 00:35:22
The diagonal are not only
constant, they loop around.
620
00:35:22 --> 00:35:24
And you use the word periodic.
621
00:35:24 --> 00:35:29
Now for me, that's
the periodic matrix.
622
00:35:29 --> 00:35:35
See, a circulant matrix comes
from a periodic problem.
623
00:35:35 --> 00:35:38
Because it loops around.
624
00:35:38 --> 00:35:42
It brings numbers, zero
is the same as number
625
00:35:42 --> 00:35:45
four or something.
626
00:35:45 --> 00:35:51
And why is that not invertible?
627
00:35:51 --> 00:35:55
The thing is can
you find a vector?
628
00:35:55 --> 00:35:57
Because matrices multiply
vectors, that's
629
00:35:57 --> 00:35:59
their whole point.
630
00:35:59 --> 00:36:03
Can you see a vector
that it takes to zero?
631
00:36:03 --> 00:36:05
Can you see a solution to Cu=0?
632
00:36:06 --> 00:36:11
I'm looking for a u with
four entries so that
633
00:36:11 --> 00:36:18
I get four zeros.
634
00:36:18 --> 00:36:20
Do you see it?
635
00:36:20 --> 00:36:21
All ones.
636
00:36:21 --> 00:36:23
All ones.
637
00:36:23 --> 00:36:25
That will do it.
638
00:36:25 --> 00:36:33
So that's a nice, natural
entry, a constant.
639
00:36:33 --> 00:36:37
And do you see why when I--
we haven't spoken about
640
00:36:37 --> 00:36:42
multiplying matrices
times vectors.
641
00:36:42 --> 00:36:44
And most people will
do it this way.
642
00:36:44 --> 00:36:46
And let's do this one this way.
643
00:36:46 --> 00:36:48
You take row one times
that, you get two, minus
644
00:36:48 --> 00:36:49
one, zero, minus one.
645
00:36:51 --> 00:36:53
You get the zero because
of that new number.
646
00:36:53 --> 00:36:58
Here we always got zero from
the all ones vector and now
647
00:36:58 --> 00:37:04
over here that minus one,
you see it's just right.
648
00:37:04 --> 00:37:09
If all the rows add to zero
then this vector of all ones
649
00:37:09 --> 00:37:14
will be, I would use the word
in the null space if you
650
00:37:14 --> 00:37:18
wanted a fancy word, a
linear algebra word.
651
00:37:18 --> 00:37:19
What does that mean?
652
00:37:19 --> 00:37:21
It solves Cu=0.
653
00:37:21 --> 00:37:24
654
00:37:24 --> 00:37:29
And why does that show that
the matrix isn't invertible?
655
00:37:29 --> 00:37:31
Because that's our point here.
656
00:37:31 --> 00:37:32
I have a solution to Cu=0.
657
00:37:35 --> 00:37:39
I claim that the existence of
such a solution has wiped out
658
00:37:39 --> 00:37:45
the possibility that the matrix
is invertible because if it
659
00:37:45 --> 00:37:49
was invertible, what
would this lead to?
660
00:37:49 --> 00:37:56
If invertible, if C inverse
exists what would I do to that
661
00:37:56 --> 00:38:04
equation that would show me
that C inverse can't exist?
662
00:38:04 --> 00:38:08
Multiply both sides
by C inverse.
663
00:38:08 --> 00:38:11
So you're seeing, just this
first day you're seeing some
664
00:38:11 --> 00:38:14
of the natural steps
of linear algebra.
665
00:38:14 --> 00:38:17
Row reduction, multiply when
you want to see what's
666
00:38:17 --> 00:38:21
happening, multiply both
sides by C inverse.
667
00:38:21 --> 00:38:25
That's the same as in ordinary
language, Do the same thing
668
00:38:25 --> 00:38:27
to all the equations.
669
00:38:27 --> 00:38:30
So I multiply both sides
by the same matrix.
670
00:38:30 --> 00:38:31
And here I would get (C
inverse)(Cu)=(C inverse)(0).
671
00:38:31 --> 00:38:36
672
00:38:36 --> 00:38:40
So what does that tell me?
673
00:38:40 --> 00:38:43
I made it long, I threw
in this extra step.
674
00:38:43 --> 00:38:51
You were going to jump
immediately to C inverse C is I
675
00:38:51 --> 00:38:54
is the identity matrix and when
the identity matrix multiplies
676
00:38:54 --> 00:38:57
a vector u, you get u.
677
00:38:57 --> 00:39:00
And on the right side, C
inverse, whatever it is if
678
00:39:00 --> 00:39:05
it existed, times zero
would have to be zero.
679
00:39:05 --> 00:39:09
So this would say that if C
inverse exists, then the only
680
00:39:09 --> 00:39:13
solution is u equals u.
681
00:39:13 --> 00:39:15
That's a good way to recognize
invertible matrices.
682
00:39:15 --> 00:39:20
If it is invertible then the
only solution to Cu=0 u=0.
683
00:39:21 --> 00:39:24
And that wasn't true here.
684
00:39:24 --> 00:39:28
So we conclude C is
not invertible.
685
00:39:28 --> 00:39:32
C is therefore not invertible.
686
00:39:32 --> 00:39:36
Now can I even jump in.
687
00:39:36 --> 00:39:38
I've got two more matrices
that I want to tell you
688
00:39:38 --> 00:39:43
about that are also close
cousins of K and C.
689
00:39:43 --> 00:39:50
But let me just explain
physically a little
690
00:39:50 --> 00:39:54
bit about where these
matrices are coming from.
691
00:39:54 --> 00:39:59
So maybe next to K-- so I'm not
going to put periodic there.
692
00:39:59 --> 00:40:01
Right?
693
00:40:01 --> 00:40:03
That's the one that I
would call periodic.
694
00:40:03 --> 00:40:08
This one is fixed at the ends.
695
00:40:08 --> 00:40:13
Can I draw a little picture
that aims to show that?
696
00:40:13 --> 00:40:18
Aims to show where
this is coming from.
697
00:40:18 --> 00:40:22
It's coming from I think
of this as controlling
698
00:40:22 --> 00:40:23
like four masses.
699
00:40:23 --> 00:40:28
Mass one, mass two, mass three
and mass four with springs
700
00:40:28 --> 00:40:40
attached and with
endpoints fixed.
701
00:40:40 --> 00:40:47
So if I put some weights on
those masses-- we'll do this;
702
00:40:47 --> 00:40:51
masses and springs is going to
be the very first application
703
00:40:51 --> 00:40:55
and it will connect to
all these matrices.
704
00:40:55 --> 00:41:05
And all I'm doing now is just
asking to draw the system.
705
00:41:05 --> 00:41:06
Draw the mechanical system.
706
00:41:06 --> 00:41:09
Actually I'll usually
draw it vertically.
707
00:41:09 --> 00:41:14
But anyway, it's got four
masses and the fact that this
708
00:41:14 --> 00:41:19
minus one here got chopped off,
what would I call that end?
709
00:41:19 --> 00:41:21
I'd call that a fixed end.
710
00:41:21 --> 00:41:25
So this is a fixed,
fixed matrix.
711
00:41:25 --> 00:41:28
Both ends or fixed.
712
00:41:28 --> 00:41:32
And it's the matrix that would
govern and the springs and
713
00:41:32 --> 00:41:36
masses all the same is
what tells me that the
714
00:41:36 --> 00:41:38
thing is Toeplitz.
715
00:41:38 --> 00:41:42
Now what's the picture
that goes with C?
716
00:41:42 --> 00:41:46
What's the picture with C?
717
00:41:46 --> 00:41:49
Do you have an
instinct of that?
718
00:41:49 --> 00:41:52
So C is periodic.
719
00:41:52 --> 00:41:57
So again we've got four
masses connected by springs.
720
00:41:57 --> 00:42:03
But what's up with those masses
to make the problem cyclic,
721
00:42:03 --> 00:42:07
periodic, circular,
whatever word you like.
722
00:42:07 --> 00:42:13
They're arranged in a ring.
723
00:42:13 --> 00:42:16
The fourth guy comes
back to the first one.
724
00:42:16 --> 00:42:22
So the four masses would be, so
in some kind of a ring, the
725
00:42:22 --> 00:42:27
springs would connect them.
726
00:42:27 --> 00:42:31
I don't know if that's
suggestive, but I hope so.
727
00:42:31 --> 00:42:37
And what's the point of,
can we just speak about
728
00:42:37 --> 00:42:39
mechanics one moment?
729
00:42:39 --> 00:42:46
How does that system differ
from this fixed system?
730
00:42:46 --> 00:42:53
Here the whole system
can't move, right?
731
00:42:53 --> 00:42:55
If there no force, then
nothing can happen.
732
00:42:55 --> 00:43:00
Here the whole system can turn.
733
00:43:00 --> 00:43:03
They can all displace the same
amount and just turn without
734
00:43:03 --> 00:43:06
any compression of the
springs, without any force
735
00:43:06 --> 00:43:08
having to do anything.
736
00:43:08 --> 00:43:12
And that's why the solution
that kills this matrix
737
00:43:12 --> 00:43:13
is one, one, one, one.
738
00:43:15 --> 00:43:19
So one, one, one, one would
describe a case where all the
739
00:43:19 --> 00:43:21
displacements were equal.
740
00:43:21 --> 00:43:25
In a way it's like the
arbitrary constant in calculus.
741
00:43:25 --> 00:43:29
You're always adding plus C.
742
00:43:29 --> 00:43:35
So here we've got a solution of
all ones that produces zero the
743
00:43:35 --> 00:43:38
way the derivative of a
constant function is
744
00:43:38 --> 00:43:41
the zero function.
745
00:43:41 --> 00:43:49
So this is just like
an indication.
746
00:43:49 --> 00:43:51
Yes, perfect.
747
00:43:51 --> 00:43:52
I've got two more matrices.
748
00:43:52 --> 00:43:58
Are you okay for two more?
749
00:43:58 --> 00:44:03
Yes okay, what are they?
750
00:44:03 --> 00:44:10
Okay a different blackboard
for the last two.
751
00:44:10 --> 00:44:17
So one of them is going to
come by freeing up this end.
752
00:44:17 --> 00:44:24
So I'm going to take
that support away.
753
00:44:24 --> 00:44:29
And you might imagine like a
tower oscillating up and down
754
00:44:29 --> 00:44:34
or you might turn it upside
down and like a hanging spring,
755
00:44:34 --> 00:44:39
or rather four springs with
four masses hanging onto them.
756
00:44:39 --> 00:44:43
But this end is fixed and
this is not fixed anymore,
757
00:44:43 --> 00:44:46
this is now free.
758
00:44:46 --> 00:44:50
And can I tell you the matrix,
the free-fixed matrix.
759
00:44:50 --> 00:44:53
Free-fixed.
760
00:44:53 --> 00:44:56
Because it's the top end
that I changed, I'm
761
00:44:56 --> 00:44:58
going to call it T.
762
00:44:58 --> 00:45:10
So all the other guys are going
to be the same but the top one,
763
00:45:10 --> 00:45:15
the top row, the boundary row,
boundary conditions are always
764
00:45:15 --> 00:45:20
the tough part, the tricky
part, the key part of a model,
765
00:45:20 --> 00:45:23
and here the natural
boundary condition is
766
00:45:23 --> 00:45:25
to have a one there.
767
00:45:25 --> 00:45:34
That two changed to a one.
768
00:45:34 --> 00:45:37
Now if I asked you for the
properties of that matrix--
769
00:45:37 --> 00:45:41
so that's the third. shall
I do the fourth one?
770
00:45:41 --> 00:45:44
So you have them all, you'll
have the whole picture.
771
00:45:44 --> 00:45:45
The fourth one, well
you can guess.
772
00:45:45 --> 00:45:48
What's the fourth?
773
00:45:48 --> 00:45:51
What am I going to do?
774
00:45:51 --> 00:45:53
Free up the other end.
775
00:45:53 --> 00:45:59
So this guy had one free
end and the other guy
776
00:45:59 --> 00:46:01
has B for both ends.
777
00:46:01 --> 00:46:04
B for both ends are
going to be free.
778
00:46:04 --> 00:46:06
So this is free-fixed.
779
00:46:06 --> 00:46:08
This'll be free-free.
780
00:46:08 --> 00:46:13
So that means I have this free
end, the usual stuff in the
781
00:46:13 --> 00:46:23
middle, no change, and
the last row is what?
782
00:46:23 --> 00:46:25
What am I going to
put in the last row?
783
00:46:25 --> 00:46:26
Minus one, one.
784
00:46:26 --> 00:46:26
Minus one, one.
785
00:46:26 --> 00:46:29
786
00:46:29 --> 00:46:34
So I've changed the diagonal.
787
00:46:34 --> 00:46:38
There I put a single one in
because I freed up one end.
788
00:46:38 --> 00:46:41
With B I freed both ends
and I got two minus ones.
789
00:46:41 --> 00:46:44
Now what do you think?
790
00:46:44 --> 00:46:52
So we've drawn the free-fixed
one and what's your guess?
791
00:46:52 --> 00:46:55
They're all symmetric.
792
00:46:55 --> 00:46:57
That's no accident.
793
00:46:57 --> 00:47:00
They're all tridiagonal,
no accident again.
794
00:47:00 --> 00:47:02
Why are they tridiagonal?
795
00:47:02 --> 00:47:06
Physically they're tridiagonal
because that mass is only
796
00:47:06 --> 00:47:08
connected to it's two
neighbors, it's not
797
00:47:08 --> 00:47:10
connected to that mass.
798
00:47:10 --> 00:47:16
That's why we get a zero in
the two, four position.
799
00:47:16 --> 00:47:19
Because two is not
connected to four.
800
00:47:19 --> 00:47:21
So it's tridiagonal.
801
00:47:21 --> 00:47:25
And it's not Toeplitz
anymore, right?
802
00:47:25 --> 00:47:27
Toeplitz says constant
diagonals and these are
803
00:47:27 --> 00:47:29
not quite constant.
804
00:47:29 --> 00:47:34
I would create K, I would take
T equal K if I was going to
805
00:47:34 --> 00:47:37
create this matrix and then I
would say T of one,
806
00:47:37 --> 00:47:40
one equal one.
807
00:47:40 --> 00:47:49
That command would fix
up the first entry.
808
00:47:49 --> 00:47:50
Yeah, that's a
serious question.
809
00:47:50 --> 00:47:53
Maybe, can I hang on
until Friday, and
810
00:47:53 --> 00:47:54
even maybe next week.
811
00:47:54 --> 00:47:56
Because it's very important.
812
00:47:56 --> 00:48:00
When I said boundary
conditions are the key to
813
00:48:00 --> 00:48:02
problems, I'm serious.
814
00:48:02 --> 00:48:07
If I had to think okay, what do
people come in my office ask
815
00:48:07 --> 00:48:09
about questions, I say
right away, What's the
816
00:48:09 --> 00:48:10
boundary condition?
817
00:48:10 --> 00:48:12
Because I know that's
where the problem is.
818
00:48:12 --> 00:48:16
And so here we'll see
these guys clearly.
819
00:48:16 --> 00:48:23
Fixed and free, very important.
820
00:48:23 --> 00:48:26
But also let me say two more
words, I never can resist.
821
00:48:26 --> 00:48:30
So fixed means the
displacement is zero.
822
00:48:30 --> 00:48:32
Something was set to zero.
823
00:48:32 --> 00:48:36
The fifth guy, the fifth over
here, that fifth column
824
00:48:36 --> 00:48:39
was knocked out.
825
00:48:39 --> 00:48:44
Free means that in here it
could mean that the fifth guy
826
00:48:44 --> 00:48:49
is the same as the fourth.
827
00:48:49 --> 00:48:52
The slope is zero.
828
00:48:52 --> 00:48:55
Fixed is u is zero.
829
00:48:55 --> 00:48:59
Free is slope is zero.
830
00:48:59 --> 00:49:05
So here I have a slope of
zero at that end, here
831
00:49:05 --> 00:49:05
I have it at both ends.
832
00:49:05 --> 00:49:09
So maybe that's a
sort of part answer.
833
00:49:09 --> 00:49:12
Now I wanted to get to
the difference between
834
00:49:12 --> 00:49:15
these two matrices.
835
00:49:15 --> 00:49:19
And the main properties.
836
00:49:19 --> 00:49:19
So what are we see?
837
00:49:19 --> 00:49:23
Symmetric again, tridiagonal
again, not quite Toeplitz,
838
00:49:23 --> 00:49:27
but almost, sort of
morally Toeplitz.
839
00:49:27 --> 00:49:32
But then the key question
was invertible or not.
840
00:49:32 --> 00:49:34
Key question was
invertible or not.
841
00:49:34 --> 00:49:35
Right.
842
00:49:35 --> 00:49:37
And what's your
guess on these two?
843
00:49:37 --> 00:49:41
Do you think that one's
invertible or not?
844
00:49:41 --> 00:49:41
Make a guess.
845
00:49:41 --> 00:49:46
You're allowed to guess.
846
00:49:46 --> 00:49:47
Yeah it is.
847
00:49:47 --> 00:49:48
Why's that?
848
00:49:48 --> 00:49:52
Because this thing has
still got a support.
849
00:49:52 --> 00:49:56
It's not free to shift forever.
850
00:49:56 --> 00:49:57
It's held in there.
851
00:49:57 --> 00:50:01
So that gives you a
hint about this guy.
852
00:50:01 --> 00:50:04
Invertible or not for B?
853
00:50:04 --> 00:50:06
No.
854
00:50:06 --> 00:50:09
And now prove that it's not.
855
00:50:09 --> 00:50:14
Physically you were saying,
well this free guy with this
856
00:50:14 --> 00:50:19
thing gone now, this
is now free-free.
857
00:50:19 --> 00:50:21
Physically we're saying
the whole thing can move,
858
00:50:21 --> 00:50:24
there's nothing holding it.
859
00:50:24 --> 00:50:27
But now, for linear algebra,
that's not the proper language.
860
00:50:27 --> 00:50:31
You have to say something
about that matrix.
861
00:50:31 --> 00:50:37
Maybe tell me something
about Bu=0. u What are
862
00:50:37 --> 00:50:38
you going to take for u?
863
00:50:38 --> 00:50:39
Yeah.
864
00:50:39 --> 00:50:41
Same u.
865
00:50:41 --> 00:50:45
We're lucky in this course, u
equal is the
866
00:50:45 --> 00:50:48
guilty main vector many times.
867
00:50:48 --> 00:50:55
Because again the rows are all
adding to zero and the all ones
868
00:50:55 --> 00:51:02
vector is in the null space.
869
00:51:02 --> 00:51:06
If I could just close
with one more word.
870
00:51:06 --> 00:51:07
Because it's the
most important.
871
00:51:07 --> 00:51:10
Two words, two words.
872
00:51:10 --> 00:51:12
Because they're the most
important words, they're the
873
00:51:12 --> 00:51:15
words that we're leading
to in this chapter.
874
00:51:15 --> 00:51:18
And I'm assuming that for
most people they will be
875
00:51:18 --> 00:51:21
new words, but not for all.
876
00:51:21 --> 00:51:24
It's a further property
of this matrix.
877
00:51:24 --> 00:51:25
So we've got, how many?
878
00:51:25 --> 00:51:27
Four properties, or five?
879
00:51:27 --> 00:51:29
I'm going to go for one more.
880
00:51:29 --> 00:51:33
And I'm just going to
say that name first so
881
00:51:33 --> 00:51:36
you know it's coming.
882
00:51:36 --> 00:51:38
And then I'll say, I
can't resist saying
883
00:51:38 --> 00:51:41
a tiny bit about it.
884
00:51:41 --> 00:51:45
I'll use a whole
blackboard for this.
885
00:51:45 --> 00:51:54
So I'm going to say that K and
T are, here it comes, take
886
00:51:54 --> 00:52:07
a breath; positive
definite matrices.
887
00:52:07 --> 00:52:10
So if you don't know what
that means, I'm happy.
888
00:52:10 --> 00:52:10
Right?
889
00:52:10 --> 00:52:14
Because well, I can tell
you one way to recognize a
890
00:52:14 --> 00:52:16
positive definite matrix.
891
00:52:16 --> 00:52:21
And while we're at it, let
me tell you about C and B.
892
00:52:21 --> 00:52:30
Those are positive
semi-definite because
893
00:52:30 --> 00:52:32
they hit zero somehow.
894
00:52:32 --> 00:52:35
Positive means up there,
greater than zero.
895
00:52:35 --> 00:52:40
And what is greater than zero
that we've already seen?
896
00:52:40 --> 00:52:42
And we'll say more.
897
00:52:42 --> 00:52:44
The pivots were.
898
00:52:44 --> 00:52:50
So if I have a symmetric matrix
and the pivots are all positive
899
00:52:50 --> 00:52:55
then that matrix is not only
invertible, because I'm in good
900
00:52:55 --> 00:52:59
shape, the determinant isn't
zero, I can go backwards and do
901
00:52:59 --> 00:53:03
everything, those positive
numbers are telling me that
902
00:53:03 --> 00:53:07
more than that, the matrix
is positive definite.
903
00:53:07 --> 00:53:11
So that's a test.
904
00:53:11 --> 00:53:14
We'll say more about positive
definite, but one way to
905
00:53:14 --> 00:53:18
recognize it is compute the
pivots by elimination.
906
00:53:18 --> 00:53:20
Are they positive?
907
00:53:20 --> 00:53:23
We'll see that all the
eigenvalues are positive.
908
00:53:23 --> 00:53:27
The word positive definite
just brings the whole of
909
00:53:27 --> 00:53:29
linear algebra together.
910
00:53:29 --> 00:53:33
It connects to pivots, it
connects to eigenvalues, it
911
00:53:33 --> 00:53:36
connects to least squares,
it's all over the place.
912
00:53:36 --> 00:53:39
Determinants too.
913
00:53:39 --> 00:53:42
Questions or discussion.
914
00:53:42 --> 00:53:44
It's a big class and we're just
meeting for the first time
915
00:53:44 --> 00:53:49
but there's lots of time
to, chance to ask me.
916
00:53:49 --> 00:53:52
I'll always be
here after class.
917
00:53:52 --> 00:53:53
So shall we stop today?
918
00:53:53 --> 00:53:58
I'll see you Friday
or this afternoon.
919
00:53:58 --> 00:54:03
If this wasn't familiar, this
afternoon would be a good idea.
920
00:54:03 --> 00:54:05
Thank you.
| 677.169 | 1 |
Mathematics for High School Teachers An Advanced Perspective
9780130449412
ISBN:
0130449415
Pub Date: 2002 Publisher: Prentice Hall
Summary: Mathematics for High School Teachers-An Advanced Perspectiveis intended as a text for mathematics courses for prospective or experienced secondary school mathematics teachers and all others who wish to examine high school mathematics from a higher point of view. Preliminary versions of the book have been used in a variety of ways, ranging from junior and senior (capstone) or graduate mathematics courses for pre-servi...ce secondary mathematics education majors to graduate professional development courses for teachers. Some courses included both undergraduate and graduate students and practicing teachers with good success. There is enough material in this book for at least a full year (two semesters) of study under normal conditions, even if only about half of the problems are assigned. With a few exceptions, the chapters are relatively independent and an instructor may choose from them. However, some chapters contain more sophisticated content than others. Here are four possible sequences for a full semester's work: Algebra emphasis: Chapters 1-6 Geometry emphasis: Chapters 1, 7-11 Introductory emphasis: Chapters 1, 3, 4, 7, 8,10 More advanced emphasis: Chapters 1, 2, 5, 6, 9,11. In each sequence we suggest beginning with Chapter 1 so that students are aware of the features of this book and of some of the differences between it and other mathematics texts they may have used. More information and suggestions in this regard can be found in the Instructor's Notes. Additional instructional resources are also at the web site . The presentation assumes the student has had at least one year of calculus and a post-calculus mathematics course (such as real analysis, linear algebra, or abstract algebra) in which proofs were required and algebraic structures were discussed. The term "from an advanced standpoint" is taken to mean that the text examines high school mathematical ideas from a perspective appropriate for college mathematics majors, and makes use of the kind of mathematical knowledge and sophistication the student is gaining or has gained in other courses. Two basic characteristics ofMathematics for High School Teachers-An Advanced Perspective,taken together, distinguish courses taught from this book from many current courses. First, the material is rooted in the core mathematical content and problems of high school mathematics courses before calculus. Specifically, the development emanates from the major concepts found in high school mathematics: numbers, algebra, geometry, and functions. Second, the concepts and problems are treated from a mathematically advanced standpoint, and differ considerably from materials designed for high school students. The authors feel that the mathematical content in this book lies in an area of mathematics that is of great benefit to all those interested in mathematics at the secondary school level, but is rarely seen by them. Specifically, we have endeavored to include: analyses of alternate definitions, language, and approaches to mathematical ideas extensions and generalizations of familiar theorems discussions of the historical contexts in which concepts arose and have changed over time applications of the mathematics in a wide range of settings analyses of common problems of high school mathematics from a deeper mathematical level demonstrations of alternate ways of approaching problems, including ways with and without calculator and computer technology connections between ideas that may have been studied separately in different courses relationships of ideas studied in school to ideas students may encounter in later study. There are many reasons why we believe a teacher or other person interested in high school mathematics
Usiskin, Zalman is the author of Mathematics for High School Teachers An Advanced Perspective, published 2002 under ISBN 9780130449412 and 0130449415. Four hundred twenty two Mathematics for High School Teachers An Advanced Perspective textbooks are available for sale on ValoreBooks.com, one hundred thirty six used from the cheapest price of $46.59, or buy new starting at $98 gives readers a comprehensive look at the most important concepts in the mathematics taught in grades 9-12. Real numbers, functions, congruence, similarity, area [more]
This book gives readers a comprehensive look at the most important concepts in the mathematics taught in grades 9-12. Real numbers, functions, congruence, similarity, area and volume, trigonometry and more. For high sch
| 677.169 | 1 |
So far we have discussed equations of various kinds. vow we shall consider someinequalities; not of the social kind, but between real nuhbers. A mathematical inequality is amathematical expression of the condition that of two quantities one is greater than, greaterthan or equal to, less than or less than or equal to the other. An inequality that holds for everyreal number is called an absolute inequality. In this unit we shall restrict ourselves to suchinequalities.We will discuss six famous absolute inequalities. We have divided them into two sections-those that have been used for centuries and those that were discovered by some famousnineteenth century European mathematicians. These inequalities have several applicationsalso. We will discuss a few of them. You may come across some applications in other coursestoo, at which time we hope that you will find that you didn't study this unit in vain!Let us list
our
unit objectives now.
Objectives
After reading this unit you should be able to prove and applythe inequalities of the means;the triangle inequality;the Cauchy-Schwan. (Bunyakovskii) nequality:Weierstrass' inequalities:Tchebychev's inequalities.Let us discuss the ineqyalities one by one.
3.2
INEQUALITIES KNOWN TO THE ANCIENTS
In this section we shall discuss two inequalities handed down to us by ancientmathematicians. But first we will give a list of some properties of inequalities you must befmiliar with. They are the following:for
a,
b, c, d
E
R
i)
aZb,cZO*acZbc
ii)
a2be-as-b
11
iii)
a2b~-5-provideda;tO,bzO.
abiv) a2b,c2d*a+c>b+dv) an>-bn,a2O*a2b,wherene
N.
We will often use these properties implicitly while proving the inequalities mentioned in theunit objectives.Now let us discuss the inequality that relates three averages.
i
3.2.1
Inequality
of
the Means
An important part of arithmetic that can be traced back to the Babylonians and Pythagoreans(approximately 6th century B.C.) is the theory of means or averages. The word "average"comes from the Latin word "havaria", which was theinsurance paid to compensate for damageto goods
in
transit in the olden days. All of us are familiar with compensate for damage togoods in transit
in
the olden days. All of us are familiar with the term "average". In fact, all ofus must have often calculated fie average of a fmite set of numbers by addhg them up anddividing the sum by the total number of these numbers. But this is only one
of
Inany types ofaverages. We will discuss three of these types here. Let us start with the "usual" average.
Definition:
the
arithmetic mean
(AM)
ofn real numbers x,, x2,
...,
X,
is
XI
+
X2
+
...
+
X"
-
,
hat is
n
[A
=~
i]
.
11
1
-1
A--
+o
1
For example, the AM of
-
-
nd 0 is
2-
-
33
18
'
The AM is often used in statistics for studying data.Another type of average is the geometric mean. This is the best mean to use if we want to findthe mean of any finite set of positive numbers that follow geometric progression. Thus, thismean is very useful for studyingpopulation'growth. Let us see how the geometric mean isdefmed.
Definition:
The
geometric mean
(GM)
of n positive real numbers
XI,
%,
......,
Xn
1sFor example, the GM of
3
and4 is
.=
fi
nd the
GM
of 2,4 and
8
is(2x4x8)lB=4.
.
Yet another kind of average of numbers is their harmonic mean, which we now define.
Definition:
The
harmonic mean
0
f n non-zero real numbers
lnequnlities
Eauations and Incaualiticr
1For example, the
HM
f
-2,
-
and
7
is
3
-
42
-
-
3
1 1
37
'
---+7+-
The HM is the most appropriate type of average to use when we want to find the average rateof a set of varying rates. Thus, it is the best average to use for obtaining the average velocityof
a
vehicle covering various distances at different speeds..At this point we would like to make a remark.
Note:
We can obtain the AM of
any
n real numbers. But, we only define the GM of n
positive
This
proof
is
due
to
Cauchy]
real numbers; and the
HM
f n
non-zero
real numbers.
who you
will
meet
again
in
See.
6.3.
Now let us look at the three different means together. To do so, we clearly need to restrictourselves to positive real numbers. What is the AM of 2,4 and
8?
How is it related to theirGM? And, how is their GM related to their HM? The following result answers these questions.
Theorem
1:
Let
{x,,
xz,
...,
x,
)
be any finite set of positive real numbers, and let A, G and Hdenote their arithmetic, geometric and harmonic means, respectively. ThenWe will only give a broad outline of the proof here. The inequality A
2
G is first proved byinduction (see Unit 2) for all those integers n that are powers of
| 677.169 | 1 |
UltimaCalc 2.1.255
Description of UltimaCalc 2.1.255: UltimaCalc is a scientific and mathematical calculator designed to occupy minimum screen area, making it immediately available for use. UltimaCalc can stay on top of other windows. Type a calculation as plain text, evaluate it, maybe edit it and re-calculate. Has a comprehensive context-sensitive help system.
CalculatesFunctions available include logarithms to base 2, exp, two-argument inverse tangent; cube root (even of negative numbers); factorials, combinations, permutations, powers, modulus, GCD. Also floor and ceiling functions, absolute value, min, max, extract the fractional part of a number. Calculate the slope of a line given its end points. Even calculate definite integrals.
Define your own functions and constants, saved as a plain text file.
Find the date of Easter. Calculate future or past dates. Julian day numbers.
Calculate the mean, median and standard deviation of a sample and its population.
Create bar, line, pie charts. Add title, subtitle, labels. Adjust the layout, choose colours and hatching, save as an image.
Regression: Plot a scatter chart and regression line. Fit a polynomial to data. Analyse the effects of multiple variables on data. Absolute deviation fit minimises the effects of outlying values.
Plot functions: Specify starting and ending conditions and how variables change. Choose axis locations. Combine multiple plots. Save as an image.
Solve triangles: Given one side and two other facts, calculate the unknowns. View the result graphically.
Solve Simultaneous Linear Equations, and do Navigational calculations.
Log calculations to a text file, or copy and paste into other programs. Save specialised calculations with notes in data files.
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Roadmap to 8th Grade Math Description
If Students Need to Know It, It's in This Book This book develops the Math skills of eighth graders. It builds skills that will help them succeed in school and on the New York State test. Why The Princeton Review? We have more than 20 years of experience helping students master the skills needed to excel on standardized tests. Each year, we help more than 2 million students score higher and earn better grades. We Know the New York State Testing Program Our experts at The Princeton Review have analyzed the New York State test, and this book provides the most up-to-date, thoroughly researched practice possible for the Grade 8 Mathematics test. We break down the test into individual skills to familiarize students with the test's structure, while increasing their overall skill level. We Get Results We know what it takes to succeed in the classroom and on tests. This book includes strategies that are proven to improve student performance. We provide - content review, detailed lessons, and practice exercises modeled on the skills tested by the New York State Grade 8 Mathematics test - engaging puzzles and word problems - 2 complete practice New York State Math tests
Discussion : Roadmap to 8th Grade Math
| 677.169 | 1 |
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Educational/OtherMatheAss -
Multimedia & Design/Other Related Tools ... MatheAss (former Math-Assist) is a computer program for the numerical solution of many problems in school mathematics. It finds wide use in Germany in high school mathematics, for schools in the federal state of Hessen (Germany) exists a state license, which allows all secondary schools using MatheAss. "MatheAss" means Mathematical Assistant and that's exactly what this program is designed to be. It's for secondary level or high school students and teachers and anybody else who has ...
8.
pmp -
Utilities ... pmp is a performance monitor for remote/multiple Windows servers. You can monitor the real time CPU, memory, disk and network usage values from the several networked servers and can be notified for specific server status. In addition to the basic alarming methods, pmp provides email notification, automatic/manual remote command execution. pmp also provides HTML reporting and you can share it with other staff.
Version 8.0.1 includes Windows Vista compatibility, new visual notification method, ...
9.
GEUP 3D -
Multimedia & Design/Graphic & Design ... GEUP 3D is an interactive solid geometry software for math calculation and visualization. It allows to create dynamic and general constructions/applications visually by defining math elements. GEUP 3D allows the modification of the construction visually (directly in screen) and it calculates each one of its particular cases in real time. GEUP 3D is the equivalent of GEUP 4 in the 3D space.With GEUP 3D you can study graphically a general problem and obtain dynamically its particular cases. This ...
10.
InterReg -
Educational/Mathematics ... Inter The applications are many and reach from education over science to productive use in your company: in example
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Math 9H
We use the same textbooks as regular Math 9 courses, so the topics learned and sequence of lessons is generally the same. Some key differences include:
1) Contests and Challenge Problems
In Honours classes, a portion of our class time will be spent on math contests, problem-solving topics and challenge problems. We will also practice past math contests to prepare for this year's upcoming contests.
2) Depth of Study
In Honours classes, as we teach each topic, we go a bit further and introduce some problems that would be found at the next grade level. For example, a Math 9 Principles student would learn basic linear relations, but a Math 9 Honours student would learn basic linear relations AND the more advanced material taught in Math 10.
3) Homework
In regular classes, students are assigned more of the easier "Practice" questions, while in Honours classes, students will be assigned more of the harder "Problems and Applications" questions. Occasionally, honours assignments and math contests will be assigned for homework as well.
4) Tests and Evaluation
The tests in Honours classes will reflect the notes, examples, and extra material that we cover in our lessons, and are generally a bit more difficult than the corresponding tests in regular classes. However, there will also be more bonus questions on tests, and students will also have opportunities to get extra marks from participation in math contests.
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97808218 Course in Convexity (Graduate Studies in Mathematics, V. 54)
Convexity is a simple idea that manifests itself in a surprising variety of places. This fertile field has an immensely rich structure and numerous applications. Barvinok demonstrates that simplicity, intuitive appeal, and the universality of applications make teaching (and learning) convexity a gratifying experience. The book will benefit both teacher and student: It is easy to understand, entertaining to the reader, and includes many exercises that vary in degree of difficulty. Overall, the author demonstrates the power of a few simple unifying principles in a variety of pure and applied problems.
The notion of convexity comes from geometry. Barvinok describes here its geometric aspects, yet he focuses on applications of convexity rather than on convexity for its own sake. Mathematical applications range from analysis and probability to algebra to combinatorics to number theory. Several important areas are covered, including topological vector spaces, linear programming, ellipsoids, and lattices. Specific topics of note are optimal control, sphere packings, rational approximations, numerical integration, graph theory, and more. And of course, there is much to say about applying convexity theory to the study of faces of polytopes, lattices and polyhedra, and lattices and convex bodies.
The prerequisites are minimal amounts of linear algebra, analysis, and elementary topology, plus basic computational skills. Portions of the book could be used by advanced undergraduates. As a whole, it is designed for graduate students interested in mathematical methods, computer science, electrical engineering, and operations research. The book will also be of interest to research mathematicians, who will find some results that are recent, some that are new, and many known results that are discussed from a new
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...
More About
This Book
a great way to spark student interest in math. Grades 5-12.
Editorial Reviews
Booknews
Each chapter of this new supplemental mathematics text for grades 5-12 presents the historical background of a mathematician or problem, along with an activity based upon the problem and its solution. The projects are designed to be incorporated into a variety of curriculums, and they can be used as introductions into new topics or as a fun way to make mathematics relevant for young people. Johnson, a mathematics teacher at Nashua High School in New Hampshire, demonstrates how the solutions to these age old problems can be applied to everyday situations. Annotation c. Book News, Inc., Portland, OR (booknews.com)
From The Critics
Famous Problems and Their Mathematicians is an excellent compilation of the more challenging problems one encounters at the end of a chapter in most high school and some junior high school mathematics textbooks. Each of the volume's 61 chapters represents a specific type of problem. All are written in the same format. A problem that has been given a name is followed by some background information on the creator and the problem. Next come directions, followed by steps to a solution, or an actual solution, to the problem. Also included for each problem is a page or section called the "Teacher Page," which gives further information to help teach the concept represented by the problem. Finally, an "Extension Section" takes the concepts covered to new levels in the classroom. Many different disciplines of mathematics are covered, including arithmetic, algebra, geometry, trigonometry, and logic. The problems are multicultural, reflecting the mathematics of many nations. Among the familiar problems presented are those of Fermat, Agnesi, and Pythagoras. Also included are problems by lesser known mathematicians, such as Aryabhata (Indian), Abu'l-Wafa' (Islamic), and Brahmagupta (Hindu). A time line giving the name, country of origin, dates, and page number of each problem is also included. This volume is a great source of new and interesting problems; it should be a supplement to all junior and senior high school mathematics courses. Highly Recommended, Grades 7-College, Teaching Professional. REVIEWER: Robert Burchell (Paramus Catholic High School
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simple introductory treatment emphasizes fundamentals and is easily comprehensible to readers with some college algebra. Topics include rudiments; arbitrary sets and their cardinal numbers; ordered sets and their ordered types; and well-ordered sets and their ordinal numbers. "Exceptionally well written." — School Science and Mathematics. "A very fine book." — Mathematics Teacher
{"currencyCode":"USD","itemData":[{"priceBreaksMAP":null,"buyingPrice":8.06,"ASIN":"0486601412","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":3.99,"ASIN":"0486678709","isPreorder":0}],"shippingId":"0486601412::2Wz%2BCu3qTqSUkeoibE2HuwD%2BEL8u9jCZV%2BfTa3VZOJWODK7dPfdGlUf5gE9jruz7vj9nuhaqr0LzK7puFToBtf9USmJAzbw2WljtZdwHX8M%3D,0486678709::lJcUMBqM30w4M6yqzPqIITEuELp7GqUVW%2BYsxtzhXvEAxNdC6c2TXUlXL6jJMIAvBntj3dwXbv6exactURH2FETLbWTLdch8LhtKdT6x very good introduction to Set theory. It starts with basics, and takes one through to The Axiom of Choice or well-ordering theorem. Set theory being a rather abstract subject, most books do not give a motivation for the various terminology in set theory (which enhances understanding of why many concepts had to be introduced. The book however uses strange archaic Math typography, because of which through out the book I kept on trying to figure out whether it is a srcipt pi or script R. This sometimes confuses the thought process. But overall, the book is quite easy to read and manages to convey to the reader some of the beauty of set theory. At points, one can't help but smile at the ingenuity and the audacity of human thought. I recommened it highly.
Kamke, along with Suppes and Halmos, is one of the classic texts in the theory of sets. Kamke is a bit unique, I think, in that he brings up terms such as jump, cut, and gap. Those are currently out-of-fashion but are in fact more intuitive and give you a direct philosophical grasp. A must-read.
This is a great little book of math for all of those interested in the finer points of set theory. I'm an undergrad math major and found the material challenging yet rewarding. If you are as young as I am, or an experienced mathematician, I believe that this book has something for you to think about. From the first you'll be taken along at a steady pace covering theorems that are quite interesting.
I agree completely with RJ's review, including the his/her frustration with the typography in which capital letters in what I believe is German script are used to denominate sets. Still, once I had that sorted out this was a straightforward and rewarding read. (Other than the capital German letters used to denominate sets, the notation is relatively easy to follow as well, and contrary to another review, there is a perfectly good index to the places where each symbol is introduced.) I intend to supplement its intuitive approach by reading Suppes' presentation of axiomatic set theory.
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Math homework help - answers to math problems - hotmath, Math homework help. hotmath explains math textbook homework problems with step-by-step math answers for algebra, geometry, and calculus. online tutoring available for.
Clauses: the essential building-blocks - commnet, Dependent clauses. dependent clauses cannot stand by themselves and make good sense. they must be combined with an independent clause so that they become part of a.
Aplia :: engage. prepare. educate., Over 5,300 instructors at more than 1,700 institutions have used aplia..
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Mathematics in Our World
"Mathematics in Our World" is designed for mathematics survey courses for non-science majors. The text covers a variety of topics designed to foster ...Show synopsis"Mathematics in Our World" is designed for mathematics survey courses for non-science majors. The text covers a variety of topics designed to foster interest in and show the applicability of mathematics. The book is written by our successful statistics author, Allan Bluman. His easy-going writing style and step-by-step approach make this text very readable and accessible to lower-level students. The text contains many pedagogical features designed to both aid the student and instill a sense that mathematics is not just adding and subtracting73311820 GREAT shape! May have some minor physical...Very Good. 00733118 Mathematics in Our World with MathZone. This book is in...Good. Mathematics in Our World with MathZone. This book is in Good condition. Buy with confidence. We ship from multiple location.
Description:Good. 0073311820 USED BOOK in good condition| No supplements|...Good. 0073311820Reviews of Mathematics in Our World
This book was in terrific condition and came sooner than expected for my husband's college course. Even better, the book is the teacher's edition and had some great helps for those of you who are rusty on math procedures for different real-world applications.
If you need tyo brush up your math skills in any way, this is the book for it, especially if you get the teacher's
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71
Offering
10+ subjects
including precalculus
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How do you effectively study mathematics? How does one read a maths book instead or just staring at it for hours?
(Apologies in advance if the question is ill-posed or too subjective in its current ...
This sort of question is probably in bad taste for math.stackexchange, but is probably in high demand. (I tried to start a site on Area 51 to house questions like this, but my request was closed due ...
I'm hoping the userbase here doesn't mind if I do a little crowd-sourcing.
I'm curious to find out about popular general-interest mathematics or statistics classes that are offered universities that ...
I'm a undergraduate who wishes to learn category theory but I only have basic knowledge of linear algebra and set theory, I've also had a short course on number theory which used some basic concepts ...
With a few colleagues, we're trying to design an (intermediate) algebra course (US terminology) where we stress the interplay between algebra and geometry. The algebraic topics we would like to cover ...
This is a tall order, but since this site is for everything mathematical, here goes:
What are some nice games/puzzles/problems/problem-like activities that would be appropriate for a three-five year ...
It is well known that one of the first books Ramanujan studied was "Synopsis of Pure and Applied Mathematics" and that it shaped the way Ramanujan thought and wrote about mathematics. Being interested ...
I throw two coins (simultaneously). A student (very much a beginner in both math and probability theory) thought that the following 3 outcomes are equally likely: "two heads", "two tails", "a head and ...
The Mathematical Contest in Modeling (MCM) is coming up in a couple of weeks. With the idea that the competition is ultimately about teaching students to become better at mathematical modeling, I was ...
What are some great math social activities for students? I'm looking for things that bring people together with a "light" mathematical touch. The goal is to create a stronger mathematical community in ...
I'm searching for some good reading material on multifractal analysis. Preferably something accessible that doesn't put the stress too much on mathematical proofs but rather on applications. As long ...
I am studying for the GRE Mathematics subject exam [1]. I am looking for tips regarding how to more effectively study for it. Does anyone know of any good study materials or have any tips in general?
...
We would like to generate simple mathematical formulas on the fly (e.g. $\sin^2 \theta$). The above function seems to be a MathJaX representation. Most of the solutions depend on writing mathematical ...
So, I am a graduate student who is certain that he does not want to do analysis (I think...). What are the most exciting fields in mathematics right now? It seems to me that very generally, they are ...
I wanted to ask these questions here, feeling that this place place is without doubt is better than anywhere else, feeling hesitant and shy to ask such question.
For several reasons, the environment ...
I am an ESL learner and most of the time I am able to understand English but In my courses, we don't study mathematics, and I think that it is a mistake or wrong.
1-What are really the best sources ...
I am currently a mathematics graduate student at Western Kentucky University in Bowling Green, KY. I am looking for some kind of summer opportunity to participate in during summer 2011.
Does anyone ...
YARFMO (Yet another reposting from Mathoverflow) ;-)
The more you know about math the more you find conceptions previously thought correct to be false:
1.) math is not as exact as many believe - in
As it is required for most students who wish to do a Ph.D in maths in the US to sit the GRE subject specific mathematics exam, I hope this question will be of interest to the mathematical community
Frequently, we introduce a new concept with a formal definition, then immediately say "Intuitively, what this means is..." What are the absolute best metaphors you've seen (for concepts of any level)?
...
I am tutoring several talented students, middle school level and early high school level, in mathematics. I am always looking for new sources from which to draw questions. Can anyone recommend books, ...
I was wondering if anyone had any experience with an undergraduate course that emphasized the building of mathematical theories or if they'd ever heard of this being done? How did the class work (did ...
I like to use physical demonstrations when teaching mathematics (putting physics in the service of mathematics, for once, instead of the other way around), and it'd be great to get some more ideas to ...
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Provides an introduction to the development of numbers and their applications. This book includes material with more challenging ideas about infinity and complex numbers and offers examples ranging from recreational puzzles to very real world scenarios. It is for those who are curious about the mathematics behind real-life questions. more...
Made of two mutually explanatory parts, this book provides information on the general, historical and cultural background, and the development of each subdiscipline that together comprise Chinese mathematics. It is organised topically rather than chronologically, and tells how to interpret the contextual setting, both mathematical and sinologicalThe book analyzes the mathematical tablets from the private collection of Martin Schoyen. It includes analyses of tablets which have never been studied before. This provides new insight into Babylonian understanding of sophisticated mathematical objects. The book is carefully written and organized. The tablets are classified according to mathematical... more...
Dirac operators are used in physics, differential geometry, and group-theoretic settings. Using Dirac operators as a unifying theme, this work demonstrates how some of the important results in representation theory fit together when viewed from this perspective. It presents the important ideas on Dirac operators and Dirac cohomology. more...
This book examines the development of mathematics from the late 16th Century to the end of the 19th Century. Each chapter will focus on a particular topic and outline its history with the provision of facsimiles of primary source material along with explanatory notes and modern interpretations. - ;Aimed at students and researchers in Mathematics, History... more...
Universal Algebra has become the most authoritative, consistently relied on text in a field with applications in other branches of algebra and other fields such as combinatorics, geometry, and computer science. Each chapter is followed by an extensive list of exercises and problems. The "state of the art" account also includes new appendices... more...
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Discrete Mathematics Help
Discrete Mathematics is also called finite Mathematics or Decision Mathematics. It is the study of Mathematical structures that are fundamentally discrete in the sense of not supporting or requiring the notion of continuity. Objects studied in finite mathematics are largely countable sets such as integers, finite graphs, and formal languages. It has become popular in recent decades due to its application in Computer Science. It is used in software developments, cryptography, programming language, etc. It covers set theory, graph theory, logic, permutation and combination, etc.
Get discrete math help here and enhance your knowledge in mathematics. Students can join our online tutoring programs and get required help from an expert tutor anytime.
Learn basics of discrete math and for that, get online help anytime by staying at home. Our personalized and one-on-one tutoring sessions are quite useful for students. Our tutors undergo extensive training and keep themselves abreast with the current requirements measured by the educational boards. Moreover, we follow same textbooks to maintain the same academic standard offered by the educational boards. Moreover, our online sessions are well-organized, relevant to topics and worthwhile for students in all manners. Experience this mathematics help today.
Discrete Math Online Tutoring
Through online tutoring and homework help, offered by TutorVista, students can improve their grades and boost up their confidence. We help you to solve problems based on Set theory, Graph theory, Number theory, Logic, Permutations and Combinations in an easy manner. Our online tutors have expertise in these respective topics. The topics included in the study of discrete math are given below.
Set Theory: Set is define as the collection of objects, which are arranged in a group. The set of numbers or objects can be denoted in the symbol braces {}. For example, the set of first 4 even numbers is {2,4,6,8}.
Graph Theory:It is the study of the graph. Graph is a mathematical structure, which contains a set points.
Logic: Logic in mathematics can be define as reasoning. There are basically three types of logic gates, AND($\wedge$), NOT(~), and OR($\vee$)
Permutations: The different arrangements that can be made with a given number of things taking some or all of them at a time are called permutations. Order does matter for the permutation. For example, first, second, third, etc.
Combinations: The selection of a number of things taking some or all of them at a time are called combinations. Order does not matter for the combination.
Sequence: A set of numbers arranged in a definite order according to some definite rule is called a sequence. A sequence is a function whose domain is the set N of natural numbers.
Series: Indicated sum of the terms in a sequence is called a series. The result of performing the additions is the sum of the series.
You can also get solved problems, answers and worksheets from TutorVista. Make learning Discrete Mathematics easy by choosing tutoring sessions with TutorVista.
Discrete Math Problems
Question: In how many ways can 6 men and 5 women be seated in a line so that no two women sit together?
Answer: Let the women be denoted by W and the men be denoted by M.
Now the women can sit at the places marked by '.' . We have seven places for 5 women. The women can sit in 7P5 ways. Also 6 men can be arranged in 6! ways.
Question: 2012 was a leap year. create a truth table for negation.
Solution: 2012 was not a leap
Truth table for NOT
Input p
Output $\widetilde{p}$
T
F
F
T
Question: In how many ways can three prizes be distributed among 4 boys when
i) No one gets more than one prize. ii) A boy can get any number of prizes.
Answer: i) The first prize can be given in 4 ways as one cannot get more than one prize, the remaining two prizes can be given in 3 and 2 ways respectively. The total number of ways = 4 x 3 x 2 = 24.
ii) As there is no restriction, each prize can be given in 4 ways. The total number of ways = 43 = 64.
Question: From a class of 32 students, 4 are to be chosen for a competition. In how many ways can this be done?
Answer: We are to select 4 students from 32. This selection can done in
Question: A sports team of 11 students is to be constituted choosing at least 5 from class XI and 5 at least from XII. If there are 20 students in each of these classes, in how many ways can the team be constituted?
Answer: Number of students in each class is 20.
Total number of selections = 2 x 600935040 = 1201870080
Question: Given five different green dyes, four different blue dyes and three different red dyes, how many combinations of dyes can be chosen taking at least one green and one blue dye?
Answer: The least number of dyes that a combination can have is 2. (one blue and one green).
Maximum number of dyes that a combination can have is 12 (5G, 4B, 6R).
At least one green dye can be selected out of 5 green dyes. The number of ways is
After selecting one or more green dyes, we can select at least one blue dye out of 4 different blue dyes. The number of ways is
After selecting at least one green dye and at least one blue dye, at least one red dye or no red dye can be selected in = 1 + 3 + 3 + 1 = 8 ways
The total number of combinations (31)(15)(8) = 3720.
Question: A letter lock consists of three rings, each marked with fifteen different letters. Find in how many ways it is possible to make an unsuccessful attempt to open the lock.
Answer: 1st ring can be attempted in 15 ways. 1st and 2nd ring can be attempted in 15 x 15 = 152 ways 1st and 2nd and 3d ring can be attempted in 15 x 15 x 15 = 153 ways Among these 153 attempts, one attempt will be a successful attempt. Hence the number of unsuccessful attempts is 153-1= 3375 -1 = 3374
Question: Find the sum of all four digit numbers formed by using 2, 3, 6, 9 in which no digit is repeated.
Answer: If 2 occupies unit's place, the remaining 3 digits can be arranged in 3!= 6ways. Similarly, if 2 occupies ten's place, hundred's place, thousand's place, in each of these cases we get 3! numbers. Thus, the positional value contributed by 2 to the sum when it occupies different values is
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bought this book to accompany Strang's online lectures from the MIT open courseware series, which I am completing as an independent study. The lectures are great--Strang is engaging and easy to understand. The book--not so much. It doesn't explain concepts fully and jumps through examples without explaining the steps
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Advanced Modern Engineering Mathematics
Building on the foundations laid in the companion text Modern Engineering Mathematics, this book gives an extensive treatment of some of the advanced ...Show synopsisBuilding on the foundations laid in the companion text Modern Engineering Mathematics, this book gives an extensive treatment of some of the advanced areas of mathematics that have applications in various fields of engineering, particularly as tools for computer-based system modelling, analysis and design. The philosophy of learning by doing helps students develop the ability to use mathematics with understanding to solve engineering problems. A wealth of engineering examples and the integration of MATLAB and MAPLE further support students Building on the foundations laid in the companion text...New. Building on the foundations laid in the companion text Modern Engineering Mathematics 3e, this book gives an extensive treatment of some of the advanced areas of mathematics that have applications in various fields of engineering, particularly as tools for computer-based system modelling, analysis and design. Printed Pages: 972 Advanced Modern Engineering Mathematics
The series make learning engineering maths simpler. Unfortunately there are little questions with answers within this book. Its the "new teaching method," where nothing can be known, unless your the lecturer with the answers booklet.
Still it is a great series, worth it as it covers many fields of mathematic study relevant for engineers. Fourier transforms, vectors, statistics, ect. Please find a chapter index and see for yourself.
***Remember this is a two part series. Modern Eng.. Math.. and Advanced Modern Eng.. Math.. I would advise people to buy both
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What Can You Do with Sage Math?
Sage Beginner's Guide — Save 50%
Unlock the full potential of Sage for simplifying and automating mathematical computing with this book and eBook
$29.99
$15.00
by
Craig Finch | May 2011 |
Open Source
Sage is a powerful tool—but you don't have to take my word for it. This article will showcase a few of the things that Sage can do to enhance your work. Look at the things Sage can do, and start to think about how Sage might be useful to you.
Getting started
You don't have to install Sage to try it out! In this article, we will use the notebook interface to showcase some of the basics of Sage so that you can follow along using a public notebook server. These examples can also be run from an interactive session if you have installed Sage. Go to and sign up for a free account. You can also browse worksheets created and shared by others. The notebook interface should look like this:
Create a new worksheet by clicking on the link called New Worksheet:
Type in a name when prompted, and click Rename. The new worksheet will look like this:
Enter an expression by clicking in an input cell and typing or pasting in an expression:
Click the evaluate link or press Shift-Enter to evaluate the contents of the cell.
A new input cell will automatically open below the results of the calculation. You can also create a new input cell by clicking in the blank space just above an existing input cell.
Using Sage as a powerful calculator
Sage has all the features of a scientific calculator—and more. If you have been trying to perform mathematical calculations with a spreadsheet or the built-in calculator in your operating system, it's time to upgrade. Sage offers all the built-in functions you would expect. Here are a few examples:
If you have to make a calculation repeatedly, you can define a function and variables to make your life easier. For example, let's say that you need to calculate the Reynolds number, which is used in fluid mechanics:
When you type the code into an input cell and evaluate the cell, your screen will look like this:
Now, you can change the value of one or more variables and re-run the calculation:
Sage can also perform exact calculations with integers and rational numbers. Using the pre-defined constant pi will result in exact values from trigonometric operations. Sage will even utilize complex numbers when needed. Here are some examples:
Symbolic mathematics
Much of the difficulty of higher mathematics actually lies in the extensive algebraic manipulations that are required to obtain a result. Sage can save you many hours, and many sheets of paper, by automating some tedious tasks in mathematics. We'll start with basic calculus. For example, let's compute the derivative of the following equation:
The following code defines the equation and computes the derivative:
var('x') f(x) = (x^2 - 1) / (x^4 + 1) show(f) show(derivative(f, x))
The results will look like this:
The first line defines a symbolic variable x (Sage automatically assumes that x is always a symbolic variable, but we will define it in each example for clarity). We then defined a function as a quotient of polynomials. Taking the derivative of f(x) would normally require the use of the quotient rule, which can be very tedious to calculate. Sage computes the derivative effortlessly.
Now, we'll move on to integration, which can be one of the most daunting tasks in calculus. Let's compute the following indefinite integral symbolically:
The code to compute the integral is very simple:
f(x) = e^x * cos(x) f_int(x) = integrate(f, x) show(f_int)
The result is as follows:
To perform this integration by hand, integration by parts would have to be done twice, which could be quite time consuming. If we want to better understand the function we just defined, we can graph it with the following code:
f(x) = e^x * cos(x) plot(f, (x, -2, 8))
Sage will produce the following plot:
Sage can also compute definite integrals symbolically:
To compute a definite integral, we simply have to tell Sage the limits of integration:
This would have required the use of a substitution if computed by hand.
Have a go hero
There is actually a clever way to evaluate the integral from the previous problem without doing any calculus. If it isn't immediately apparent, plot the function f(x) from 0 to 1 and see if you recognize it. Note that the aspect ratio of the plot may not be square.
The partial fraction decomposition is another technique that Sage can do a lot faster than you. The solution to the following example covers two full pages in a calculus textbook —assuming that you don't make any mistakes in the algebra!
We'll use partial fractions again when we talk about solving ordinary differential equations symbolically.
Linear algebra
Linear algebra is one of the most fundamental tasks in numerical computing. Sage has many facilities for performing linear algebra, both numerical and symbolic. One fundamental operation is solving a system of linear equations:
Although this is a tedious problem to solve by hand, it only requires a few lines of code in Sage:
Notice that Sage provided an exact answer with integer values. When we created matrix A, the argument QQ specified that the matrix was to contain rational values. Therefore, the result contains only rational values (which all happen to be integers for this problem).
Unlock the full potential of Sage for simplifying and automating mathematical computing with this book and eBook
Solving an ordinary differential equation
Solving ordinary differential equations by hand can be time consuming. Although many differential equations can be handled with standard techniques such as the Laplace transform, other equations require special methods of solution. For example, let's try to solve the following equation:
It turns out that the equation we solved is known as Bessel's equation. This example illustrates that Sage knows about special functions, such as Bessel and Legendre functions. It also shows that you can use the assume function to tell Sage to make specific assumptions when solving problems.
More advanced graphics
Sage has sophisticated plotting capabilities. By combining the power of the Python programming language with Sage's graphics functions, we can construct detailed illustrations. To demonstrate a few of Sage's advanced plotting features, we will solve a simple system of equations algebraically:
var('x') f(x) = x^2 g(x) = x^3 - 2 * x^2 + 2
solutions=solve(f == g, x, solution_dict=True)
for s in solutions: show(s)
The result is as follows:
We used the keyword argument solution_dict=True to tell the solve function to return the solutions in the form of a Python list of Python dictionaries. We then used a for loop to iterate over the list and display the three solution dictionaries. Let's illustrate our answers with a detailed plot:
We created a plot of each function in a different colour, and labelled the axes. We then used another for loop to iterate through the list of solutions and annotate each one.
Visualising a three-dimensional surface
Sage does not restrict you to making plots in two dimensions. To demonstrate the 3D capabilities of Sage, we will create a parametric plot of a mathematical surface known as the "figure 8" immersion of the Klein bottle. You will need to have Java enabled in your web browser to see the 3D plot.
In the Sage notebook interface, the 3D plot is fully interactive. Clicking and dragging with the mouse over the image changes the viewpoint. The scroll wheel zooms in and out, and right-clicking on the image brings up a menu with further options.
Typesetting mathematical expressions
Sage can be used in conjunction with the LaTeX typesetting system to create publication-quality typeset mathematical expressions. In fact, all of the mathematical expressions in this article were typeset using Sage and exported as graphics.
A practical example: analysing experimental data
One of the most common tasks for an engineer or scientist is analysing data from an experiment. Sage provides a set of tools for loading, exploring, and plotting data. The following series of examples shows how a scientist might analyse data from a population of bacteria that are growing in a fermentation tank. Someone has measured the optical density (abbreviated OD) of the liquid in the tank over time as the bacteria are multiplying. We want to analyse the data to see how the size of the population of bacteria varies over time. Please note that the examples in this section must be run in order, since the later examples depend upon results from the earlier ones.
Time for action – fitting the standard curve
The optical density is correlated to the concentration of bacteria in the liquid. To quantify this correlation, someone has measured the optical density of a number of calibration standards of known concentration. In this example, we will fit a "standard curve" to the calibration data that we can use to determine the concentration of bacteria from optical density readings:
What just happened?
We introduced some new concepts in this example. On the first line, the statement import numpy allows us to access functions and classes from a module called NumPy. NumPy is based upon a fast, efficient array class, which we will use to store our data. We created a NumPy array and hard-coded the data values for OD, and created another array to store values of concentration (in practice, we would read these values from a file) We then defined a Python function called standard_curve, which we will use to convert optical density values to concentrations. We used the find_fit function to fit the slope and intercept parameters to the experimental data points. Finally, we plotted the data points with the scatter_plot function and the plotted the fitted line with the plot function. Note that we had to use a function called zip to combine the two NumPy arrays into a single list of points before we could plot them with scatter_plot.
Time for action – plotting experimental data
Now that we've defined the relationship between the optical density and the concentration of bacteria, let's look at a series of data points taken over the span of an hour. We will convert from optical density to concentration units, and plot the data.
What just happened?
We defined one NumPy array of sample times, and another NumPy array of optical density values. As in the previous example, these values could easily be read from a file. We used the standard_curve function and the fitted parameter values from the previous example to convert the optical density to concentration. We then plotted the data points using the scatter_plot function.
Time for action – fitting a growth model
Now, let's fit a growth model to this data. The model we will use is based on the Gompertz function, and it has four parameters:
What just happened?
We defined another Python function called gompertz to model the growth of bacteria in the presence of limited resources. Based on the data plot from the previous example, we estimated values for the parameters of the model to use an initial guess for the fitting routine. We used the find_fit function again to fit the model to the experimental data, and displayed the fitted values. Finally, we plotted the fitted model and the experimental data on the same axes.
Summary
This article has given you a quick, high-level overview of some of the many things that Sage can do for you.
Specifically, we looked at:
Using Sage as a sophisticated scientific and graphing calculator
Speeding up tedious tasks in symbolic mathematics
Solving a system of linear equations, a system of algebraic equations, and an ordinary differential equation
About the Author :
Craig Finch is a Ph. D. candidate in the Modeling and Simulation program at the University of Central Florida (UCF). He earned a Bachelor of Science degree from the University of Illinois at Urbana-Champaign and a Master of Science degree from UCF, both in electrical engineering. Craig worked as a design engineer for TriQuint Semiconductor, and currently works as a research assistant in the Hybrid Systems Lab at the UCF NanoScience Technology Center. Craig's professional goal is to develop tools for computational science and engineering and use them to solve difficult problems. In particular, he is interested in developing tools to help biologists study living systems. Craig is committed to using, developing, and promoting open-source software. He provides documentation and "how-to" examples on his blog at
I would like to thank my advisers, Dr. J. Hickman and Dr. Tom Clarke, for giving me the opportunity to pursue my doctorate. I would also like to thank my parents for buying the Apple IIGS computer that started it all.
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books.google.com - Expanded coverage of essential math, including integral equations, calculus of variations, tensor analysis, and special integralsMath Refresher for Scientists and Engineers, Third Edition is specifically designed as a self-study guide to help busy professionals and students in science and engineering... Refresher for Scientists and Engineers
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The graphing calculators that are most commonly used in pre-calculus and
Calculus I are the TI-82/83/84 series and the TI-85/86 series. Specifically, the standard
TI-86 will be what this manual is based on, and is shown below on the right.
TI-83 TI-86
These calculators are similar to each other and both have the functionality needed
for mathematics up through Calculus I. The main difference between them is the TI-83 is
generally more user friendly, while the TI-86 has more features and functionality. Both
calculators have a black and white screen on the top half and a large keypad on the
bottom half. The TI-82 and TI-84 are very similar to the TI-83, and the TI-85 is very
similar to the TI-86. Keep in mind that there are some subtle differences between
calculators of the same series.
Below is a closer look at the keypads. Both have the same number of keys, but
Some are in different places and some are on one keypad and not the other. Both
keypads have three "levels" of keys. As you can see, there is white writing on the keys
themselves, yellow to the upper-left of the keys, and green (TI-83) or blue (TI-86) to the
upper right of the keys.
TI-83 Keypad TI-86 Keypad
To use the white level you simply type the key. For example, to enter a "5"
simply press the [ 5 ] located in the middle of the keypad. However, if you want to use
the yellow functions, press the [ 2nd ] key followed by the button directly below it. For
example, to enter the number "Pi", press the yellow [ 2nd ] and then press the [ ^ ] key
located on the right middle side of the keypad. The blue level of the keypad gives access
to the alphabet. For upper case letters, press the [ ALPHA ] key followed by the key
directly below the letter. And, for lower case letters, press [ 2nd ] [ ALPHA ] then the
key directly below the letter. For example, to enter an "E" using the TI-86, press
[ ALPHA ] [ ^ ]. To enter an "e" using the TI-86, press [ 2nd ] [ ALPHA ] [ ^ ].
Graphing – Inputting Equations (TI-86)
Overview
To graph a function on the TI-86, the equation must be written in "y = f(x)" form.
Once this is accomplished, a graph can be made. The graph will be displayed in
the window dimensions that are currently set. To change the window dimensions
refer to the section titled "Window Adjust".
Procedure
1. From the home screen press [ GRAPH ] to display the graphing options
2. Press [ F1 ] to select " y(x)= "
3. Press [ CLEAR ] if the " y1=" line is not blank
4. Enter the equation, pressing [ x-VAR ] whenever an "x" is needed
5. Once the equation is entered in, press [ 2nd ] [ F5 ] to select "GRAPH"
6. Your equation will be graphed in the window
Example
Graph the function y = x3 – 2x2 – x + 1.3
1.) 2.)
3, 4.) 5, 6.) Graphed in (-5,5) x (-5,5).
Graphing – Multiple Equations / Line Style (TI-86)
Overview.
The TI-86 allows multiple equations to be displayed simultaneously. The line
style can also be changed to help distinguish between graphs.
Procedure
1. Complete steps one through four of "Inputting Equations"
2. Press the down arrow
3. Complete steps one through four of "Inputting Equations" with a different
equation
4. Press [ MORE ], and move the cursor back to the "y1=" line
5. Press [ F3 ] to select "STYLE". Notice the diagonal line, on the left, is now
bold. This is what the first graph will be graphed as. Pressing "STYLE"
multiple time will shuffle through different style options
6. Press [ 2nd ] [ F5 ] to graph the equations
Example
Graph the function y = sin (x) (bold), and y = cos (x) (not-bold)
1, 2, 3, 4.) 5, 6.)
Window Adjust (TI-86)
Overview
When viewing the graph of a function, there may be a "best" view of the graph.
Too close and the general shape of the graph may not be seen, and too far and the
details of the graph may go unnoticed. The TI graphing calculators allow the user
to look very closely at certain parts of a graph, similar to looking under a
microscope. Also, the user is allowed to look at a much bigger area of the graph.
The four most basic and most used methods to adjust the viewing window are 1.)
Manually setting the x and y intervals in the "WIND" function, 2.) "ZIN" (zoom
in) under the "ZOOM" function, 3.) "ZOUT" (zoom out) under the "ZOOM"
function, and 4.) "BOX" under the "ZOOM" function.
Procedure
Manually setting the x and y intervals in the "WIND" function
1. Graph the function and view it
2. Press [ GRAPH ] then [ F2 ] to select "WIND"
3. Using the up/down arrow keys, select the field to set and press [ CLEAR ]
4. Type in the value wanted for each field keeping in mind that xMin must be a
smaller number than xMax, and yMin must be smaller that yMax. xScl and
yScl stand for x-scale and y scale, this is the distance between tick marks
5. Press [ F5 ], when finished, to graph it
"ZIN"
1. Graph the function and view it
2. Press [ F3 ] to select "ZOOM"
3. Move the cursor to the center of the area to zoom in on
4. Press [ F2 ] to select "ZIN" and press [ ENTER ]
"ZOUT"
1. Graph the function and view it
2. Press [ F3 ] to select "ZOOM"
3. Move the cursor to the center of the area to zoom in on
4. Press [ F3 ] to select "ZOUT"and press [ ENTER ]
"BOX" - This uses a user defined section of the window and blows it up to fill the
entire window.
1. Graph the function and view it
2. Press [ F3 ] to select "ZOOM"
3. Press [ F1 ] to select "BOX"
4. Use the arrow keys to move the cursor to one corner of the box that will fill the
custom window and press [ ENTER ]
5. Use the arrow keys to move the cursor to the opposite corner of the box and
press [ ENTER ]
(Examples on next page)
Examples
Examine the function y = sin (1/(x-2)) + 1
Manual "WIND"
1.) This is how the function 2.) This is the graph in x->(-10,10),
should be entered. y-> (-10,10) viewing window.
3.) Enter in new window dimensions. 4.) Here's the graph in the new
window.
"ZIN"
1.) From the same screen as #2 2.) Press [ F2 ] to select "ZIN",
above, press [ F3 ] to select "ZOOM". move the cursor to center of where most
activity appears, press [ ENTER ].
3.) Repeat to zoom in even closer.
"ZOUT"
1.) From #3 in the previous 2.) Press [ F3 ] again to select "ZOUT".
example, press [ GRAPH ] [ F3 ].
3.) Move the cursor to center 4.) Repeat to Zoom out even further.
of location and press [ ENTER ].
"BOX"
1.) From the same screen as #2 in first 2.) Press [ F1 ] to select "BOX",
example, press [ F3 ] to select "ZOOM". position the cursor near x = 1, y = 2,
press [ ENTER ]
3.) Position the cursor near 4.) The box is now blown up in the
x = 3, y = -1, press [ ENTER ]. window.
FMIN, Finding the Minimum Value of a Function (TI-86)
Overview
Often in Calculus, the minimum and maximum values of functions are of interest.
This page shows how to find the local minimum of a function on an interval. To
find the local maximum, simply repeat this example but instead use the "FMAX"
function instead of the "FMIN" function in #3 below.
Procedure
1. Graph the function and view it
2. Press [ MORE ]and then press F1 to select "MATH"
3. Press [ F4 ] to select "FMIN"
4. Use the left/right arrow keys to select the left bound of the interval containing
the minimum value and press [ ENTER ]
5. Use the left/right arrow keys to select the right bound of the interval
containing the minimum value and press [ ENTER ]
6. Use the left/right arrow keys to select a guess close to the minimum value and
press [ ENTER ]
7. Your minimum value will be returned in x-y coordinates
Example
Find the local minimum value of the equation y = x3 - 2x2 - x + 1.3
1, 2.) Graphed in (-5,5) x (-5,5). 3.)
4.) Select the left bound. 6.) Select a guess.
5.) Select the right bound. 7.) Minimum value is displayed.
ISECT (intersect) TI-86
Overview
To find the intersection point of two graphs, the "ISECT" function can be useful.
Although, the "TRACE" function can do a similar job, the "ISECT" function can
return the result faster and more accurately. To use this function, at least two
equations must be entered and graphed with the intersection point in the viewing
window. The "ISECT" function asks for the first function and then the second
function it intersects and then asks for a guess. (The reason for the guess is that,
in case there are two intersection points, the calculator returns the right one.)
Procedure
1. Graph at least two functions and view them with the intersection point in the
viewing window
2. Press [ MORE ] and [ F1 ] to select "MATH"
3. Press [ MORE ] and [ F3 ] to select [ ISECT ]
4. Use the up/down arrows to move the cursor to the first function and press [
ENTER ]
5. Use the up/down arrows to move the cursor to the second function and press [
ENTER ]
6. Use the left/right arrows to move the cursor near the intersection point and
press [ ENTER ]
Example
Find the first intersection point of Sin (x) and Cos (x) on the positive x-axis.
1.) Sin (x) and Cos (x) are graphed on 2.)
the x-interval from 0 to 2Pi, Sin (x) is
the bold curve.
3.) First curve 4.) Second curve
(continued on next page)
5.) Guess 6.) Note: 1/sqrt (2) = .707106.....
and Pi/4 = .78539....
dy/dx (TI-86)
Overview
Checking your derivatives using the TI graphing calculators can be very helpful
when doing homework.
Procedure
1. Graph the function and view it
2. Press [ MORE ] and [ F1 ] to select "MATH"
3. Press [ F2 ] to select dy/dx
4. Move the cursor or type in the x-value of the point where the derivative is to
be evaluated and press [ ENTER ]
Example
Verify that the top hemisphere of the unit circle, y = √ (1 - x2), has a derivative of
-1 at the point on the circumference 45 degrees up from the x-axis. This is the
point (1/√ 2, 1/√ 2) or approximately (.707, .707).
1.) 2.)
3.) Moving the cursor. 4.) Type in the x value.
5.)
Integrals (TI-86)
Overview
It is often convenient to check answers to homework using the graphing
calculator. Though it's no substitute for doing integrals, the TI graphing
calculators allow the user to find numerical values of definite integrals.
Procedure
1. Graph the function and view it, making sure the lower and upper bounds of
the integral are within the viewing window
2. Press [ MORE ] and then [ F1 ] to select "MATH"
3. Press [ F3 ] to select the "integrate" function
4. Move the cursor or type in the lower bound, press [ ENTER ]
5. Move the cursor or type in the upper bound, press [ ENTER ]
Example
Find the surface area of the top hemisphere of a circle with radius 1. From
geometry we know this is (Pi)*(r2)/2 = (Pi)*(12)/2 ≈ 1.57079 ... , now we'll check
using the calculator's integrate function.
1.) x2 + y2 = 12 2, 3.)
4.) 5.)
6.) The calculator's numerical answer of
1.57193.... is off by a couple thousandths.
Glossary (TI-86)
Home Screen The starting point for doing anything with the TI graphing calculators. To
get to the Home Screen use the QUIT function. Using the TI-83 press [
2nd ] [MODE ], with the TI-86 press [ 2nd ] [ EXIT ].
Inflection Point The point were a graph changes from concave up to concave down or
vice versa. For example, it can be shown that the inflection point of the
equation y = x3 is at (0,0); here, the graph changes from concave down to
concave up.
int (x) The function on the TI calculators that returns the greatest integer less than
or equal to the value within the parentheses. For example, int (3.3) returns
3 and int (-3.3) returns "-4". If this is not clear, draw a number line and
demonstrate it to yourself.
Integral In simple terms, the area above the x-axis and below the graph of a
function, between two endpoints.
Radian Mode When working with trigonometric functions, the user has to be aware of
whether degrees or radians are being used. If the user assumes the
calculator is in degree mode, but it's really in radian mode, the calculator
will give "bad" answers. To change from degree to radian mode or vice-
versa, from the home screen press [ 2nd ] [ MORE ] to select the "MODE"
function. Then, highlight the "Radian" option. For most situations in Pre-
Calculus and Calculus I, you will want the entire left column of the
"MODE" screen highlighted.
Style The function on the TI graphing calculators that changes the style of the
line of the graph. Standard line style is a line one-pixel thick. Other
options are dotted lines, bold lines, lines with shading below or above.
This is used to distinguish between multiple functions displayed at the
same time.
Trace The function on the TI calculators that uses a cursor to trace over the
graph displaying the x and y coordinates. The left/right arrow keys move
the cursor position along the graph of an equation, the up/down arrow
keys move the cursors to another graph when multiple equations are
displayed.
Window The defined area displayed on the screen of the TI calculators. The
window function will let the user adjust the x and y intervals that will
define the boundaries of the displayed graphs. Example x and y intervals
on the TI graphing calculators would be: x-min = -10, x-max = 10, y-min
= -10, y-max = 10. This would set the window boundaries to a 20 by 20
rectangle centered at the origin (0,0).
"y=f(x)" form The required form of the equation needed, to be entered into the TI
graphing calculators, when graphing in the x-y plane. For example
" y - 2x = x2 + 2" is not in y = f(x) form, whereas "y = x2 + 2x + 2" is in
"y = f(x)"
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Math is often viewed as a completely neutral subject; 1+1=2 is the same around the world, isn't it? Yet this handy book reveals how the rules of a covenant-keeping God allow math to function, as it points out the evidence's of God's hand in creation, the functionality of everyday, practical math, and how to view and teach math biblically. A list of resources and curriculum to use is provided. 101 pages, softcover.
This second edition features an updated curriculums and resources chapter, as well as additional details on teaching math biblically.
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The book provides a bridge from courses in general physics to the intermediate-level courses in classical mechanics, electrodynamics and quantum mechanics. The author bases the mathematical discussions on specific physical problems to provide a basis for developing mathematical intuition.
Inside This Book(Learn More)
Browse and search another edition of this book.
First Sentence
Our aim in this book is to introduce you to some mathematical applications that you will encounter in intermediate and advanced undergraduate courses in classical mechanics, electromagnetism, and quantum physics. Read the first page
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More About
This Textbook
Overview
Undergraduate Algebra is a text for the standard undergraduate algebra course. It concentrates on the basic structures and results of algebra, discussing groups, rings, modules, fields, polynomials, finite fields, Galois Theory, and other topics. The author has also included a chapter on groups of matrices which is unique in a book at this level. Throughout the book, the author strikes a balance between abstraction and concrete results, which enhance each other. Illustrative examples accompany the general theory. Numerous exercises range from the computational to the theoretical, complementing results from the main text.
Editorial Reviews
Booknews
Splendid undergraduate text, intended to function as a companion to the distinguished author's Linear algebra and to provide young mathematicians with a secure command of the fundamentals of groups, rings, fields, and related structures. Ten chapters, many excellent problems, written with exemplary clarity and with exceptional sensitivity to what young readers might on first encounter consider to be "scary". Departs from the previous edition (1987) by the inclusion of some new material and exercises. The author has been very well served by the production people at Springer, who have produced a physically beautiful book at a reasonable price. (NW) Annotation c. Book News, Inc., Portland, OR (booknews.com)
From the Publisher
From the reviews of the third edition:
"As is very typical for Professor Lang's self demand and style of publishing, he has tried to both improve and up-date his already well-established text. … Numerous examples and exercises accompany this now already classic primer of modern algebra, which as usual, reflects the author's great individuality just as much as his unrivalled didactic mastery and his care for profound mathematical education at any level. … The present textbook … will remain one of the great standard introductions to the subject for beginners." (Werner Kleinert, Zentralblatt MATH, Vol. 1063
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Note I said "shouldn't". A high school physics class has little need for numbers. Conceptual questions are more important, and the calculations can be abstract; for example, "A box is at rest on a slope. The coefficient of static friction between the box and the surface is μ. What is the greatest possible angle of inclination?" This sort of question sidesteps the numerics issue, meanwhile better testing understanding than if I had asked for the minimum coefficient of static friction such that a 2kg box would stay at rest on a slope of inclination 37 degrees from horizontal. As well, the first question makes obvious the dependencies of the problem, such as the fact that the answer doesn't depend on the gravitational strength or the weight of the box.
So basically ignoring all of the practical side of physics, which involves actally doing experiments. What you have just described was basically my high school maths mechanics class. Physics is related to maths, but it is not maths, half of it is experimenting and calculating stuff in the real world, all of which can serve to re-enforce the theoretical ideas.
Oh, sorry, I should have been more clear. I'm referring primarily to exams and testing theoretical material. By all means, yes, calculators/computers should be used in experimentation and lab settings.
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Precalculus: Graphical, Numerical, Algebraic
In Precalculus, the authors encourage graphical, numerical, and algebraic modeling of functions as well as a focus on problem solving, conceptual ...Show synopsisIn Precalculus, the authors encourage graphical, numerical, and algebraic modeling of functions as well as a focus on problem solving, conceptual understanding, and facility with technology. They have created a book that is designed for instructors and written for students making this the most effective precalculus text available
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Mathematics Placement
Based on your previous coursework and experience in mathematics, you can determine which of the following initial mathematics courses would be most appropriate if you choose to take a Mathematics course to fulfill the Quantitative and Analytical Reasoning requirement. Read the descriptions of these courses carefully, mindful of your prior math preparation, and choose the level that matches your interests and abilities. First-year students typically choose their first mathematics course from among the four options listed below. Several majors require specific quantitative courses as seen in the table on Quantitative and Analytical Reasoning.
Students who are primarily seeking to obtain a broad background and to fulfill the quantitative requirement will be best served by options 1 through 4. Students desiring a more technical quantitative background, particularly for use in mathematics or quantitative science, will be better served by options 4 through 9, courses in the main calculus sequence. Option 4 does indeed fit both categories of students. All of the courses in options 1 to 4 have the (QA) or (QA*) designation as indicated.
If you opt to take calculus, which course in the sequence is for you? Information for placement within the calculus sequence is provided below. Your background and previous calculus experience will place you into the most suitable course. You may also contact a member of the Mathematics Department for advice – see
Statistics (MATH 138) (QA*) - An introduction to descriptive and inferential statistics. Emphasizes everyday applications and practical skills. This course is an excellent preparation for dealing with the statistics one encounters every day in our society, and is particularly recommended for students who neither need nor desire a calculus background. Prerequisite: two years of high school algebra.
Discrete Mathematics (MATH 163) (QA) – An introduction to basic techniques and modes of reasoning in combinatorial problem-solving. Topics will be chosen from combinatorial mathematics, logic and Boolean algebra, difference equations, graph theory and applied algebra. Prerequisite: two years of high school algebra. Note this course is offered only in spring semester.
Modeling with Calculus (MATH 140) (QA*) – Modeling with Calculus introduces and applies the concept of calculus to solve open-ended, real-word problems, especially those in the natural and social sciences. The emphasis is on developing and interpreting mathematical models. Topics include differential calculus, linear algebra, and differential equations. This course takes advantage of computational tools so that the focus can be on calculus concepts useful in applied work. This course is appropriate for students with no prior calculus experience. Prerequisite: High school math beyond Algebra II recommended. Students who have taken a full year of high school calculus should begin calculus study with MATH 152, MATH 153, or MATH 249; see calculus placement advice below.
Accelerated Calculus I (MATH 151) (QA*)(0.5 cr) - A first course in calculus for students with some previous exposure to the subject. Topics covered include limits; continuity; derivatives of algebraic, trigonometric, and exponential functions; implicit differentiation; the Mean Value Theorem; and optimization.
Calculus Placement Advice
In the Advising and Course Preferences Questionnaire you indicated if you have previous calculus experience. This data will assist the Registration Advisor in placing you into the most suitable course based on your background. While each specific situation is different, we generally follow the placement advice outlined below.
Students with no calculus background should take MATH 140 Modeling with Calculus. This course is designed for students who are likely to only take one course in calculus. We recommend that students have high school math beyond Algebra II. Students with some exposure to calculus who are planning to take more than one calculus course should consider taking Accelerated Calculus I MATH 151, a 0.5 credit course.
Preparation for Calculus: In most cases, students who want to take calculus but who have not had calculus before should take MATH 140, which includes significant pre-calculus review. Students who feel they need more review also have the option to sign up for MATH 135 Preparation for Calculus. Please note that MATH 135 does not carry a QA or QA* designation. Please contact a member of the Math Department if you are not sure which course is best for you.
Students with High School Calculus:
AP credit
A score of 4 on the Calculus AB exam earns credit for MATH 151 and places students into MATH 249, MATH 152, or MATH 153.
A score of 5 on the Calculus AB exam or a score of 4 on the Calculus BC exam earns credit for MATH 151 and MATH 152, and places students into MATH 249.
A score of 5 on the Calculus BC exam earns credit for MATH 151, MATH 152, and MATH 153 and places students into MATH 249.
Students with high school calculus but no AP credit
Calculus taken
Grades
Place into
Comments
Full year AP (A/B version)
A's or A/B
MATH 153 or MATH 249
MATH 153 is a half semester course offered in the second half semester.
Chemistry, Mathematics, and Physics majors use sequences and series.
Full year AP (B/C versions)
A's or A/B
MATH 249
Full year non-AP
A's
MATH 153 or MATH 249
MATH 153 is a half semester course offered in the second half semester.
Chemistry, Mathematics, and Physics majors use sequences and series.
Full year AP (A/B or B/C versions)
B's or B/C
MATH 152
MATH 152 is a half semester course offered in both first and
second half semesters
Full year non-AP
A/B or B's
MATH 152 or MATH 153
Semester only or full year with lower grades
MATH 140 or MATH 151
MATH 140 is designed for students who'll only take one course in calculus.
MATH 151 is appropriate for students with exposure to calculus, but who
need a review.
Students wishing to place lower than recommended in this table will need departmental approval. Students may also seek departmental approval to enroll at a higher level than recommended.
General Calculus Placement advice: As a rule, we recommend that students aim high in their calculus placement. If students get in over their heads, we can help them change to a lower level course in the sequence. If students find themselves unchallenged after three weeks in a lower-level course, it is often too late to change to a higher level. If in doubt, please contact the department personally.
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The second year undergraduate physics course consists of 13 lectures. The combination of vector calculus and wave physics can often be difficult when first encountered. A good way to gain confidence is to apply them in problems. Mathematica is used as the vehicle for the course notes and assignments and is used both as a calculation and visualization tool.
Materials
Recommended reading:
Introduction to Electrodynamics by Griffiths
Electromagnetism: Principles and Applications by Lorrain and Corson
Volume 2 of The Feynman Lectures on Physics
Description
This course provides both an introduction to Electromagnetism and to Mathematica. A large proportion of the students taking this course use Mathematica to complete the assignments.
Topics: These lecture notes are based on notes by Dr Tom R. Marsh of the Department of Physics and Astronomy at the University of Southampton. Introduction:
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Webster, TX Precalculus portion focuses on reading comprehension and analysis of certain passages asking for the purpose of the author's intention in the passage, and also inference of what is said in the passage. The math portion deals with simple algebra and number operations, percents, graphs and word pr...
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According to The Orange Grove, this text is "written for high school students. CK-12 Foundation's Trigonometry FlexBook is an...
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According to The Orange Grove, this text is "written for high school students. CK-12 Foundation's Trigonometry FlexBook is an introduction to trigonometry. Topics include: Trigonometric Identities & Equations, Circular Functions, and Polar Equations & Complex Number.״
'Don;t want an expensive calculator with functions you would never use?Do you want to solve your trigonometric problems in a...
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'Don;t want an expensive calculator with functions you would never use?Do you want to solve your trigonometric problems in a faster and more pleasing way?Well then , Designer Trigonometric Calculator is the right thing for you!With al the trigonometric functions in just one application , you'll be solving your problems in no time.!!!Choose the way you like your calculator to be!Calculate Sin , Cos , Tan etc. in degrees or radians as per your wish in this packaged application!'This is a free app
״Math.Trig is great for students learning how to solve triangles with it's simple interface, clear, brightly colored diagrams...
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Mathematics is not typically considered (by mathematicians) to be a solo sport; on the contrary, some amount of mathematical interaction with others is often deemed crucial. Courses are the student's ...
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Functions, Statistics and
Trigonometry (FST)
Course Outcome Summary
Information
Credits 1
Organization Hartford Union High School:
Mission/Description
Mission Statement: Our mathematics program is designed to enable all students to function successfully
in a number related world, to give students a basis to increase their mathematical knowledge and improve
their quality of life, and to develop an understanding and appreciation of mathematics and technology.
Course Description: This course is designed as a college preparatory course for the student who is not
necessarily majoring in a mathematics or science field. Successful completion of this course should
prepare the student for Pre-Calculus. The topics will include functions, logarithms, trigonometry,
probability, and statistics. The student will be required to have a graphing calculator (TI83 Plus or TI83
recommended. Cost approximately $100.)
Textbooks
Advanced Mathematical Concepts
Units and Timelines
Linear Relations and Functions - 10 days
Exploring Graphs of Functions - 15 days
Solving Polynomial and Rational Functions - 20 days
Exploring Conic Sections - 15 days
Exploring Exponential and Logarithmic Functions - 15 days
Sequences and Series - 15 days
The Trigonometric functions - 15 days
Graphs and Inverses of the Trigonometric functions - 15 days
Trigonometric Identities and Equations - 15 days
Vectors and Parametric Equations - 10 days
Combinatorics and Probability - 15 days
Statistics and Data Analysis - 20 days
Core Abilities
Analyze divergent views to resolve conflicts.
Assess personal goals, values, and motivations.
Assume responsibility for own actions.
Balance a lifestyle that is safe and healthy.
Demonstrate appropriate communication skills.
Devise solutions to complex problems.
Initiate positive leadership skills.
Integrate self awareness and attitudes appropriate in diverse social situations.
Organize data into useable forms.
Utilize appropriate behaviors and attitudes in the work environment.
Utilize technology to improve efficiency.
Competencies, Linked Standards, Objectives and
Performance Standards
1. Use statistics to describe data sets.
Properties
Domain: Cognitive
Level: Application
Difficulty: Medium
Importance: Important
Linked External Standards
WI.MA.A.12.1 Use reason and logic
WI.MA.A.12.2 Communicate logical arguments and clearly show why a result does or
does not make sense, why the reasoning is or is not valid, and an understanding of the
difference between examples that support a conjecture and a proof of the conjectureA.12.6 Read and understand mathematical texts and other instructional
materials, writing about mathematics (e.g., articles in journals), and mathematical ideas
as they are used in other contexts2 Organize and display data from statistical investigations
WI.MA.E.12.3 Interpret and analyze information from organized and displayed data
when given measures of dispersion, including standard deviation and variance; measures
of reliability; and measures of correlation
WI.MA.E.12.4 Analyze, evaluate, and critique the methods and conclusions of
statistical experiments reported in journals, magazines, news media, advertising, etc.
Performance Standards
o learner calculates statistics from a data set and display the results in a graph with 75%
accuracy.
o learner determines the probability that a randomly selected member of a sample lies
within a specified number of standard deviations from the mean.
o learner determines the standard error of the mean to predict the true mean of a
population with a 1% and 5% level of confidence.
Learning objectives
a. Calculate measures of central tendency and spread for data sets.
b. Describe relations between measures of central tendency or measures of spread.
c. Use sigma notation to represent a sum, mean, mean deviation, standard deviation or
variance.
d. Interpret data presented in a frequency distribution.
e. Read and interpret histograms, stem-and-leaf plots, and box-and-whisker plots.
f. Draw a histogram, frequency polygon, stem-and-leaf plot, and box-and-whisker plot, to
display data.
g. Use the normal distribution curve to represent data.
h. Use samples to make inferences about populations.
2. Describe the nature of graphs by examining transformations, symmetry, continuity,
extrema, end behavior, and asymptotes.
Properties
Domain: Cognitive
Level: Comprehension
Difficulty: Low
Importance: Important
Linked External Standards
WI.MA.A.12.1 Use reason and logic
WI.MA.A.12.4 Develop effective oral and written presentations employing correct
mathematical terminology, notation, symbols, and conventions for mathematical
arguments and display of dataPerformance Standards
o learner describes how transformations affect important features of graphs and how those
effects are represented in the changes in equations with 75% accuracy.
o learner analyzes all aspects of a function using appropriate tests with 75% accuracy.
Learning objectives
a. Describe the symmetries of graphs.
b. Identify transformations of the constant, identity, polynomial, square root, absolute
value, greatest integer, and rational functions.
c. Graph nonlinear inequalities.
d. Solve absolute value inequalities.
e. Determine the inverse of a function.
f. Graph functions and their inverses.
g. Determine whether a function is continuous or discontinuous.
h. Identify the end behavior of functions .
i. Determine whether a function is increasing or decreasing on an interval.
j. Find the critical points of a function.
k. Use the first and second derivative tests to determine whether a critical point is a
maximum, minimum, or inflection point.
l. Graph rational functions.
m. Determine vertical, horizontal, and slant asymptotes of a function.
3. Solve exponential and logarithmic equations.
Properties
Domain: Cognitive
Level: Application
Difficulty: High
Importance: Important
Linked External StandardsB.12.3 Perform and explain operations on real numbers
WI.MA.B.12.4 Select and use appropriate procedures in problem solving situations
involving the application of different number systems (natural, integers, rational, and
real)
WI.MA.B.12.5 Create and critically evaluate numerical arguments presented in a
variety of classroom and real-world situations (e.g., political, economic, scientific, social
WI.MA.E.12.1 Work with data in the context of real-world situations
WI.MA.F.12.1 Analyze and generalize patterns of change (e.g., direct and inverse
variation) and numerical sequences, and then represent them with algebraic expressions
and equationsWI.MA.F.12.4 Model and solve a variety of mathematical and real-world problems by
using algebraic expressions, equations, and inequalities
Performance Standards
o learner applies the properties of logarithms to solve equations with 75% accuracy.
Learning objectives
a. Find the value of logarithmic expressions.
b. Describe properties of exponential and logarithmic functions.
c. Graph exponential and logarithmic functions.
d. Use e and natural logarithms to solve applied problems.
e. Use logarithmic and exponential functions to model data.
4. Use trigonometry to find sides or angles in triangles.
Properties
Domain: Cognitive
Level: Application
Difficulty: Medium
Importance: Essential
Linked External Standards
WI.MA.A.12.1 Use reason and logicWI.MA.D.12.3 Determine measurements indirectly
WI.MA.F.12.4 Model and solve a variety of mathematical and real-world problems by
using algebraic expressions, equations, and inequalities
Performance Standards
o learner uses properties of trigonometry to solve for missing dimensions in a triangle with
75% accuracy.
Learning objectives
a. Convert decimal degree measures to degrees, minutes, and seconds and vice versa.
b. Find sines, cosines, and tangents of acute angles using the trigonometric ratios.
c. Find missing side and/or angle measures of a triangle using the trigonometric ratios.
d. Solve problems involving the Law of Sines and the Law of Cosines.
e. Determine the area of a triangle given the measure of two sides and the included angle,
the measure of two angles and a side, or the measure of the three sides.
5. Solve problems involving circular functions and their graphs.
Properties
Domain: Cognitive
Level: Application
Difficulty: Medium
Importance: Important4 Model and solve a variety of mathematical and real-world problems by
using algebraic expressions, equations, and inequalities
Performance Standards
o learner solves equations involving circular functions with 75% accuracy.
o learner graphs the 6 circular functions and be able to graph their image under a linear
transformation with 75% accuracy.
o learner uses sinusoidal functions to solve problems with 75% accuracy.
Learning objectives
a. Identify the amplitude, period, frequency, vertical translation, and phase shift of circular
functions.
b. Evaluate inverse and reciprocal circular functions.
c. State properties of inverse and reciprocal circular functions.
d. Graph or identify graphs of circular functions.
e. Graph or identify graphs of inverse functions.
f. Model real-world data using sine and cosine functions.
6. Use the unit circle to find the values of the trigonometric functions for any angle.
Linked External Standards
WI.MA.B.12.4 Select and use appropriate procedures in problem solving situations
involving the application of different number systems (natural, integers, rational, and
real)Performance Standards
o learner labels the unit circle with 100% accuracy.
o learner uses properties of the unit circle to find the value of the six trigonometric ratios
with 75% accuracy.
Learning objectives
a. Identify angles that are conterminal.
b. Identify the key points on the unit circle.
c. Convert between degrees and radians.
d. Find the values of the six trigonometric functions of an angle in standard position given
a point on its terminal side.
e. Identify the number of degrees or radians in a given number of rotations.
7. Solve problems using combinatorics and probability.
Properties
Domain: Cognitive
Level: Application
Difficulty: High
Importance: Essential
Linked External Standards
WI.MA.A.12.1 Use reason and logic5 Determine the likelihood of occurrence of complex events
Performance Standards
o learner applies the properties and formulas for counting to solve probability problems
with 75% accuracy.
Learning objectives
a. Solve problems related to the Basic Counting Principle.
b. Distinguish between dependent and independent events.
c. Use combinations to compute the number of ways of selecting objects.
d. Solve problems involving permutations and/or combinations.
e. Solve problems involving permutations with repetitions.
f. Solve problems involving circular permutations.
g. Find the probability of an event.
h. Find the odds for an event.
i. Find the probability of independent and dependent events.
j. Identify mutually exclusive events.
k. Find the probability of mutually exclusive and inclusive events.
l. Find the probability of an event given the occurrence of another event.
8. Calculate or approximate zeros of polynomial functions.
Properties
Domain: Cognitive
Level: Application
Difficulty: Medium
Importance: Essential uses the techniques of solving polynomials to approximate all the zeroes of a
function with 75 % accuracy.
Learning objectives
a. Determine roots of polynomial equations.
b. Apply the Fundamental Theorem of Algebra.
c. Solve quadratic equations by factoring and using the Quadratic Formula.
d. Describe the roots of a quadratic function using the discriminant.
e. Find factors using the Remainder and Factor Theorems.
f. Apply the Rational Root Theorem to identify all possible rational roots.
g. Determine the number of positive and negative real roots of a function.
h. Approximate the real zeros of a polynomial function.
i. Solve rational equations and inequalities.
j. Decompose a fraction into partial fractions.
k. Solve radical equations and inequalities.
l. Model real-world data using polynomial functions.
m. Interpret real-world data using polynomial functions.
9. Examine the conic sections.
Properties
Domain: Cognitive
Level: Analysis
Difficulty: High
Importance: Important
Linked External Standards
WI.MA.A.12.1 Use reason and logic coordinate identifies a conic section from its graph or equation with 75% accuracy.
o learner writes an equation to represent a graph of a conic section with 75% accuracy.
o learner solves a system of second degree equations with 75% accuracy.
Learning objectives
a. State the properties of each of the conic sections.
b. Use properties of conic sections (circles, ellipses, hyperbolas, and parabolas) to write
equations describing them.
c. Rewrite the general equations of conic sections in standard form.
d. Identify conic sections by their equations in standard form.
e. Identify conic sections by their equations in general form.
f. Graph the conic sections.
g. Graph systems of second degree equations and inequalities.
h. Solve systems of second degree equations and inequalities.
10. Study the polar coordinate system.
Properties
Domain: Cognitive
Level: Analysis
Difficulty: Low
Importance: Essential
Performance Standards
o learner converts between the rectangular coordinate system and the polar coordinate
system by using the conversion formulas with 75% accuracy.
o learner graphs in the polar coordinate system with 75% accuracy.
Learning objectives
a. Graph points in polar coordinates.
b. Determine the distance between two points with polar coordinates.
c. Graph polar equations.
d. Convert between polar and rectangular coordinates.
e. Write the polar from of a linear equation.
f. Graph the polar form of a linear equation.
11. Calculate powers and roots of complex numbers.
Properties
Domain: Cognitive
Level: Application
Difficulty: Medium
Importance: Useful
Performance Standards
o learner finds powers and roots of complex numbers with 75% accuracy.
Learning objectives
a. Add, subtract, multiply and divide complex numbers.
b. Graph complex numbers.
c. Represent complex numbers in different forms.
d. Explore geometric relationships in the complex plane.
e. Find the product and quotient of complex numbers in polar form.
f. Use De Moivre's Theorem to find powers and roots of complex numbers in polar form.
12. Perform operations with and solve problems using vectors.
Linked External Standards
WI.MA.D.12.2 Select and use tools with appropriate degree of precision to determine
measurements directly within specified degrees of accuracy and error (tolerance)
WI.MA.D.12.3 Determine measurements indirectly
Performance Standards
o learner performs operations and solves problems using vectors with 75% accuracy.
Learning objectives
a. Find equal, opposite, and parallel vectors.
b. Add and subtract vectors geometrically.
c. Find ordered pairs that represent vectors.
d. Add, subtract, multiply, and find the magnitude of vectors algebraically.
e. Add, subtract,and find the magnitude of the vectors in three-dimensional space.
f. Find the inner and cross products of two vectors.
g. Determine whether two vectors are perpendicular.
h. Solve problems using vectors and trigonometry
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Houston Preal complex numbers, and radical and rational functions. botany, zoology, anatomy, physi...
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Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more
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The Algebra 1: The Complete Course DVD Series will help students build confidence in their ability to understand and solve algebraic problems.
In this episode, concrete examples and practical applications show how the mastery of fundamental algebraic concepts is the key to success in today's technologically advanced world. Students will also learn the development of algebraic symbolism as well as the geometric and numeric currents. Grades 5-9. 30 minutes on DVD.
Customer Reviews for Algebra 1 - The Complete Course: An Overview DVD
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About this course
This distance learning course provides the information you will need to prepare for the AQA A-Level in Maths with Statistics. In this home study course, you will focus on four core topics of algebra, geometry, trigonometry and calculus, which make up two-thirds of the A-Level qualification. The remaining third is focused on the study of statistics, including estimation, probability and distributions. The course is optimized for students studying at home and includes full tutor support via email.
A-Level Maths with Statistics is a valuable complement to other A-Level courses with a statistical element, such as biology, sociology or psychology, and for those wishing to study these subjects at a higher level. A-Level Maths with Statistics is also applicable to many jobs and careers and is a well-respected qualification that can be used for career progression and further training whilst in employment.
Entry requirements
English reading and writing skills, and maths to at least GCSE grade C or equivalent are required. You will need to have general skills and knowledge base associated with a GCSE course or equivalent standard.
This specification is designed to:
develop the student's understanding of mathematics and mathematical processes in a way that promotes confidence and fosters enjoyment
develop abilities to reason logically and to recognise incorrect reasoning, to generalise and to construct mathematical proofs
extend their range of mathematical skills and techniques and use them in more difficult unstructured problems
use mathematics as an effective means of communication
acquire the skills needed to use technology such as calculators and computers effectively, to recognise when such use may be inappropriate and to be aware of limitations
develop an awareness of the relevance of mathematics to other fields of study, to the world of work and to society in general
On this course you will study six units:
AS Level
Unit 1 MPC1 Core 1
Unit 2 MPC2 Core 2
Unit 3 MS1B Statistics 1B
A2 Level
Unit 4 MPC3 Core 3
Unit 5 MPC4 Core 4
Unit 6 MS2B Statistics 2
Each unit has 1 written paper of 1 hour 30 minutes.
Course Content
AS Level
Unit 1 MPC1 Core 1
Co-ordinate Geometry
Quadratic functions
Differentiation
Integration
Unit 2 MPC2 Core 2
Algebra and Functions
Sequences and Series
Trigonometry
Exponentials and logarithms
Differentiation
Integration
Unit 3 MS1B Statistics 1B
Statistical Measures
Probability
Discrete Random Variables
Normal Distribution
Estimation
A2 Level
Unit 4 MPC3 Core 3
Algebra and Functions
Trigonometry
Exponentials and Logarithms
Differentiation
Integration
Numerical Methods
Unit 5 MPC4 Core 4
Algebra and Functions
Coordinate Geometry in the (x, y) plane
Sequences and Series
Trigonometry
Exponentials and Logarithms
Differentiation and Integration
Vectors
Unit 6 MS2B Statistics 2
Poisson distribution
Continuous random variables
The t-distribution
Hypothesis Testing
Chi-squared tests
AS +A2 = A Level in Maths with Statistics. Both AS and A2 level courses and examinations must be successfully completed to gain a full A Level.
AQA Specification 6360
The course comes to you as a paper-based packExams are taken at an AQA centre and we can provide an extensive list of centres for you. Please read our FAQs for further information
Our A Levels come with tutor support for 24 months.
You will have access to a tutor, via email, who will mark your work and guide you through the course to help you be ready for your examinations. In addition you will be supplied with a comprehensive Study Guide which will help you through the study and assessment process.
Through UK Distance Learning & Publishing, the fascinating subject of A-Level Biology is now available to distance learning students for study at home using our online learning platform. online learningThis course can be taken at anytime and anywhere using our online learning platform.
Through UK Distance Learning & Publishing, the fascinating subject of A-Level Biology is now available to distance learning students for study at home. Closely home study
Psychology is an engaging and challenging subject that offers fascinating insights into the workings of the human mind. With this distance learning online course in A-Level Psychology, you can study at home with UK Distance Learning & Publishing, confident that our experienced tutors are always on hand to offer you guidance and support.In this online home study A-Level Psychology course, you will study a broad range of topics. These include important contemporary issues such as memory, attachment, stress, sleep and celebrity. Students will also study psychopathology, specialising in schizophrenia.Through studying psychology, we can learn more about how we think, whilst simultaneously developing our own powers of critical and evaluative thought. This well-respected A-Level subject is ideal for those who aspire to study psychology at university or to work in a related field. It is also suitable for those who simply want to further their own knowledge of this demanding discipline
Following the popular AQA English Literature B specification, this home study course offers students the chance to study a variety of texts, including three of their own choice. This distance learning course aims to engage students in the study of literature, encouraging a multi-layered approach to the reading of literary texts. Focusing on the central place of narrative in the construction of texts, encouraging critical debate and fostering a recognition of the role of genre, this comprehensive home study course guides students through the study of A-Level English Literature, including full tutor support via email.
An understanding and appreciation of literature can bring an enhanced enjoyment of reading and a lifelong love of language to all students. This distance learning course is suitable for students wishing to increase their general knowledge of English literature or for those seeking to progress to the study English literature or related subjects at a higher level.
This online distance learning course in A-Level Religious Studies approaches the subject of religion as an academic discipline, allowing students to develop their own views and values through a critical, reflective and evaluative approach. The course is optimized for study at home and tpotally online, according to the AQA A-Level specification. The units studied include: Introduction to Religious Studies; Studies in Religion; and Religion and Human Experience. Particular attention is given to the topics of World Religions: Buddhism and to the Christian New Testament. This distance learning course is suitable for those who want to learn more about religious studies for their own interest or for those who wish to study for an A-Level qualification as preparation for study at a higher level. A-Level Religious Studies is a particularly good complement for Philosophy or History A-Levels and is suitable for those who intend to study Religious Studies or Theology at university level. The course is suitable for students or all religions or none and no prior knowledge of religious studies is required.
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ISBN: 0321448049 / ISBN-13: 9780321448040
Mathematics for Elementary School Teachers
Future elementary and middle school teachers need a clear, coherent presentation of the mathematical concepts, procedures, and processes they will be ...Show synopsisFuture elementary and middle school teachers need a clear, coherent presentation of the mathematical concepts, procedures, and processes they will be called upon to teach. This text uniquely balances "what" they will teach (concepts and content) with "how" to teach (processes and communication). As a result, students using "Mathematics for Elementary School Teachers"Hide synopsis
...Show more To see available supplements that will enliven your course with activities, classroom videos, and professional development for future teachers, visit ... Please read description before purchase >>> instructor...Good. Please read description before purchase >>> instructor annotated version printed on cover with all identical Students content with teaching tips, and all solutions text only no access code. Used has highlighting throughout book good binding Quicker shipper with tracking #
Description:Like New. U.S. Students hardcover edition in plastic wrap text...Like New. U.S. Students hardcover edition in plastic wrap text only no supplements >light scratches on cover otherwise New
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Provides an examination of the theory of super-recursive algorithms and its ramifications for the computer industry, networks, artificial intelligence, embedded systems, and the Internet. This book demonstrates how these algorithms are more appropriate as mathematical models for modern computers. more...
Pyramid Algorithms presents a unique approach to understanding, analyzing, and computing the most common polynomial and spline curve and surface schemes used in computer-aided geometric design, employing a dynamic programming method based on recursive pyramids. The recursive pyramid approach offers the distinct advantage of revealing the entire... more...
This work is a self-contained elementary exposition of the theory of recursive functionals, that also includes a number of advanced results. Although aiming basically at a theory of higher order computability, attention is restricted to second order functionals, where the arguments are numerical functions and the values, when defined, are natural numbers.... more...
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Concise Introduction to MATLAB
9780073385839
ISBN:
0073385832
Pub Date: 2007 Publisher: McGraw-Hill Companies, The
Summary: A Concise Introduction to Matlab is a simple, concise book designed to cover all the major capabilities of MATLAB that are useful for beginning students. Thorough coverage of Function handles, Anonymous functions, and Subfunctions. In addition, key applications including plotting, programming, statistics and model building are also all covered. MATLAB is presently a globally available standard computational tool for ...engineers and scientists. The terminology, syntax, and the use of the programming language are well defined and the organization of the material makes it easy to locate information and navigate through the textbook.
Palm, William J., III is the author of Concise Introduction to MATLAB, published 2007 under ISBN 9780073385839 and 0073385832. Five hundred fifty four Concise Introduction to MATLAB textbooks are available for sale on ValoreBooks.com, one hundred seventeen used from the cheapest price of $21.93, or buy new starting at $63.96.[read more]Writing programs to do tedious computations quickly and making a graphical representation of that data. One example is the differential equation solver. Very versatile built in functions to do that task.
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{"currencyCode":"USD","itemData":[{"priceBreaksMAP":null,"buyingPrice":57.47,"ASIN":"0521575400","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":54.28,"ASIN":"0521347939","isPreorder":0}],"shippingId":"0521575400::UOAb4jeOgm62AS5HZ1Tl08R32OvSi74Ft%2B6x3UCbYSnMChW0sisajC9%2BWg%2B8GwievH%2B6bks9FjD33SYUseWDwohKrRXUWIFpv%2FAaKiNMgFw%3D,0521347939::Knapk7ufFWs7bFAYhnhSJuFWOMTlzZvKB%2FxJMSGjxzcS%2FCh8zgH3Z0yBkYJOtgsTEfUmWVJU8hG12UmOZaUoMBpYJf5XY3KehqkTcehB'm pleased to report there is a new edition of R. Burn's A Pathway into Number Theory, a book that takes readers quickly and painlessly from simple facts about whole numbers to the wonders of the quadratic forms, Pell's equation and Minkowski's theorem.' Ian Stewart, New Scientist
'... admirably suitable for those meeting number theory for the first time and for unsupported individual study.' Nick Lord, The Mathematical Gazette
Book Description
Now in its second edition, this book consists of a sequence of exercises that will lead readers from simple number work to the point where they can prove algebraically the classical results of elementary number theory for themselves. A high-school course in mathematics is the only requirement.
This book is a carefully sequenced set of problems along with answers and a few comments. Burn uses those problems to introduce important number theory ideas. I enjoyed working through the problems to learn more about number theory. Most problems are accessible to those with a good high school mathematics background.
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LoviOtvet Calculator
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Universal Calculator or solver - as any name, and unique to the application is not found.
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working through this Study Guide you will definitely improve your results - whether you are working towards being the top performer in your class or whether you regularly break out in a sweat when you have to present your test scores or school report at home! This marvellous resource provides you with: • Introductions to and discussions of the various themes and topics relevant to Grade 10 Mathematics • Fully worked-out examples with their answers • Loads of exercises and questions to practise your newly gained skills • Answers to these exercises at the back of the book • Exemplar examination papers for you to work through and their answers This Study & Master Guide is written according to the NCS for Mathematics.
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Overview
Editorial Reviews
Booknews
Bittinger (mathematics, Indiana U. and Purdue U.) uses a five step problem solving approach with real data applications to make algebra both straightforward and connected to everyday life. Detailed graphs and color drawings and photographs also help students to visualize mathematical concepts. The book is designed to assist in every step of curriculum, from review exercises with answers, to pre and post-tests. There are also a number of supplemental materials available for the instructor to use in conjunction with this text. Annotation c. Book News, Inc., Portland, OR (booknews.com)
Systems of Equations in Two Variables. Solving by Substitution or Elimination. Solving by Applications: Systems of Two Equations. Systems of Equations inThree Variables. Solving Applications: Systems of Three Equations. Systems of Linear Inequalities in Two Variables. Business and Economic
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More About
This Textbook
Overview
Although extensively revised, this new edition continues in the fine tradition of its predecessor. Major changes include: a notation that formalizes the distinction between equality and congruence and between line, ray and line segment; a completely rewritten chapter on mathematical logic with inclusion of truth tables and the logical basis for the discovery of non-Euclidean geometries; expanded coverage of analytic geometry with more theorems discussed and proved with coordinate geometry; two distinct chapters on parallel lines and parallelograms; a condensed chapter on numerical trigonometry; more problems; expansion of the section on surface areas and volume; and additional review exercises at the end of each chapter. Concise and logical, it will serve as an excellent review of high school geometry
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PDF Geometry Curriculum Map Table Of Contents Unit 1 Basic build on rigid motions as a familiar starting point for development of concept of geometric proof use the definition of congruence in terms of rigid motions to PDF Geometry Curriculum Map Table Of Contents Unit 1 Basic
PDF The role of abduction in realizing geometric invariants the role of abduction in realizing geometric invariants nam nguyen-danh1 1 didactics of mathematics university of w 252 rzburg am hubland 97074 w 252 rzburg germany PDF The role of abduction in realizing geometric invariants
PDF Translation of terms used on the statewide mathematics translation of terms used on the statewide mathematics assessments spanish over english each year as oregon presents more mathematics items in PDF Translation of terms used on the statewide mathematics
PDF Transformations in the Coordinate Plane - Gaston County 518 chapter 10 geometric figures computer graphicscomputer and video game programmers use transformations to create patterns and animations how can you use PDF Transformations in the Coordinate Plane - Gaston County
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MATH TREK Algebra 1
04/01/04
For curriculum-based algebra instruction, teachers and students can use MATH TREK Algebra 1. The multimedia program includes tutorials, assessments and student tracking. Students can use the program's scientific calculator, glossary and journal to help them complete the various exercises and activities. The assessment and student-tracking features provide immediate feedback to students so that they can stay on top of their progress. This engaging program, complete with sound, animation and graphics, can be used on stand-alone computers or a network. NECTAR Foundation, (613) 224-3031
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