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This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely some geometric objects and operations can be expressed in terms of complex numbers. For example, the real numbers form the real line, which is pictured as the horizontal axis of the complex plane, while real multiples of
$$
i
$$
are the vertical axis. A complex number can also be defined by its geometric polar coordinates: the radius is called the absolute value of the complex number, while the angle from the positive real axis is called the argument of the complex number. The complex numbers of absolute value one form the unit circle. Adding a fixed complex number to all complex numbers defines a translation in the complex plane, and multiplying by a fixed complex number is a similarity centered at the origin (dilating by the absolute value, and rotating by the argument). The operation of complex conjugation is the reflection symmetry with respect to the real axis.
The complex numbers form a rich structure that is simultaneously an algebraically closed field, a commutative algebra over the reals, and a Euclidean vector space of dimension two.
## Definition and basic operations
A complex number is an expression of the form , where and are real numbers, and is an abstract symbol, the so-called imaginary unit, whose meaning will be explained further below. For example, is a complex number.
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## Definition and basic operations
A complex number is an expression of the form , where and are real numbers, and is an abstract symbol, the so-called imaginary unit, whose meaning will be explained further below. For example, is a complex number.
For a complex number , the real number is called its real part, and the real number (not the complex number ) is its imaginary part. The real part of a complex number is denoted ,
$$
\mathcal{Re}(z)
$$
, or
$$
\mathfrak{R}(z)
$$
; the imaginary part is ,
$$
\mathcal{Im}(z)
$$
, or
$$
\mathfrak{I}(z)
$$
: for example,
$$
\operatorname{Re}(2 + 3i) = 2
$$
,
$$
\operatorname{Im}(2 + 3i) = 3
$$
.
A complex number can be identified with the ordered pair of real numbers
$$
(\Re (z),\Im (z))
$$
, which may be interpreted as coordinates of a point in a Euclidean plane with standard coordinates, which is then called the complex plane or Argand diagram. The horizontal axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical axis, with increasing values upwards.
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A complex number can be identified with the ordered pair of real numbers
$$
(\Re (z),\Im (z))
$$
, which may be interpreted as coordinates of a point in a Euclidean plane with standard coordinates, which is then called the complex plane or Argand diagram. The horizontal axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical axis, with increasing values upwards.
A real number can be regarded as a complex number , whose imaginary part is 0. A purely imaginary number is a complex number , whose real part is zero. It is common to write , , and ; for example, .
The set of all complex numbers is denoted by
$$
\Complex
$$
(blackboard bold) or (upright bold).
In some disciplines such as electromagnetism and electrical engineering, is used instead of , as frequently represents electric current, and complex numbers are written as or .
### Addition and subtraction
Two complex numbers
$$
a =x+yi
$$
and
$$
b =u+vi
$$
are added by separately adding their real and imaginary parts.
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In some disciplines such as electromagnetism and electrical engineering, is used instead of , as frequently represents electric current, and complex numbers are written as or .
### Addition and subtraction
Two complex numbers
$$
a =x+yi
$$
and
$$
b =u+vi
$$
are added by separately adding their real and imaginary parts. That is to say:
$$
a + b =(x+yi) + (u+vi) = (x+u) + (y+v)i.
$$
Similarly, subtraction can be performed as
$$
a - b =(x+yi) - (u+vi) = (x-u) + (y-v)i.
$$
The addition can be geometrically visualized as follows: the sum of two complex numbers and , interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices , and the points of the arrows labeled and (provided that they are not on a line). Equivalently, calling these points , , respectively and the fourth point of the parallelogram the triangles and are congruent.
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That is to say:
$$
a + b =(x+yi) + (u+vi) = (x+u) + (y+v)i.
$$
Similarly, subtraction can be performed as
$$
a - b =(x+yi) - (u+vi) = (x-u) + (y-v)i.
$$
The addition can be geometrically visualized as follows: the sum of two complex numbers and , interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices , and the points of the arrows labeled and (provided that they are not on a line). Equivalently, calling these points , , respectively and the fourth point of the parallelogram the triangles and are congruent.
### Multiplication
The product of two complex numbers is computed as follows:
$$
(a+bi) \cdot (c+di) = ac - bd + (ad+bc)i.
$$
For example,
$$
(3+2i)(4-i) = 3 \cdot 4 - (2 \cdot (-1)) + (3 \cdot (-1) + 2 \cdot 4)i = 14 +5i.
$$
In particular, this includes as a special case the fundamental formula
$$
i^2 = i \cdot i = -1.
$$
This formula distinguishes the complex number i from any real number, since the square of any (negative or positive) real number is always a non-negative real number.
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Equivalently, calling these points , , respectively and the fourth point of the parallelogram the triangles and are congruent.
### Multiplication
The product of two complex numbers is computed as follows:
$$
(a+bi) \cdot (c+di) = ac - bd + (ad+bc)i.
$$
For example,
$$
(3+2i)(4-i) = 3 \cdot 4 - (2 \cdot (-1)) + (3 \cdot (-1) + 2 \cdot 4)i = 14 +5i.
$$
In particular, this includes as a special case the fundamental formula
$$
i^2 = i \cdot i = -1.
$$
This formula distinguishes the complex number i from any real number, since the square of any (negative or positive) real number is always a non-negative real number.
With this definition of multiplication and addition, familiar rules for the arithmetic of rational or real numbers continue to hold for complex numbers. More precisely, the distributive property, the commutative properties (of addition and multiplication) hold. Therefore, the complex numbers form an algebraic structure known as a field, the same way as the rational or real numbers do.
### Complex conjugate, absolute value, argument and division
The complex conjugate of the complex number is defined as
$$
\overline z = x-yi.
$$
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Therefore, the complex numbers form an algebraic structure known as a field, the same way as the rational or real numbers do.
### Complex conjugate, absolute value, argument and division
The complex conjugate of the complex number is defined as
$$
\overline z = x-yi.
$$
It is also denoted by some authors by
$$
z^*
$$
. Geometrically, is the "reflection" of about the real axis. Conjugating twice gives the original complex number:
$$
\overline{\overline{z}}=z.
$$
A complex number is real if and only if it equals its own conjugate. The unary operation of taking the complex conjugate of a complex number cannot be expressed by applying only the basic operations of addition, subtraction, multiplication and division.
For any complex number , the product
$$
z \cdot \overline z = (x+iy)(x-iy) = x^2 + y^2
$$
is a non-negative real number. This allows to define the absolute value (or modulus or magnitude) of z to be the square root
$$
|z|=\sqrt{x^2+y^2}.
$$
By Pythagoras' theorem,
$$
|z|
$$
is the distance from the origin to the point representing the complex number z in the complex plane.
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For any complex number , the product
$$
z \cdot \overline z = (x+iy)(x-iy) = x^2 + y^2
$$
is a non-negative real number. This allows to define the absolute value (or modulus or magnitude) of z to be the square root
$$
|z|=\sqrt{x^2+y^2}.
$$
By Pythagoras' theorem,
$$
|z|
$$
is the distance from the origin to the point representing the complex number z in the complex plane. In particular, the circle of radius one around the origin consists precisely of the numbers z such that
$$
|z| = 1
$$
. If
$$
z = x = x + 0i
$$
is a real number, then
$$
|z|= |x|
$$
: its absolute value as a complex number and as a real number are equal.
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In particular, the circle of radius one around the origin consists precisely of the numbers z such that
$$
|z| = 1
$$
. If
$$
z = x = x + 0i
$$
is a real number, then
$$
|z|= |x|
$$
: its absolute value as a complex number and as a real number are equal.
Using the conjugate, the reciprocal of a nonzero complex number
$$
z = x + yi
$$
can be computed to be
$$
\frac{1}{z}
= \frac{\bar{z}}{z\bar{z}}
= \frac{\bar{z}}{|z|^2}
= \frac{x - yi}{x^2 + y^2}
= \frac{x}{x^2 + y^2} - \frac{y}{x^2 + y^2}i.
$$
More generally, the division of an arbitrary complex number
$$
w = u + vi
$$
by a non-zero complex number
$$
z = x + yi
$$
equals
$$
\frac{w}{z}
= \frac{w\bar{z}}{|z|^2}
= \frac{(u + vi)(x - iy)}{x^2 + y^2}
= \frac{ux + vy}{x^2 + y^2} + \frac{vx - uy}{x^2 + y^2}i.
$$
This process is sometimes called "rationalization" of the denominator (although the denominator in the final expression may be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.
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If
$$
z = x = x + 0i
$$
is a real number, then
$$
|z|= |x|
$$
: its absolute value as a complex number and as a real number are equal.
Using the conjugate, the reciprocal of a nonzero complex number
$$
z = x + yi
$$
can be computed to be
$$
\frac{1}{z}
= \frac{\bar{z}}{z\bar{z}}
= \frac{\bar{z}}{|z|^2}
= \frac{x - yi}{x^2 + y^2}
= \frac{x}{x^2 + y^2} - \frac{y}{x^2 + y^2}i.
$$
More generally, the division of an arbitrary complex number
$$
w = u + vi
$$
by a non-zero complex number
$$
z = x + yi
$$
equals
$$
\frac{w}{z}
= \frac{w\bar{z}}{|z|^2}
= \frac{(u + vi)(x - iy)}{x^2 + y^2}
= \frac{ux + vy}{x^2 + y^2} + \frac{vx - uy}{x^2 + y^2}i.
$$
This process is sometimes called "rationalization" of the denominator (although the denominator in the final expression may be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator. Extract of page 37
The argument of (sometimes called the "phase" ) is the angle of the radius with the positive real axis, and is written as , expressed in radians in this article.
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Using the conjugate, the reciprocal of a nonzero complex number
$$
z = x + yi
$$
can be computed to be
$$
\frac{1}{z}
= \frac{\bar{z}}{z\bar{z}}
= \frac{\bar{z}}{|z|^2}
= \frac{x - yi}{x^2 + y^2}
= \frac{x}{x^2 + y^2} - \frac{y}{x^2 + y^2}i.
$$
More generally, the division of an arbitrary complex number
$$
w = u + vi
$$
by a non-zero complex number
$$
z = x + yi
$$
equals
$$
\frac{w}{z}
= \frac{w\bar{z}}{|z|^2}
= \frac{(u + vi)(x - iy)}{x^2 + y^2}
= \frac{ux + vy}{x^2 + y^2} + \frac{vx - uy}{x^2 + y^2}i.
$$
This process is sometimes called "rationalization" of the denominator (although the denominator in the final expression may be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator. Extract of page 37
The argument of (sometimes called the "phase" ) is the angle of the radius with the positive real axis, and is written as , expressed in radians in this article. The angle is defined only up to adding integer multiples of
$$
2\pi
$$
, since a rotation by
$$
2\pi
$$
(or 360°) around the origin leaves all points in the complex plane unchanged.
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Extract of page 37
The argument of (sometimes called the "phase" ) is the angle of the radius with the positive real axis, and is written as , expressed in radians in this article. The angle is defined only up to adding integer multiples of
$$
2\pi
$$
, since a rotation by
$$
2\pi
$$
(or 360°) around the origin leaves all points in the complex plane unchanged. One possible choice to uniquely specify the argument is to require it to be within the interval
$$
(-\pi,\pi]
$$
, which is referred to as the principal value.
The argument can be computed from the rectangular form by means of the arctan (inverse tangent) function.
### Polar form
For any complex number z, with absolute value
$$
r = |z|
$$
and argument
$$
\varphi
$$
, the equation
$$
z=r(\cos\varphi +i\sin\varphi)
$$
holds. This identity is referred to as the polar form of z. It is sometimes abbreviated as
$$
z = r \operatorname\mathrm{cis} \varphi
$$
.
In electronics, one represents a phasor with amplitude and phase in angle notation:
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It is sometimes abbreviated as
$$
z = r \operatorname\mathrm{cis} \varphi
$$
.
In electronics, one represents a phasor with amplitude and phase in angle notation:
If two complex numbers are given in polar form, i.e., and , the product and division can be computed as
$$
z_1 z_2 = r_1 r_2 (\cos(\varphi_1 + \varphi_2) + i \sin(\varphi_1 + \varphi_2)).
$$
$$
\frac{z_1}{z_2} = \frac{r_1}{r_2} \left(\cos(\varphi_1 - \varphi_2) + i \sin(\varphi_1 - \varphi_2)\right), \text{if }z_2 \ne 0.
$$
(These are a consequence of the trigonometric identities for the sine and cosine function.)
In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. The picture at the right illustrates the multiplication of
$$
(2+i)(3+i)=5+5i.
$$
Because the real and imaginary part of are equal, the argument of that number is 45 degrees, or (in radian).
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In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. The picture at the right illustrates the multiplication of
$$
(2+i)(3+i)=5+5i.
$$
Because the real and imaginary part of are equal, the argument of that number is 45 degrees, or (in radian). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan(1/3) and arctan(1/2), respectively. Thus, the formula
$$
\frac{\pi}{4} = \arctan\left(\frac{1}{2}\right) + \arctan\left(\frac{1}{3}\right)
$$
holds. As the arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of :
$$
\frac{\pi}{4} = 4 \arctan\left(\frac{1}{5}\right) - \arctan\left(\frac{1}{239}\right)
$$
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Thus, the formula
$$
\frac{\pi}{4} = \arctan\left(\frac{1}{2}\right) + \arctan\left(\frac{1}{3}\right)
$$
holds. As the arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of :
$$
\frac{\pi}{4} = 4 \arctan\left(\frac{1}{5}\right) - \arctan\left(\frac{1}{239}\right)
$$
### Powers and roots
The n-th power of a complex number can be computed using de Moivre's formula, which is obtained by repeatedly applying the above formula for the product:
$$
z^{n}=\underbrace{z \cdot \dots \cdot z}_{n \text{ factors}} = (r(\cos \varphi + i\sin \varphi ))^n = r^n \, (\cos n\varphi + i \sin n \varphi).
$$
For example, the first few powers of the imaginary unit i are
$$
i, i^2 = -1, i^3 = -i, i^4 = 1, i^5 = i, \dots
$$
.
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As the arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of :
$$
\frac{\pi}{4} = 4 \arctan\left(\frac{1}{5}\right) - \arctan\left(\frac{1}{239}\right)
$$
### Powers and roots
The n-th power of a complex number can be computed using de Moivre's formula, which is obtained by repeatedly applying the above formula for the product:
$$
z^{n}=\underbrace{z \cdot \dots \cdot z}_{n \text{ factors}} = (r(\cos \varphi + i\sin \varphi ))^n = r^n \, (\cos n\varphi + i \sin n \varphi).
$$
For example, the first few powers of the imaginary unit i are
$$
i, i^2 = -1, i^3 = -i, i^4 = 1, i^5 = i, \dots
$$
.
The th roots of a complex number are given by
$$
z^{1/n} = \sqrt[n]r \left( \cos \left(\frac{\varphi+2k\pi}{n}\right) + i \sin \left(\frac{\varphi+2k\pi}{n}\right)\right)
$$
for .
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### Powers and roots
The n-th power of a complex number can be computed using de Moivre's formula, which is obtained by repeatedly applying the above formula for the product:
$$
z^{n}=\underbrace{z \cdot \dots \cdot z}_{n \text{ factors}} = (r(\cos \varphi + i\sin \varphi ))^n = r^n \, (\cos n\varphi + i \sin n \varphi).
$$
For example, the first few powers of the imaginary unit i are
$$
i, i^2 = -1, i^3 = -i, i^4 = 1, i^5 = i, \dots
$$
.
The th roots of a complex number are given by
$$
z^{1/n} = \sqrt[n]r \left( \cos \left(\frac{\varphi+2k\pi}{n}\right) + i \sin \left(\frac{\varphi+2k\pi}{n}\right)\right)
$$
for . (Here
$$
\sqrt[n]r
$$
is the usual (positive) th root of the positive real number .) Because sine and cosine are periodic, other integer values of do not give other values.
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(Here
$$
\sqrt[n]r
$$
is the usual (positive) th root of the positive real number .) Because sine and cosine are periodic, other integer values of do not give other values. For any
$$
z \ne 0
$$
, there are, in particular n distinct complex n-th roots. For example, there are 4 fourth roots of 1, namely
$$
z_1 = 1, z_2 = i, z_3 = -1, z_4 = -i.
$$
In general there is no natural way of distinguishing one particular complex th root of a complex number. (This is in contrast to the roots of a positive real number x, which has a unique positive real n-th root, which is therefore commonly referred to as the n-th root of x.) One refers to this situation by saying that the th root is a -valued function of .
### Fundamental theorem of algebra
The fundamental theorem of algebra, of Carl Friedrich Gauss and Jean le Rond d'Alembert, states that for any complex numbers (called coefficients) , the equation
$$
a_n z^n + \dotsb + a_1 z + a_0 = 0
$$
has at least one complex solution z, provided that at least one of the higher coefficients is nonzero.
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One refers to this situation by saying that the th root is a -valued function of .
### Fundamental theorem of algebra
The fundamental theorem of algebra, of Carl Friedrich Gauss and Jean le Rond d'Alembert, states that for any complex numbers (called coefficients) , the equation
$$
a_n z^n + \dotsb + a_1 z + a_0 = 0
$$
has at least one complex solution z, provided that at least one of the higher coefficients is nonzero. This property does not hold for the field of rational numbers
$$
\Q
$$
(the polynomial does not have a rational root, because is not a rational number) nor the real numbers
$$
\R
$$
(the polynomial does not have a real root, because the square of is positive for any real number ).
Because of this fact,
$$
\Complex
$$
is called an algebraically closed field. It is a cornerstone of various applications of complex numbers, as is detailed further below.
There are various proofs of this theorem, by either analytic methods such as Liouville's theorem, or topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of odd degree has at least one real root.
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It is a cornerstone of various applications of complex numbers, as is detailed further below.
There are various proofs of this theorem, by either analytic methods such as Liouville's theorem, or topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of odd degree has at least one real root.
## History
The solution in radicals (without trigonometric functions) of a general cubic equation, when all three of its roots are real numbers, contains the square roots of negative numbers, a situation that cannot be rectified by factoring aided by the rational root test, if the cubic is irreducible; this is the so-called casus irreducibilis ("irreducible case"). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545 in his Ars Magna, though his understanding was rudimentary; moreover, he later described complex numbers as being "as subtle as they are useless". Cardano did use imaginary numbers, but described using them as "mental torture." This was prior to the use of the graphical complex plane. Cardano and other Italian mathematicians, notably Scipione del Ferro, in the 1500s created an algorithm for solving cubic equations which generally had one real solution and two solutions containing an imaginary number.
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This was prior to the use of the graphical complex plane. Cardano and other Italian mathematicians, notably Scipione del Ferro, in the 1500s created an algorithm for solving cubic equations which generally had one real solution and two solutions containing an imaginary number. Because they ignored the answers with the imaginary numbers, Cardano found them useless.
Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.
Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician Rafael Bombelli. A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.
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The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician Rafael Bombelli. A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.
The earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of the Greek mathematician Hero of Alexandria in the 1st century AD, where in his Stereometrica he considered, apparently in error, the volume of an impossible frustum of a pyramid to arrive at the term
$$
\sqrt{81 - 144}
$$
in his calculations, which today would simplify to
$$
\sqrt{-63} = 3i\sqrt{7}
$$
. Negative quantities were not conceived of in Hellenistic mathematics and Hero merely replaced the negative value by its positive
$$
\sqrt{144 - 81} = 3\sqrt{7}.
$$
The impetus to study complex numbers as a topic in itself first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (Niccolò Fontana Tartaglia and Gerolamo Cardano).
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The earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of the Greek mathematician Hero of Alexandria in the 1st century AD, where in his Stereometrica he considered, apparently in error, the volume of an impossible frustum of a pyramid to arrive at the term
$$
\sqrt{81 - 144}
$$
in his calculations, which today would simplify to
$$
\sqrt{-63} = 3i\sqrt{7}
$$
. Negative quantities were not conceived of in Hellenistic mathematics and Hero merely replaced the negative value by its positive
$$
\sqrt{144 - 81} = 3\sqrt{7}.
$$
The impetus to study complex numbers as a topic in itself first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (Niccolò Fontana Tartaglia and Gerolamo Cardano). It was soon realized (but proved much later) that these formulas, even if one were interested only in real solutions, sometimes required the manipulation of square roots of negative numbers. In fact, it was proved later that the use of complex numbers is unavoidable when all three roots are real and distinct.
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It was soon realized (but proved much later) that these formulas, even if one were interested only in real solutions, sometimes required the manipulation of square roots of negative numbers. In fact, it was proved later that the use of complex numbers is unavoidable when all three roots are real and distinct. However, the general formula can still be used in this case, with some care to deal with the ambiguity resulting from the existence of three cubic roots for nonzero complex numbers. Rafael Bombelli was the first to address explicitly these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic, trying to resolve these issues.
The term "imaginary" for these quantities was coined by René Descartes in 1637, who was at pains to stress their unreal nature:
A further source of confusion was that the equation
$$
\sqrt{-1}^2 = \sqrt{-1}\sqrt{-1} = -1
$$
seemed to be capriciously inconsistent with the algebraic identity
$$
\sqrt{a}\sqrt{b} = \sqrt{ab}
$$
, which is valid for non-negative real numbers and , and which was also used in complex number calculations with one of , positive and the other negative.
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The term "imaginary" for these quantities was coined by René Descartes in 1637, who was at pains to stress their unreal nature:
A further source of confusion was that the equation
$$
\sqrt{-1}^2 = \sqrt{-1}\sqrt{-1} = -1
$$
seemed to be capriciously inconsistent with the algebraic identity
$$
\sqrt{a}\sqrt{b} = \sqrt{ab}
$$
, which is valid for non-negative real numbers and , and which was also used in complex number calculations with one of , positive and the other negative. The incorrect use of this identity in the case when both and are negative, and the related identity
$$
\frac{1}{\sqrt{a}} = \sqrt{\frac{1}{a}}
$$
, even bedeviled Leonhard Euler. This difficulty eventually led to the convention of using the special symbol in place of
$$
\sqrt{-1}
$$
to guard against this mistake. Extract of page 32 Even so, Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.
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Extract of page 32 Even so, Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.
In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be re-expressed by the following de Moivre's formula:
$$
(\cos \theta + i\sin \theta)^{n} = \cos n \theta + i\sin n \theta.
$$
In 1748, Euler went further and obtained Euler's formula of complex analysis:
$$
e ^{i\theta } = \cos \theta + i\sin \theta
$$
by formally manipulating complex power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.
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In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be re-expressed by the following de Moivre's formula:
$$
(\cos \theta + i\sin \theta)^{n} = \cos n \theta + i\sin n \theta.
$$
In 1748, Euler went further and obtained Euler's formula of complex analysis:
$$
e ^{i\theta } = \cos \theta + i\sin \theta
$$
by formally manipulating complex power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.
The idea of a complex number as a point in the complex plane was first described by Danish–Norwegian mathematician Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's A Treatise of Algebra.
Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed.
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The idea of a complex number as a point in the complex plane was first described by Danish–Norwegian mathematician Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's A Treatise of Algebra.
Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra. Carl Friedrich Gauss had earlier published an essentially topological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1". It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology:
If one formerly contemplated this subject from a false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, −1, positive, negative, or imaginary (or even impossible) units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness.
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If one formerly contemplated this subject from a false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, −1, positive, negative, or imaginary (or even impossible) units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness.
In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée, Mourey, Warren, Français and his brother, Bellavitis.
The English mathematician G.H. Hardy remarked that Gauss was the first mathematician to use complex numbers in "a really confident and scientific way" although mathematicians such as Norwegian Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.
Augustin-Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complex analysis to a high state of completion, commencing around 1825 in Cauchy's case.
The common terms used in the theory are chiefly due to the founders.
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Augustin-Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complex analysis to a high state of completion, commencing around 1825 in Cauchy's case.
The common terms used in the theory are chiefly due to the founders. Argand called the direction factor, and
$$
r = \sqrt{a^2 + b^2}
$$
the modulus; Cauchy (1821) called the reduced form (l'expression réduite) and apparently introduced the term argument; Gauss used for
$$
\sqrt{-1}
$$
, introduced the term complex number for , and called the norm. The expression direction coefficient, often used for , is due to Hankel (1867), and absolute value, for modulus, is due to Weierstrass.
Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others. Important work (including a systematization) in complex multivariate calculus has been started at beginning of the 20th century. Important results have been achieved by Wilhelm Wirtinger in 1927.
## Abstract algebraic aspects
While the above low-level definitions, including the addition and multiplication, accurately describe the complex numbers, there are other, equivalent approaches that reveal the abstract algebraic structure of the complex numbers more immediately.
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## Abstract algebraic aspects
While the above low-level definitions, including the addition and multiplication, accurately describe the complex numbers, there are other, equivalent approaches that reveal the abstract algebraic structure of the complex numbers more immediately.
### Construction as a quotient field
One approach to
$$
\C
$$
is via polynomials, i.e., expressions of the form
$$
p(X) = a_nX^n+\dotsb+a_1X+a_0,
$$
where the coefficients are real numbers. The set of all such polynomials is denoted by
$$
\R[X]
$$
. Since sums and products of polynomials are again polynomials, this set
$$
\R[X]
$$
forms a commutative ring, called the polynomial ring (over the reals). To every such polynomial p, one may assign the complex number
$$
p(i) = a_n i^n + \dotsb + a_1 i + a_0
$$
, i.e., the value obtained by setting
$$
X = i
$$
.
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Since sums and products of polynomials are again polynomials, this set
$$
\R[X]
$$
forms a commutative ring, called the polynomial ring (over the reals). To every such polynomial p, one may assign the complex number
$$
p(i) = a_n i^n + \dotsb + a_1 i + a_0
$$
, i.e., the value obtained by setting
$$
X = i
$$
. This defines a function
$$
\R[X] \to \C
$$
This function is surjective since every complex number can be obtained in such a way: the evaluation of a linear polynomial
$$
a+bX
$$
at
$$
X = i
$$
is
$$
a+bi
$$
. However, the evaluation of polynomial
$$
X^2 + 1
$$
at i is 0, since
$$
i^2 + 1 = 0.
$$
This polynomial is irreducible, i.e., cannot be written as a product of two linear polynomials. Basic facts of abstract algebra then imply that the kernel of the above map is an ideal generated by this polynomial, and that the quotient by this ideal is a field, and that there is an isomorphism
$$
\R[X] / (X^2 + 1) \stackrel \cong \to \C
$$
between
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This polynomial is irreducible, i.e., cannot be written as a product of two linear polynomials. Basic facts of abstract algebra then imply that the kernel of the above map is an ideal generated by this polynomial, and that the quotient by this ideal is a field, and that there is an isomorphism
$$
\R[X] / (X^2 + 1) \stackrel \cong \to \C
$$
between the quotient ring and
$$
\C
$$
. Some authors take this as the definition of
$$
\C
$$
.
Accepting that
$$
\Complex
$$
is algebraically closed, because it is an algebraic extension of
$$
\mathbb{R}
$$
in this approach,
$$
\Complex
$$
is therefore the algebraic closure of
$$
\R.
$$
### Matrix representation of complex numbers
Complex numbers can also be represented by matrices that have the form
$$
\begin{pmatrix}
a & -b \\
b & \;\; a
\end{pmatrix}.
$$
Here the entries and are real numbers. As the sum and product of two such matrices is again of this form, these matrices form a subring of the ring of matrices.
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### Matrix representation of complex numbers
Complex numbers can also be represented by matrices that have the form
$$
\begin{pmatrix}
a & -b \\
b & \;\; a
\end{pmatrix}.
$$
Here the entries and are real numbers. As the sum and product of two such matrices is again of this form, these matrices form a subring of the ring of matrices.
A simple computation shows that the map
$$
a+ib\mapsto \begin{pmatrix}
a & -b \\
b & \;\; a
\end{pmatrix}
$$
is a ring isomorphism from the field of complex numbers to the ring of these matrices, proving that these matrices form a field. This isomorphism associates the square of the absolute value of a complex number with the determinant of the corresponding matrix, and the conjugate of a complex number with the transpose of the matrix.
The geometric description of the multiplication of complex numbers can also be expressed in terms of rotation matrices by using this correspondence between complex numbers and such matrices. The action of the matrix on a vector corresponds to the multiplication of by .
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The geometric description of the multiplication of complex numbers can also be expressed in terms of rotation matrices by using this correspondence between complex numbers and such matrices. The action of the matrix on a vector corresponds to the multiplication of by . In particular, if the determinant is , there is a real number such that the matrix has the form
$$
\begin{pmatrix}
\cos t & - \sin t \\
\sin t & \;\; \cos t
\end{pmatrix}.
$$
In this case, the action of the matrix on vectors and the multiplication by the complex number _ BLOCK3_ are both the rotation of the angle .
## Complex analysis
The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example).
Unlike real functions, which are commonly represented as two-dimensional graphs, complex functions have four-dimensional graphs and may usefully be illustrated by color-coding a three-dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.
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Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example).
Unlike real functions, which are commonly represented as two-dimensional graphs, complex functions have four-dimensional graphs and may usefully be illustrated by color-coding a three-dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.
### Convergence
The notions of convergent series and continuous functions in (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said to converge if and only if its real and imaginary parts do. This is equivalent to the (ε, δ)-definition of limits, where the absolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view,
$$
\mathbb{C}
$$
, endowed with the metric
$$
\operatorname{d}(z_1, z_2) = |z_1 - z_2|
$$
is a complete metric space, which notably includes the triangle inequality
$$
|z_1 + z_2| \le |z_1| + |z_2|
$$
for any two complex numbers and .
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This is equivalent to the (ε, δ)-definition of limits, where the absolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view,
$$
\mathbb{C}
$$
, endowed with the metric
$$
\operatorname{d}(z_1, z_2) = |z_1 - z_2|
$$
is a complete metric space, which notably includes the triangle inequality
$$
|z_1 + z_2| \le |z_1| + |z_2|
$$
for any two complex numbers and .
### Complex exponential
Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the exponential function , also written , is defined as the infinite series, which can be shown to converge for any z:
$$
\exp z:= 1+z+\frac{z^2}{2\cdot 1}+\frac{z^3}{3\cdot 2\cdot 1}+\cdots = \sum_{n=0}^{\infty} \frac{z^n}{n!}.
$$
For example,
$$
\exp (1)
$$
is Euler's number
$$
e \approx 2.718
$$
.
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From a more abstract point of view,
$$
\mathbb{C}
$$
, endowed with the metric
$$
\operatorname{d}(z_1, z_2) = |z_1 - z_2|
$$
is a complete metric space, which notably includes the triangle inequality
$$
|z_1 + z_2| \le |z_1| + |z_2|
$$
for any two complex numbers and .
### Complex exponential
Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the exponential function , also written , is defined as the infinite series, which can be shown to converge for any z:
$$
\exp z:= 1+z+\frac{z^2}{2\cdot 1}+\frac{z^3}{3\cdot 2\cdot 1}+\cdots = \sum_{n=0}^{\infty} \frac{z^n}{n!}.
$$
For example,
$$
\exp (1)
$$
is Euler's number
$$
e \approx 2.718
$$
. Euler's formula states:
$$
\exp(i\varphi) = \cos \varphi + i\sin \varphi
$$
for any real number .
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### Complex exponential
Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the exponential function , also written , is defined as the infinite series, which can be shown to converge for any z:
$$
\exp z:= 1+z+\frac{z^2}{2\cdot 1}+\frac{z^3}{3\cdot 2\cdot 1}+\cdots = \sum_{n=0}^{\infty} \frac{z^n}{n!}.
$$
For example,
$$
\exp (1)
$$
is Euler's number
$$
e \approx 2.718
$$
. Euler's formula states:
$$
\exp(i\varphi) = \cos \varphi + i\sin \varphi
$$
for any real number . This formula is a quick consequence of general basic facts about convergent power series and the definitions of the involved functions as power series. As a special case, this includes Euler's identity
$$
\exp(i \pi) = -1.
$$
### Complex logarithm
For any positive real number t, there is a unique real number x such that
$$
\exp(x) = t
$$
.
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### Complex logarithm
For any positive real number t, there is a unique real number x such that
$$
\exp(x) = t
$$
. This leads to the definition of the natural logarithm as the inverse
$$
\ln \colon \R^+ \to \R ; x \mapsto \ln x
$$
of the exponential function. The situation is different for complex numbers, since
$$
\exp(z+2\pi i) = \exp z \exp (2 \pi i) = \exp z
$$
by the functional equation and Euler's identity.
For example, , so both and are possible values for the complex logarithm of .
In general, given any non-zero complex number w, any number z solving the equation
$$
\exp z = w
$$
is called a complex logarithm of , denoted
$$
\log w
$$
. It can be shown that these numbers satisfy
$$
z = \log w = \ln|w| + i\arg w,
$$
where
$$
\arg
$$
is the argument defined above, and
$$
\ln
$$
the (real) natural logarithm. As arg is a multivalued function, unique only up to a multiple of , log is also multivalued.
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It can be shown that these numbers satisfy
$$
z = \log w = \ln|w| + i\arg w,
$$
where
$$
\arg
$$
is the argument defined above, and
$$
\ln
$$
the (real) natural logarithm. As arg is a multivalued function, unique only up to a multiple of , log is also multivalued. The principal value of log is often taken by restricting the imaginary part to the interval . This leads to the complex logarithm being a bijective function taking values in the strip
$$
\R^+ + \; i \, \left(-\pi, \pi\right]
$$
(that is denoted _ BLOCK9_ in the above illustration)
$$
\ln \colon \; \Complex^\times \; \to \; \; \; \R^+ + \; i \, \left(-\pi, \pi\right] .
$$
If
$$
z \in \Complex \setminus \left( -\R_{\ge 0} \right)
$$
is not a non-positive real number (a positive or a non-real number), the resulting principal value of the complex logarithm is obtained with .
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This leads to the complex logarithm being a bijective function taking values in the strip
$$
\R^+ + \; i \, \left(-\pi, \pi\right]
$$
(that is denoted _ BLOCK9_ in the above illustration)
$$
\ln \colon \; \Complex^\times \; \to \; \; \; \R^+ + \; i \, \left(-\pi, \pi\right] .
$$
If
$$
z \in \Complex \setminus \left( -\R_{\ge 0} \right)
$$
is not a non-positive real number (a positive or a non-real number), the resulting principal value of the complex logarithm is obtained with . It is an analytic function outside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number
$$
z \in -\R^+
$$
, where the principal value is .
Complex exponentiation is defined as
$$
z^\omega = \exp(\omega \ln z),
$$
and is multi-valued, except when is an integer. For , for some natural number , this recovers the non-uniqueness of th roots mentioned above.
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Complex exponentiation is defined as
$$
z^\omega = \exp(\omega \ln z),
$$
and is multi-valued, except when is an integer. For , for some natural number , this recovers the non-uniqueness of th roots mentioned above. If is real (and an arbitrary complex number), one has a preferred choice of
$$
\ln x
$$
, the real logarithm, which can be used to define a preferred exponential function.
Complex numbers, unlike real numbers, do not in general satisfy the unmodified power and logarithm identities, particularly when naïvely treated as single-valued functions; see failure of power and logarithm identities. For example, they do not satisfy
$$
a^{bc} = \left(a^b\right)^c.
$$
Both sides of the equation are multivalued by the definition of complex exponentiation given here, and the values on the left are a subset of those on the right.
### Complex sine and cosine
The series defining the real trigonometric functions sine and cosine, as well as the hyperbolic functions sinh and cosh, also carry over to complex arguments without change. For the other trigonometric and hyperbolic functions, such as tangent, things are slightly more complicated, as the defining series do not converge for all complex values.
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The series defining the real trigonometric functions sine and cosine, as well as the hyperbolic functions sinh and cosh, also carry over to complex arguments without change. For the other trigonometric and hyperbolic functions, such as tangent, things are slightly more complicated, as the defining series do not converge for all complex values. Therefore, one must define them either in terms of sine, cosine and exponential, or, equivalently, by using the method of analytic continuation.
### Holomorphic functions
A function
$$
f: \mathbb{C}
$$
→
$$
\mathbb{C}
$$
is called holomorphic or complex differentiable at a point
$$
z_0
$$
if the limit
$$
\lim_{z \to z_0} {f(z) - f(z_0) \over z - z_0 }
$$
exists (in which case it is denoted by
$$
f'(z_0)
$$
). This mimics the definition for real differentiable functions, except that all quantities are complex numbers. Loosely speaking, the freedom of approaching
$$
z_0
$$
in different directions imposes a much stronger condition than being (real) differentiable.
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This mimics the definition for real differentiable functions, except that all quantities are complex numbers. Loosely speaking, the freedom of approaching
$$
z_0
$$
in different directions imposes a much stronger condition than being (real) differentiable. For example, the function
$$
f(z) = \overline z
$$
is differentiable as a function
$$
\R^2 \to \R^2
$$
, but is not complex differentiable.
A real differentiable function is complex differentiable if and only if it satisfies the Cauchy–Riemann equations, which are sometimes abbreviated as
$$
\frac{\partial f}{\partial \overline z} = 0.
$$
Complex analysis shows some features not apparent in real analysis. For example, the identity theorem asserts that two holomorphic functions and agree if they agree on an arbitrarily small open subset of
$$
\mathbb{C}
$$
. Meromorphic functions, functions that can locally be written as with a holomorphic function , still share some of the features of holomorphic functions. Other functions have essential singularities, such as at .
## Applications
Complex numbers have applications in many scientific areas, including signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis. Some of these applications are described below.
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## Applications
Complex numbers have applications in many scientific areas, including signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis. Some of these applications are described below.
Complex conjugation is also employed in inversive geometry, a branch of geometry studying reflections more general than ones about a line. In the network analysis of electrical circuits, the complex conjugate is used in finding the equivalent impedance when the maximum power transfer theorem is looked for.
### Geometry
#### Shapes
Three non-collinear points
$$
u, v, w
$$
in the plane determine the shape of the triangle
$$
\{u, v, w\}
$$
. Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic as
$$
S(u, v, w) = \frac {u - w}{u - v}.
$$
The shape
$$
S
$$
of a triangle will remain the same, when the complex plane is transformed by translation or dilation (by an affine transformation), corresponding to the intuitive notion of shape, and describing similarity. Thus each triangle
$$
\{u, v, w\}
$$
is in a similarity class of triangles with the same shape.
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Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic as
$$
S(u, v, w) = \frac {u - w}{u - v}.
$$
The shape
$$
S
$$
of a triangle will remain the same, when the complex plane is transformed by translation or dilation (by an affine transformation), corresponding to the intuitive notion of shape, and describing similarity. Thus each triangle
$$
\{u, v, w\}
$$
is in a similarity class of triangles with the same shape.
#### Fractal geometry
The Mandelbrot set is a popular example of a fractal formed on the complex plane. It is defined by plotting every location
$$
c
$$
where iterating the sequence _ BLOCK1_ does not diverge when iterated infinitely. Similarly, Julia sets have the same rules, except where
$$
c
$$
remains constant.
#### Triangles
Every triangle has a unique Steiner inellipse – an ellipse inside the triangle and tangent to the midpoints of the three sides of the triangle. The foci of a triangle's Steiner inellipse can be found as follows, according to Marden's theorem: Denote the triangle's vertices in the complex plane as , , and .
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#### Triangles
Every triangle has a unique Steiner inellipse – an ellipse inside the triangle and tangent to the midpoints of the three sides of the triangle. The foci of a triangle's Steiner inellipse can be found as follows, according to Marden's theorem: Denote the triangle's vertices in the complex plane as , , and . Write the cubic equation
$$
(x-a)(x-b)(x-c)=0
$$
, take its derivative, and equate the (quadratic) derivative to zero. Marden's theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse.
### Algebraic number theory
As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in
$$
\mathbb{C}
$$
. A fortiori, the same is true if the equation has rational coefficients. The roots of such equations are called algebraic numbers – they are a principal object of study in algebraic number theory.
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A fortiori, the same is true if the equation has rational coefficients. The roots of such equations are called algebraic numbers – they are a principal object of study in algebraic number theory. Compared to
$$
\overline{\mathbb{Q}}
$$
, the algebraic closure of
$$
\mathbb{Q}
$$
, which also contains all algebraic numbers,
$$
\mathbb{C}
$$
has the advantage of being easily understandable in geometric terms. In this way, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, more specifically applying the machinery of field theory to the number field containing roots of unity, it can be shown that it is not possible to construct a regular nonagon using only compass and straightedge – a purely geometric problem.
Another example is the Gaussian integers; that is, numbers of the form , where and are integers, which can be used to classify sums of squares.
### Analytic number theory
Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the Riemann zeta function is related to the distribution of prime numbers.
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This is done by encoding number-theoretic information in complex-valued functions. For example, the Riemann zeta function is related to the distribution of prime numbers.
### Improper integrals
In applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this; see methods of contour integration.
### Dynamic equations
In differential equations, it is common to first find all complex roots of the characteristic equation of a linear differential equation or equation system and then attempt to solve the system in terms of base functions of the form . Likewise, in difference equations, the complex roots of the characteristic equation of the difference equation system are used, to attempt to solve the system in terms of base functions of the form .
### Linear algebra
Since
$$
\C
$$
is algebraically closed, any non-empty complex square matrix has at least one (complex) eigenvalue. By comparison, real matrices do not always have real eigenvalues, for example rotation matrices (for rotations of the plane for angles other than 0° or 180°) leave no direction fixed, and therefore do not have any real eigenvalue.
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### Linear algebra
Since
$$
\C
$$
is algebraically closed, any non-empty complex square matrix has at least one (complex) eigenvalue. By comparison, real matrices do not always have real eigenvalues, for example rotation matrices (for rotations of the plane for angles other than 0° or 180°) leave no direction fixed, and therefore do not have any real eigenvalue. The existence of (complex) eigenvalues, and the ensuing existence of eigendecomposition is a useful tool for computing matrix powers and matrix exponentials.
Complex numbers often generalize concepts originally conceived in the real numbers. For example, the conjugate transpose generalizes the transpose, hermitian matrices generalize symmetric matrices, and unitary matrices generalize orthogonal matrices.
### In applied mathematics
#### Control theory
In control theory, systems are often transformed from the time domain to the complex frequency domain using the Laplace transform. The system's zeros and poles are then analyzed in the complex plane. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane.
In the root locus method, it is important whether zeros and poles are in the left or right half planes, that is, have real part greater than or less than zero.
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The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane.
In the root locus method, it is important whether zeros and poles are in the left or right half planes, that is, have real part greater than or less than zero. If a linear, time-invariant (LTI) system has poles that are
- in the right half plane, it will be unstable,
- all in the left half plane, it will be stable,
- on the imaginary axis, it will have marginal stability.
If a system has zeros in the right half plane, it is a nonminimum phase system.
#### Signal analysis
Complex numbers are used in signal analysis and other fields for a convenient description for periodically varying signals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a sine wave of a given frequency, the absolute value of the corresponding is the amplitude and the argument is the phase.
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For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a sine wave of a given frequency, the absolute value of the corresponding is the amplitude and the argument is the phase.
If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as complex-valued functions of the form
$$
x(t) = \operatorname{Re} \{X( t ) \}
$$
and
$$
X( t ) = A e^{i\omega t} = a e^{ i \phi } e^{i\omega t} = a e^{i (\omega t + \phi) }
$$
where ω represents the angular frequency and the complex number A encodes the phase and amplitude as explained above.
This use is also extended into digital signal processing and digital image processing, which use digital versions of Fourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, and video signals.
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If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as complex-valued functions of the form
$$
x(t) = \operatorname{Re} \{X( t ) \}
$$
and
$$
X( t ) = A e^{i\omega t} = a e^{ i \phi } e^{i\omega t} = a e^{i (\omega t + \phi) }
$$
where ω represents the angular frequency and the complex number A encodes the phase and amplitude as explained above.
This use is also extended into digital signal processing and digital image processing, which use digital versions of Fourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, and video signals.
Another example, relevant to the two side bands of amplitude modulation of AM radio, is:
$$
\begin{align}
\cos((\omega + \alpha)t) + \cos\left((\omega - \alpha)t\right)
_BLOCK0_\end{align}
$$
### In physics
#### Electromagnetism and electrical engineering
In electrical engineering, the Fourier transform is used to analyze varying electric currents and voltages.
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### In physics
#### Electromagnetism and electrical engineering
In electrical engineering, the Fourier transform is used to analyze varying electric currents and voltages. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. This approach is called phasor calculus.
In electrical engineering, the imaginary unit is denoted by , to avoid confusion with , which is generally in use to denote electric current, or, more particularly, , which is generally in use to denote instantaneous electric current.
Because the voltage in an AC circuit is oscillating, it can be represented as
$$
V(t) = V_0 e^{j \omega t} = V_0 \left (\cos\omega t + j \sin\omega t \right ),
$$
To obtain the measurable quantity, the real part is taken:
$$
v(t) = \operatorname{Re}(V) = \operatorname{Re}\left [ V_0 e^{j \omega t} \right ] = V_0 \cos \omega t.
$$
The complex-valued signal is called the analytic representation of the real-valued, measurable signal .
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In electrical engineering, the imaginary unit is denoted by , to avoid confusion with , which is generally in use to denote electric current, or, more particularly, , which is generally in use to denote instantaneous electric current.
Because the voltage in an AC circuit is oscillating, it can be represented as
$$
V(t) = V_0 e^{j \omega t} = V_0 \left (\cos\omega t + j \sin\omega t \right ),
$$
To obtain the measurable quantity, the real part is taken:
$$
v(t) = \operatorname{Re}(V) = \operatorname{Re}\left [ V_0 e^{j \omega t} \right ] = V_0 \cos \omega t.
$$
The complex-valued signal is called the analytic representation of the real-valued, measurable signal .
#### Fluid dynamics
In fluid dynamics, complex functions are used to describe potential flow in two dimensions.
#### Quantum mechanics
The complex number field is intrinsic to the mathematical formulations of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg's matrix mechanics – make use of complex numbers.
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#### Quantum mechanics
The complex number field is intrinsic to the mathematical formulations of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg's matrix mechanics – make use of complex numbers.
#### Relativity
In special relativity and general relativity, some formulas for the metric on spacetime become simpler if one takes the time component of the spacetime continuum to be imaginary. (This approach is no longer standard in classical relativity, but is used in an essential way in quantum field theory.) Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity.
## Characterizations, generalizations and related notions
### Algebraic characterization
The field
$$
\Complex
$$
has the following three properties:
- First, it has characteristic 0. This means that for any number of summands (all of which equal one).
- Second, its transcendence degree over
$$
\Q
$$
, the prime field of
$$
\Complex,
$$
is the cardinality of the continuum.
- Third, it is algebraically closed (see above).
It can be shown that any field having these properties is isomorphic (as a field) to
$$
\Complex.
$$
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- Second, its transcendence degree over
$$
\Q
$$
, the prime field of
$$
\Complex,
$$
is the cardinality of the continuum.
- Third, it is algebraically closed (see above).
It can be shown that any field having these properties is isomorphic (as a field) to
$$
\Complex.
$$
For example, the algebraic closure of the field
$$
\Q_p
$$
of the -adic number also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields). Also,
$$
\Complex
$$
is isomorphic to the field of complex Puiseux series. However, specifying an isomorphism requires the axiom of choice. Another consequence of this algebraic characterization is that _ BLOCK6_ contains many proper subfields that are isomorphic to
$$
\Complex
$$
.
### Characterization as a topological field
The preceding characterization of
$$
\Complex
$$
describes only the algebraic aspects of
$$
\Complex.
$$
That is to say, the properties of nearness and continuity, which matter in areas such as analysis and topology, are not dealt with.
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### Characterization as a topological field
The preceding characterization of
$$
\Complex
$$
describes only the algebraic aspects of
$$
\Complex.
$$
That is to say, the properties of nearness and continuity, which matter in areas such as analysis and topology, are not dealt with. The following description of
$$
\Complex
$$
as a topological field (that is, a field that is equipped with a topology, which allows the notion of convergence) does take into account the topological properties. _ BLOCK3_ contains a subset (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:
- is closed under addition, multiplication and taking inverses.
- If and are distinct elements of , then either or is in .
- If is any nonempty subset of , then for some in
$$
\Complex.
$$
Moreover,
$$
\Complex
$$
has a nontrivial involutive automorphism (namely the complex conjugation), such that is in for any nonzero in
$$
\Complex.
$$
Any field with these properties can be endowed with a topology by taking the sets as a base, where ranges over the field and ranges over . With this topology is isomorphic as a topological field to
$$
\Complex.
$$
The
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|
- If is any nonempty subset of , then for some in
$$
\Complex.
$$
Moreover,
$$
\Complex
$$
has a nontrivial involutive automorphism (namely the complex conjugation), such that is in for any nonzero in
$$
\Complex.
$$
Any field with these properties can be endowed with a topology by taking the sets as a base, where ranges over the field and ranges over . With this topology is isomorphic as a topological field to
$$
\Complex.
$$
The only connected locally compact topological fields are
$$
\R
$$
and
$$
\Complex.
$$
This gives another characterization of
$$
\Complex
$$
as a topological field, because
$$
\Complex
$$
can be distinguished from
$$
\R
$$
because the nonzero complex numbers are connected, while the nonzero real numbers are not.
### Other number systems
+ Number systems rational numbers real numbers complex numbers quaternions octonions sedenions complete dimension as an -vector space [does not apply] 1 2 4 8 16 ordered multiplication commutative multiplication associative normed division algebra [does not apply]
The process of extending the field
$$
\mathbb R
$$
of reals to
$$
\mathbb C
$$
is an instance of the Cayley–Dickson construction.
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|
[does not apply]
The process of extending the field
$$
\mathbb R
$$
of reals to
$$
\mathbb C
$$
is an instance of the Cayley–Dickson construction. Applying this construction iteratively to
$$
\C
$$
then yields the quaternions, the octonions, the sedenions, and the trigintaduonions. This construction turns out to diminish the structural properties of the involved number systems.
Unlike the reals,
$$
\Complex
$$
is not an ordered field, that is to say, it is not possible to define a relation that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so precludes the existence of an ordering on
$$
\Complex.
$$
Passing from
$$
\C
$$
to the quaternions
$$
\mathbb H
$$
loses commutativity, while the octonions (additionally to not being commutative) fail to be associative. The reals, complex numbers, quaternions and octonions are all normed division algebras over
$$
\mathbb R
$$
. By Hurwitz's theorem they are the only ones; the sedenions, the next step in the Cayley–Dickson construction, fail to have this structure.
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The reals, complex numbers, quaternions and octonions are all normed division algebras over
$$
\mathbb R
$$
. By Hurwitz's theorem they are the only ones; the sedenions, the next step in the Cayley–Dickson construction, fail to have this structure.
The Cayley–Dickson construction is closely related to the regular representation of
$$
\mathbb C,
$$
thought of as an
$$
\mathbb R
$$
-algebra (an
$$
\mathbb{R}
$$
-vector space with a multiplication), with respect to the basis . This means the following: the
$$
\mathbb R
$$
-linear map
$$
\begin{align}
\mathbb{C} &\rightarrow \mathbb{C} \\
z &\mapsto wz
\end{align}
$$
for some fixed complex number can be represented by a matrix (once a basis has been chosen). With respect to the basis , this matrix is
$$
\begin{pmatrix}
\operatorname{Re}(w) & -\operatorname{Im}(w) \\
\operatorname{Im}(w) & \operatorname{Re}(w)
\end{pmatrix},
$$
that is, the one mentioned in the section on matrix representation of complex numbers above.
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|
This means the following: the
$$
\mathbb R
$$
-linear map
$$
\begin{align}
\mathbb{C} &\rightarrow \mathbb{C} \\
z &\mapsto wz
\end{align}
$$
for some fixed complex number can be represented by a matrix (once a basis has been chosen). With respect to the basis , this matrix is
$$
\begin{pmatrix}
\operatorname{Re}(w) & -\operatorname{Im}(w) \\
\operatorname{Im}(w) & \operatorname{Re}(w)
\end{pmatrix},
$$
that is, the one mentioned in the section on matrix representation of complex numbers above. While this is a linear representation of
$$
\mathbb C
$$
in the 2 × 2 real matrices, it is not the only one. Any matrix
$$
J = \begin{pmatrix}p & q \\ r & -p \end{pmatrix}, \quad p^2 + qr + 1 = 0
$$
has the property that its square is the negative of the identity matrix: .
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While this is a linear representation of
$$
\mathbb C
$$
in the 2 × 2 real matrices, it is not the only one. Any matrix
$$
J = \begin{pmatrix}p & q \\ r & -p \end{pmatrix}, \quad p^2 + qr + 1 = 0
$$
has the property that its square is the negative of the identity matrix: . Then
$$
\{ z = a I + b J : a,b \in \mathbb{R} \}
$$
is also isomorphic to the field
$$
\mathbb C,
$$
and gives an alternative complex structure on
$$
\mathbb R^2.
$$
This is generalized by the notion of a linear complex structure.
Hypercomplex numbers also generalize
$$
\mathbb R,
$$
$$
\mathbb C,
$$
$$
\mathbb H,
$$
and
$$
\mathbb{O}.
$$
For example, this notion contains the split-complex numbers, which are elements of the ring
$$
\mathbb R[x]/(x^2-1)
$$
(as opposed to
$$
\mathbb R[x]/(x^2+1)
$$
for complex numbers). In this ring, the equation has four solutions.
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|
For example, this notion contains the split-complex numbers, which are elements of the ring
$$
\mathbb R[x]/(x^2-1)
$$
(as opposed to
$$
\mathbb R[x]/(x^2+1)
$$
for complex numbers). In this ring, the equation has four solutions.
The field
$$
\mathbb R
$$
is the completion of
$$
\mathbb Q,
$$
the field of rational numbers, with respect to the usual absolute value metric. Other choices of metrics on
$$
\mathbb Q
$$
lead to the fields
$$
\mathbb Q_p
$$
of -adic numbers (for any prime number ), which are thereby analogous to
$$
\mathbb{R}
$$
. There are no other nontrivial ways of completing
$$
\mathbb Q
$$
than
$$
\mathbb R
$$
and
$$
\mathbb Q_p,
$$
by Ostrowski's theorem. The algebraic closures
$$
\overline {\mathbb{Q}_p}
$$
of
$$
\mathbb Q_p
$$
still carry a norm, but (unlike
$$
\mathbb C
$$
) are not complete with respect to it.
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|
There are no other nontrivial ways of completing
$$
\mathbb Q
$$
than
$$
\mathbb R
$$
and
$$
\mathbb Q_p,
$$
by Ostrowski's theorem. The algebraic closures
$$
\overline {\mathbb{Q}_p}
$$
of
$$
\mathbb Q_p
$$
still carry a norm, but (unlike
$$
\mathbb C
$$
) are not complete with respect to it. The completion
$$
\mathbb{C}_p
$$
of
$$
\overline {\mathbb{Q}_p}
$$
turns out to be algebraically closed. By analogy, the field is called -adic complex numbers.
The fields
$$
\mathbb R,
$$
$$
\mathbb Q_p,
$$
and their finite field extensions, including
$$
\mathbb C,
$$
are called local fields.
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In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability ) or failure (with probability ). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., , the binomial distribution is a
### Bernoulli distribution
. The binomial distribution is the basis for the binomial test of statistical significance.
The binomial distribution is frequently used to model the number of successes in a sample of size drawn with replacement from a population of size . If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for much larger than , the binomial distribution remains a good approximation, and is widely used.
## Definitions
### Probability mass function
If the random variable follows the binomial distribution with parameters and , we write .
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## Definitions
### Probability mass function
If the random variable follows the binomial distribution with parameters and , we write . The probability of getting exactly successes in independent Bernoulli trials (with the same rate ) is given by the probability mass function:
$$
f(k,n,p) = \Pr(X = k) = \binom{n}{k}p^k(1-p)^{n-k}
$$
for , where
$$
\binom{n}{k} =\frac{n!}{k!(n-k)!}
$$
is the binomial coefficient. The formula can be understood as follows: is the probability of obtaining the sequence of independent Bernoulli trials in which trials are "successes" and the remaining trials result in "failure". Since the trials are independent with probabilities remaining constant between them, any sequence of trials with successes (and failures) has the same probability of being achieved (regardless of positions of successes within the sequence). There are
$$
\binom{n}{k}
$$
such sequences, since the binomial coefficient
$$
\binom{n}{k}
$$
counts the number of ways to choose the positions of the successes among the trials.
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Since the trials are independent with probabilities remaining constant between them, any sequence of trials with successes (and failures) has the same probability of being achieved (regardless of positions of successes within the sequence). There are
$$
\binom{n}{k}
$$
such sequences, since the binomial coefficient
$$
\binom{n}{k}
$$
counts the number of ways to choose the positions of the successes among the trials. The binomial distribution is concerned with the probability of obtaining any of these sequences, meaning the probability of obtaining one of them () must be added
$$
\binom{n}{k}
$$
times, hence
$$
\Pr(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
$$
.
In creating reference tables for binomial distribution probability, usually, the table is filled in up to values. This is because for , the probability can be calculated by its complement as
$$
f(k,n,p)=f(n-k,n,1-p).
$$
Looking at the expression as a function of , there is a value that maximizes it.
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In creating reference tables for binomial distribution probability, usually, the table is filled in up to values. This is because for , the probability can be calculated by its complement as
$$
f(k,n,p)=f(n-k,n,1-p).
$$
Looking at the expression as a function of , there is a value that maximizes it. This value can be found by calculating
$$
\frac{f(k+1,n,p)}{f(k,n,p)}=\frac{(n-k)p}{(k+1)(1-p)}
$$
and comparing it to 1. There is always an integer that satisfies
$$
(n+1)p-1 \leq M < (n+1)p.
$$
is monotone increasing for and monotone decreasing for , with the exception of the case where is an integer. In this case, there are two values for which is maximal: and . is the most probable outcome (that is, the most likely, although this can still be unlikely overall) of the Bernoulli trials and is called the mode.
Equivalently, . Taking the floor function, we obtain .
### Example
Suppose a biased coin comes up heads with probability 0.3 when tossed.
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Taking the floor function, we obtain .
### Example
Suppose a biased coin comes up heads with probability 0.3 when tossed. The probability of seeing exactly 4 heads in 6 tosses is
$$
f(4,6,0.3) = \binom{6}{4}0.3^4 (1-0.3)^{6-4}= 0.059535.
$$
### Cumulative distribution function
The cumulative distribution function can be expressed as:
$$
F(k;n,p) = \Pr(X \le k) = \sum_{i=0}^{\lfloor k \rfloor} {n\choose i}p^i(1-p)^{n-i},
$$
where
$$
\lfloor k\rfloor
$$
is the "floor" under , i.e. the greatest integer less than or equal to .
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The probability of seeing exactly 4 heads in 6 tosses is
$$
f(4,6,0.3) = \binom{6}{4}0.3^4 (1-0.3)^{6-4}= 0.059535.
$$
### Cumulative distribution function
The cumulative distribution function can be expressed as:
$$
F(k;n,p) = \Pr(X \le k) = \sum_{i=0}^{\lfloor k \rfloor} {n\choose i}p^i(1-p)^{n-i},
$$
where
$$
\lfloor k\rfloor
$$
is the "floor" under , i.e. the greatest integer less than or equal to .
It can also be represented in terms of the regularized incomplete beta function, as follows:
$$
\begin{align}
F(k;n,p) & = \Pr(X \le k) \\
&= I_{1-p}(n-k, k+1) \\
& = (n-k) {n \choose k} \int_0^{1-p} t^{n-k-1} (1-t)^k \, dt ,
\end{align}
$$
which is equivalent to the cumulative distribution functions of the beta distribution and of the -distribution:
$$
F(k;n,p) = F_{\text{beta-distribution}}\left(x=1-p;\alpha=n-k,\beta=k+1\right)
$$
$$
F(k;n,p) = F_{F\text{-distribution}}\left(x=\frac{1-p}{p}\frac{k+1}{n-k};d_1=2(n-k),d_2=2(k+1)\right).
$$
Some closed-form bounds for the cumulative distribution function are given below.
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### Cumulative distribution function
The cumulative distribution function can be expressed as:
$$
F(k;n,p) = \Pr(X \le k) = \sum_{i=0}^{\lfloor k \rfloor} {n\choose i}p^i(1-p)^{n-i},
$$
where
$$
\lfloor k\rfloor
$$
is the "floor" under , i.e. the greatest integer less than or equal to .
It can also be represented in terms of the regularized incomplete beta function, as follows:
$$
\begin{align}
F(k;n,p) & = \Pr(X \le k) \\
&= I_{1-p}(n-k, k+1) \\
& = (n-k) {n \choose k} \int_0^{1-p} t^{n-k-1} (1-t)^k \, dt ,
\end{align}
$$
which is equivalent to the cumulative distribution functions of the beta distribution and of the -distribution:
$$
F(k;n,p) = F_{\text{beta-distribution}}\left(x=1-p;\alpha=n-k,\beta=k+1\right)
$$
$$
F(k;n,p) = F_{F\text{-distribution}}\left(x=\frac{1-p}{p}\frac{k+1}{n-k};d_1=2(n-k),d_2=2(k+1)\right).
$$
Some closed-form bounds for the cumulative distribution function are given below.
## Properties
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|
It can also be represented in terms of the regularized incomplete beta function, as follows:
$$
\begin{align}
F(k;n,p) & = \Pr(X \le k) \\
&= I_{1-p}(n-k, k+1) \\
& = (n-k) {n \choose k} \int_0^{1-p} t^{n-k-1} (1-t)^k \, dt ,
\end{align}
$$
which is equivalent to the cumulative distribution functions of the beta distribution and of the -distribution:
$$
F(k;n,p) = F_{\text{beta-distribution}}\left(x=1-p;\alpha=n-k,\beta=k+1\right)
$$
$$
F(k;n,p) = F_{F\text{-distribution}}\left(x=\frac{1-p}{p}\frac{k+1}{n-k};d_1=2(n-k),d_2=2(k+1)\right).
$$
Some closed-form bounds for the cumulative distribution function are given below.
## Properties
### Expected value and variance
If , that is, is a binomially distributed random variable, being the total number of experiments and p the probability of each experiment yielding a successful result, then the expected value of is:
$$
\operatorname{E}[X] = np.
$$
This follows from the linearity of the expected value along with the fact that is the sum of identical Bernoulli random variables, each with expected value .
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## Properties
### Expected value and variance
If , that is, is a binomially distributed random variable, being the total number of experiments and p the probability of each experiment yielding a successful result, then the expected value of is:
$$
\operatorname{E}[X] = np.
$$
This follows from the linearity of the expected value along with the fact that is the sum of identical Bernoulli random variables, each with expected value . In other words, if
$$
X_1, \ldots, X_n
$$
are identical (and independent) Bernoulli random variables with parameter , then and
$$
\operatorname{E}[X] = \operatorname{E}[X_1 + \cdots + X_n] = \operatorname{E}[X_1] + \cdots + \operatorname{E}[X_n] = p + \cdots + p = np.
$$
The variance is:
$$
\operatorname{Var}(X) = npq = np(1 - p).
$$
This similarly follows from the fact that the variance of a sum of independent random variables is the sum of the variances.
### Higher moments
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The first 6 central moments, defined as
$$
\mu _{c}=\operatorname {E} \left[(X-\operatorname {E} [X])^{c}\right]
$$
, are given by
$$
\begin{align}
\mu_1 &= 0, \\
\mu_2 &= np(1-p),\\
\mu_3 &= np(1-p)(1-2p),\\
\mu_4 &= np(1-p)(1+(3n-6)p(1-p)),\\
\mu_5 &= np(1-p)(1-2p)(1+(10n-12)p(1-p)),\\
\mu_6 &= np(1-p)(1-30p(1-p)(1-4p(1-p))+5np(1-p)(5-26p(1-p))+15n^2 p^2 (1-p)^2).
\end{align}
$$
The non-central moments satisfy
$$
\begin{align}
\operatorname {E}[X] &= np, \\
\operatorname {E}[X^2] &= np(1-p)+n^2p^2,
\end{align}
$$
and in general
$$
\operatorname {E}[X^c] = \sum_{k=0}^c \left\{ {c \atop k} \right\} n^{\underline{k}} p^k,
$$
where
$$
\textstyle \left\{{c\atop k}\right\}
$$
are the Stirling numbers of the second kind, and
$$
n^{\underline{k}} = n(n-1)\cdots(n-k+1)
$$
is the _
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BLOCK6_th falling power of
$$
n
$$
.
A simple bound
follows by bounding the Binomial moments via the higher Poisson moments: _ BLOCK8_This shows that if
$$
c=O(\sqrt{np})
$$
, then
$$
\operatorname {E}[X^c]
$$
is at most a constant factor away from
$$
\operatorname {E}[X]^c
$$
### Mode
Usually the mode of a binomial distribution is equal to
$$
\lfloor (n+1)p\rfloor
$$
, where
$$
\lfloor\cdot\rfloor
$$
is the floor function. However, when is an integer and is neither 0 nor 1, then the distribution has two modes: and . When is equal to 0 or 1, the mode will be 0 and correspondingly. These cases can be summarized as follows:
$$
\text{mode} =
_BLOCK0_$$
Proof: Let
$$
f(k)=\binom nk p^k q^{n-k}.
$$
For
$$
p=0
$$
only
$$
f(0)
$$
has a nonzero value with
$$
f(0)=1
$$
.
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When is equal to 0 or 1, the mode will be 0 and correspondingly. These cases can be summarized as follows:
$$
\text{mode} =
_BLOCK0_$$
Proof: Let
$$
f(k)=\binom nk p^k q^{n-k}.
$$
For
$$
p=0
$$
only
$$
f(0)
$$
has a nonzero value with
$$
f(0)=1
$$
. For
$$
p=1
$$
we find
$$
f(n)=1
$$
and
$$
f(k)=0
$$
for
$$
k\neq n
$$
. This proves that the mode is 0 for
$$
p=0
$$
and
$$
n
$$
for
$$
p=1
$$
.
Let
$$
0 < p < 1
$$
. We find
$$
\frac{f(k+1)}{f(k)} = \frac{(n-k)p}{(k+1)(1-p)}
$$
.
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Let
$$
0 < p < 1
$$
. We find
$$
\frac{f(k+1)}{f(k)} = \frac{(n-k)p}{(k+1)(1-p)}
$$
.
From this follows
$$
\begin{align}
k > (n+1)p-1 \Rightarrow f(k+1) < f(k) \\
k = (n+1)p-1 \Rightarrow f(k+1) = f(k) \\
k < (n+1)p-1 \Rightarrow f(k+1) > f(k)
\end{align}
$$
So when
$$
(n+1)p-1
$$
is an integer, then
$$
(n+1)p-1
$$
and
$$
(n+1)p
$$
is a mode. In the case that
$$
(n+1)p-1\notin \Z
$$
, then only
$$
\lfloor (n+1)p-1\rfloor+1=\lfloor (n+1)p\rfloor
$$
is a mode.
### Median
In general, there is no single formula to find the median for a binomial distribution, and it may even be non-unique. However, several special results have been established:
- If is an integer, then the mean, median, and mode coincide and equal .Lord, Nick. (July 2010).
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However, several special results have been established:
- If is an integer, then the mean, median, and mode coincide and equal .Lord, Nick. (July 2010). "Binomial averages when the mean is an integer", The Mathematical Gazette 94, 331-332.
- Any median must lie within the interval
$$
\lfloor np \rfloor\leq m \leq \lceil np \rceil
$$
.
- A median cannot lie too far away from the mean:
$$
|m-np|\leq \min\{{\ln2}, \max\{p,1-p\}\}
$$
.
- The median is unique and equal to when (except for the case when and is odd).
- When is a rational number (with the exception of \ and odd) the median is unique.
- When
$$
p= \frac{1}{2}
$$
and is odd, any number in the interval
$$
\frac{1}{2} \bigl(n-1\bigr)\leq m \leq \frac{1}{2} \bigl(n+1\bigr)
$$
is a median of the binomial distribution.
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When is a rational number (with the exception of \ and odd) the median is unique.
- When
$$
p= \frac{1}{2}
$$
and is odd, any number in the interval
$$
\frac{1}{2} \bigl(n-1\bigr)\leq m \leq \frac{1}{2} \bigl(n+1\bigr)
$$
is a median of the binomial distribution. If
$$
p= \frac{1}{2}
$$
and is even, then
$$
m= \frac{n}{2}
$$
is the unique median.
### Tail bounds
For , upper bounds can be derived for the lower tail of the cumulative distribution function
$$
F(k;n,p) = \Pr(X \le k)
$$
, the probability that there are at most successes. Since
$$
\Pr(X \ge k) = F(n-k;n,1-p)
$$
, these bounds can also be seen as bounds for the upper tail of the cumulative distribution function for .
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### Tail bounds
For , upper bounds can be derived for the lower tail of the cumulative distribution function
$$
F(k;n,p) = \Pr(X \le k)
$$
, the probability that there are at most successes. Since
$$
\Pr(X \ge k) = F(n-k;n,1-p)
$$
, these bounds can also be seen as bounds for the upper tail of the cumulative distribution function for .
Hoeffding's inequality yields the simple bound
$$
F(k;n,p) \leq \exp\left(-2 n\left(p-\frac{k}{n}\right)^2\right), \!
$$
which is however not very tight. In particular, for , we have that (for fixed , with ), but Hoeffding's bound evaluates to a positive constant.
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Hoeffding's inequality yields the simple bound
$$
F(k;n,p) \leq \exp\left(-2 n\left(p-\frac{k}{n}\right)^2\right), \!
$$
which is however not very tight. In particular, for , we have that (for fixed , with ), but Hoeffding's bound evaluates to a positive constant.
A sharper bound can be obtained from the Chernoff bound:
$$
F(k;n,p) \leq \exp\left(-nD\left(\frac{k}{n}\parallel p\right)\right)
$$
where is the relative entropy (or Kullback-Leibler divergence) between an -coin and a -coin (i.e. between the and distribution):
$$
D(a\parallel p)=(a)\ln\frac{a}{p}+(1-a)\ln\frac{1-a}{1-p}. \!
$$
Asymptotically, this bound is reasonably tight; see for details.
One can also obtain lower bounds on the tail , known as anti-concentration bounds.
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A sharper bound can be obtained from the Chernoff bound:
$$
F(k;n,p) \leq \exp\left(-nD\left(\frac{k}{n}\parallel p\right)\right)
$$
where is the relative entropy (or Kullback-Leibler divergence) between an -coin and a -coin (i.e. between the and distribution):
$$
D(a\parallel p)=(a)\ln\frac{a}{p}+(1-a)\ln\frac{1-a}{1-p}. \!
$$
Asymptotically, this bound is reasonably tight; see for details.
One can also obtain lower bounds on the tail , known as anti-concentration bounds. By approximating the binomial coefficient with Stirling's formula it can be shown that
$$
F(k;n,p) \geq \frac{1}{\sqrt{8n\tfrac{k}{n}(1-\tfrac{k}{n})}} \exp\left(-nD\left(\frac{k}{n}\parallel p\right)\right),
$$
which implies the simpler but looser
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One can also obtain lower bounds on the tail , known as anti-concentration bounds. By approximating the binomial coefficient with Stirling's formula it can be shown that
$$
F(k;n,p) \geq \frac{1}{\sqrt{8n\tfrac{k}{n}(1-\tfrac{k}{n})}} \exp\left(-nD\left(\frac{k}{n}\parallel p\right)\right),
$$
which implies the simpler but looser bound
$$
F(k;n,p) \geq \frac1{\sqrt{2n}} \exp\left(-nD\left(\frac{k}{n}\parallel p\right)\right).
$$
For and for even , it is possible to make the denominator constant:
$$
F(k;n,\tfrac{1}{2}) \geq \frac{1}{15} \exp\left(- 16n \left(\frac{1}{2} -\frac{k}{n}\right)^2\right). \!
$$
## Statistical inference
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bound
$$
F(k;n,p) \geq \frac1{\sqrt{2n}} \exp\left(-nD\left(\frac{k}{n}\parallel p\right)\right).
$$
For and for even , it is possible to make the denominator constant:
$$
F(k;n,\tfrac{1}{2}) \geq \frac{1}{15} \exp\left(- 16n \left(\frac{1}{2} -\frac{k}{n}\right)^2\right). \!
$$
## Statistical inference
### Estimation of parameters
When is known, the parameter can be estimated using the proportion of successes:
$$
\widehat{p} = \frac{x}{n}.
$$
This estimator is found using maximum likelihood estimator and also the method of moments. This estimator is unbiased and uniformly with minimum variance, proven using Lehmann–Scheffé theorem, since it is based on a minimal sufficient and complete statistic (i.e.: ). It is also consistent both in probability and in MSE. This statistic is asymptotically normal thanks to the central limit theorem, because it is the same as taking the mean over Bernoulli samples.
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It is also consistent both in probability and in MSE. This statistic is asymptotically normal thanks to the central limit theorem, because it is the same as taking the mean over Bernoulli samples. It has a variance of
$$
var(\widehat{p}) = \frac{p(1-p)}{n}
$$
, a property which is used in various ways, such as in Wald's confidence intervals.
A closed form Bayes estimator for also exists when using the
### Beta distribution
as a conjugate prior distribution. When using a general
$$
\operatorname{Beta}(\alpha, \beta)
$$
as a prior, the posterior mean estimator is:
$$
\widehat{p}_b = \frac{x+\alpha}{n+\alpha+\beta}.
$$
The Bayes estimator is asymptotically efficient and as the sample size approaches infinity (), it approaches the MLE solution. The Bayes estimator is biased (how much depends on the priors), admissible and consistent in probability. Using the Bayesian estimator with the Beta distribution can be used with Thompson sampling.
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The Bayes estimator is biased (how much depends on the priors), admissible and consistent in probability. Using the Bayesian estimator with the Beta distribution can be used with Thompson sampling.
For the special case of using the standard uniform distribution as a non-informative prior,
$$
\operatorname{Beta}(\alpha=1, \beta=1) = U(0,1)
$$
, the posterior mean estimator becomes:
$$
\widehat{p}_b = \frac{x+1}{n+2}.
$$
(A posterior mode should just lead to the standard estimator.) This method is called the rule of succession, which was introduced in the 18th century by Pierre-Simon Laplace.
When relying on Jeffreys prior, the prior is
$$
\operatorname{Beta}(\alpha=\frac{1}{2}, \beta=\frac{1}{2})
$$
, which leads to the estimator:
$$
\widehat{p}_{Jeffreys} = \frac{x+\frac{1}{2}}{n+1}.
$$
When estimating with very rare events and a small (e.g.: if ), then using the standard estimator leads to
$$
\widehat{p} = 0,
$$
which sometimes is unrealistic and undesirable. In such cases there are various alternative estimators.
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When relying on Jeffreys prior, the prior is
$$
\operatorname{Beta}(\alpha=\frac{1}{2}, \beta=\frac{1}{2})
$$
, which leads to the estimator:
$$
\widehat{p}_{Jeffreys} = \frac{x+\frac{1}{2}}{n+1}.
$$
When estimating with very rare events and a small (e.g.: if ), then using the standard estimator leads to
$$
\widehat{p} = 0,
$$
which sometimes is unrealistic and undesirable. In such cases there are various alternative estimators. One way is to use the Bayes estimator
$$
\widehat{p}_b
$$
, leading to:
$$
\widehat{p}_b = \frac{1}{n+2}.
$$
Another method is to use the upper bound of the confidence interval obtained using the rule of three:
$$
\widehat{p}_{\text{rule of 3}} = \frac{3}{n}.
$$
### Confidence intervals for the parameter p
Even for quite large values of n, the actual distribution of the mean is significantly nonnormal. Because of this problem several methods to estimate confidence intervals have been proposed.
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### Confidence intervals for the parameter p
Even for quite large values of n, the actual distribution of the mean is significantly nonnormal. Because of this problem several methods to estimate confidence intervals have been proposed.
In the equations for confidence intervals below, the variables have the following meaning:
- n1 is the number of successes out of n, the total number of trials
-
$$
\widehat{p\,} = \frac{n_1}{n}
$$
is the proportion of successes
-
$$
z
$$
is the
$$
1 - \tfrac{1}{2}\alpha
$$
quantile of a standard normal distribution (i.e., probit) corresponding to the target error rate
$$
\alpha
$$
. For example, for a 95% confidence level the error
$$
\alpha
$$
= 0.05, so
$$
1 - \tfrac{1}{2}\alpha
$$
= 0.975 and
$$
z
$$
= 1.96.
#### Wald method
$$
\widehat{p\,} \pm z \sqrt{ \frac{ \widehat{p\,} ( 1 -\widehat{p\,} )}{ n } } .
$$
A continuity correction of may be added.
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For example, for a 95% confidence level the error
$$
\alpha
$$
= 0.05, so
$$
1 - \tfrac{1}{2}\alpha
$$
= 0.975 and
$$
z
$$
= 1.96.
#### Wald method
$$
\widehat{p\,} \pm z \sqrt{ \frac{ \widehat{p\,} ( 1 -\widehat{p\,} )}{ n } } .
$$
A continuity correction of may be added.
#### Agresti–Coull method
$$
\tilde{p} \pm z \sqrt{ \frac{ \tilde{p} ( 1 - \tilde{p} )}{ n + z^2 } }
$$
Here the estimate of is modified to
$$
\tilde{p}= \frac{ n_1 + \frac{1}{2} z^2}{ n + z^2 }
$$
This method works well for and . See here for
$$
n\leq 10
$$
. For use the
#### Wilson (score) method
below.
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For use the
#### Wilson (score) method
below.
#### Arcsine method
$$
\sin^2 \left(\arcsin \left(\sqrt{\widehat{p\,}}\right) \pm \frac{z}{2\sqrt{n}} \right).
$$
Wilson (score) method
The notation in the formula below differs from the previous formulas in two respects:
- Firstly, has a slightly different interpretation in the formula below: it has its ordinary meaning of 'the th quantile of the standard normal distribution', rather than being a shorthand for 'the th quantile'.
- Secondly, this formula does not use a plus-minus to define the two bounds. Instead, one may use
$$
z = z_{\alpha / 2}
$$
to get the lower bound, or use
$$
z = z_{1 - \alpha/2}
$$
to get the upper bound.
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Secondly, this formula does not use a plus-minus to define the two bounds. Instead, one may use
$$
z = z_{\alpha / 2}
$$
to get the lower bound, or use
$$
z = z_{1 - \alpha/2}
$$
to get the upper bound. For example: for a 95% confidence level the error
$$
\alpha
$$
= 0.05, so one gets the lower bound by using
$$
z = z_{\alpha/2} = z_{0.025} = - 1.96
$$
, and one gets the upper bound by using
$$
z = z_{1 - \alpha/2} = z_{0.975} = 1.96
$$
.
$$
\frac{
_BLOCK0_}{
_BLOCK1_}
$$
#### Comparison
The so-called "exact" (Clopper–Pearson) method is the most conservative. (Exact does not mean perfectly accurate; rather, it indicates that the estimates will not be less conservative than the true value.)
The Wald method, although commonly recommended in textbooks, is the most biased.
## Related distributions
### Sums of binomials
If and are independent binomial variables with the same probability , then is again a binomial variable; its distribution is :_
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## Related distributions
### Sums of binomials
If and are independent binomial variables with the same probability , then is again a binomial variable; its distribution is :_ BLOCK1_A Binomial distributed random variable can be considered as the sum of Bernoulli distributed random variables. So the sum of two Binomial distributed random variables and is equivalent to the sum of Bernoulli distributed random variables, which means . This can also be proven directly using the addition rule.
However, if and do not have the same probability , then the variance of the sum will be smaller than the variance of a binomial variable distributed as .
### Poisson binomial distribution
The binomial distribution is a special case of the Poisson binomial distribution, which is the distribution of a sum of independent non-identical Bernoulli trials .
### Ratio of two binomial distributions
This result was first derived by Katz and coauthors in 1978.
Let and be independent. Let .
Then log(T) is approximately normally distributed with mean log(p1/p2) and variance .
### Conditional binomials
If X ~ B(n, p) and Y | X ~ B(X, q) (the conditional distribution of Y, given X), then Y is a simple binomial random variable with distribution Y ~ B(n, pq).
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Then log(T) is approximately normally distributed with mean log(p1/p2) and variance .
### Conditional binomials
If X ~ B(n, p) and Y | X ~ B(X, q) (the conditional distribution of Y, given X), then Y is a simple binomial random variable with distribution Y ~ B(n, pq).
For example, imagine throwing n balls to a basket UX and taking the balls that hit and throwing them to another basket UY. If p is the probability to hit UX then X ~ B(n, p) is the number of balls that hit UX. If q is the probability to hit UY then the number of balls that hit UY is Y ~ B(X, q) and therefore Y ~ B(n, pq).
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If p is the probability to hit UX then X ~ B(n, p) is the number of balls that hit UX. If q is the probability to hit UY then the number of balls that hit UY is Y ~ B(X, q) and therefore Y ~ B(n, pq).
Since
$$
X \sim B(n, p)
$$
and
$$
Y \sim B(X, q)
$$
, by the law of total probability,
$$
\begin{align}
\Pr[Y = m] &= \sum_{k = m}^{n} \Pr[Y = m \mid X = k] \Pr[X = k] \\[2pt]
&= \sum_{k=m}^n \binom{n}{k} \binom{k}{m} p^k q^m (1-p)^{n-k} (1-q)^{k-m}
\end{align}
$$
Since
$$
\tbinom{n}{k} \tbinom{k}{m} = \tbinom{n}{m} \tbinom{n-m}{k-m},
$$
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the equation above can be expressed as
$$
\Pr[Y = m] = \sum_{k=m}^{n} \binom{n}{m} \binom{n-m}{k-m} p^k q^m (1-p)^{n-k} (1-q)^{k-m}
$$
Factoring
$$
p^k = p^m p^{k-m}
$$
and pulling all the terms that don't depend on
$$
k
$$
out of the sum now yields
$$
\begin{align}
\Pr[Y = m] &= \binom{n}{m} p^m q^m \left( \sum_{k=m}^n \binom{n-m}{k-m} p^{k-m} (1-p)^{n-k} (1-q)^{k-m} \right) \\[2pt]
&= \binom{n}{m} (pq)^m \left( \sum_{k=m}^n \binom{n-m}{k-m} \left(p(1-q)\right)^{k-m} (1-p)^{n-k} \right)
\end{align}
$$
After substituting
$$
i = k - m
$$
in the expression above, we get
$$
\Pr[Y = m] = \binom{n}{m} (pq)^m \left( \sum_{i=0}^{n-m} \binom{n-m}{i} (p - pq)^i (1-p)^{n-m - i} \right)
$$
Notice that the sum (in the parentheses) above equals
$$
(p - pq + 1 - p)^{n-m}
$$
by the binomial theorem.
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Substituting this in finally yields
$$
\begin{align}
\Pr[Y=m] &= \binom{n}{m} (pq)^m (p - pq + 1 - p)^{n-m}\\[4pt]
&= \binom{n}{m} (pq)^m (1-pq)^{n-m}
\end{align}
$$
and thus
$$
Y \sim B(n, pq)
$$
as desired.
Bernoulli distribution
The Bernoulli distribution is a special case of the binomial distribution, where . Symbolically, has the same meaning as . Conversely, any binomial distribution, , is the distribution of the sum of independent Bernoulli trials, , each with the same probability .
### Normal approximation
If is large enough, then the skew of the distribution is not too great. In this case a reasonable approximation to is given by the normal distribution
$$
\mathcal{N}(np,\,np(1-p)),
$$
and this basic approximation can be improved in a simple way by using a suitable continuity correction.
The basic approximation generally improves as increases (at least 20) and is better when is not near to 0 or 1.
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this basic approximation can be improved in a simple way by using a suitable continuity correction.
The basic approximation generally improves as increases (at least 20) and is better when is not near to 0 or 1. Various rules of thumb may be used to decide whether is large enough, and is far enough from the extremes of zero or one:
- One rule is that for the normal approximation is adequate if the absolute value of the skewness is strictly less than 0.3; that is, if
- : _ BLOCK1_This can be made precise using the Berry–Esseen theorem.
- A stronger rule states that the normal approximation is appropriate only if everything within 3 standard deviations of its mean is within the range of possible values; that is, only if
- :
$$
\mu\pm3\sigma=np\pm3\sqrt{np(1-p)}\in(0,n).
$$
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BLOCK1_This can be made precise using the Berry–Esseen theorem.
- A stronger rule states that the normal approximation is appropriate only if everything within 3 standard deviations of its mean is within the range of possible values; that is, only if
- :
$$
\mu\pm3\sigma=np\pm3\sqrt{np(1-p)}\in(0,n).
$$
This 3-standard-deviation rule is equivalent to the following conditions, which also imply the first rule above.
$$
n>9 \left(\frac{1-p}{p} \right)\quad\text{and}\quad n>9\left(\frac{p}{1-p}\right).
$$
The rule
$$
np\pm3\sqrt{np(1-p)}\in(0,n)
$$
is totally equivalent to request that
$$
np-3\sqrt{np(1-p)}>0\quad\text{and}\quad np+3\sqrt{np(1-p)}<n.
$$
Moving terms around yields:
$$
np>3\sqrt{np(1-p)}\quad\text{and}\quad n(1-p)>3\sqrt{np(1-p)}.
$$
Since
$$
0<p<1
$$
, we can apply the square power and divide by the respective factors
$$
np^2
$$
and
$$
n(1-p)^2
$$
, to obtain the desired conditions:
$$
n>9 \left(\frac{1-p}p\right) \quad\text{and}\quad n>9 \left(\frac{p}{1-p}\right).
$$
Notice that these conditions automatically imply that
$$
n>9
$$
.
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