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 Complex number
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Complex number
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# Complex number

In mathematics, a complex number is a number of the form

$a + bi \,$

where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. The real number a is called the real part of the complex number, and the real number b is the imaginary part. When the imaginary part b is 0, the complex number is just the real number a.

For example, 3 + 2i is a complex number, with real part 3 and imaginary part 2.

Complex numbers can be added, subtracted, multiplied, and divided like real numbers, but they have additional elegant properties. For example, real numbers alone do not provide a solution for every polynomial algebraic equation with real coefficients, while complex numbers do (the fundamental theorem of algebra).

In some fields (in particular, electrical engineering, where i is a symbol for current), complex numbers are written as a + bj.

## Contents

• 1 Definitions
• 1.1 Equality
• 1.2 Notation and operations
• 1.3 The complex number field
• 1.4 The complex plane
• 1.5 Absolute value, conjugation and distance
• 1.6 Complex fractions
• 1.7 Matrix representation of complex numbers
• 2 Geometric interpretation of the operations on complex numbers
• 2.2 Multiplication
• 2.3 Conjugation
• 3 Some properties
• 3.1 Real vector space
• 3.2 Solutions of polynomial equations
• 3.3 Algebraic characterization
• 3.4 Characterization as a topological field
• 4 Complex analysis
• 5 Applications
• 5.1 Control theory
• 5.2 Signal analysis
• 5.3 Frequency (spectral) domain electromagnetism
• 5.3.1 Sign convention
• 5.4 Improper integrals
• 5.5 Quantum mechanics
• 5.6 Relativity
• 5.7 Applied mathematics
• 5.8 Fluid dynamics
• 5.9 Fractals
• 6 History

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## Definitions

### Equality

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. That is, a + bi = c + di if and only if a = c and b = d.

### Notation and operations

The set of all complex numbers is usually denoted by C, or in blackboard bold by $\mathbb{C}$. The real numbers, R, may be regarded as "lying in" C by considering every real number as a complex: a = a + 0i.

Complex numbers are added, subtracted, and multiplied by formally applying the associative, commutative and distributive laws of algebra, together with the equation i 2 = −1:

$\,(a + bi) + (c + di) = (a + c) + (b + d)i$
$\,(a + bi) - (c + di) = (a - c) + (b - d)i$
$\,(a + bi)(c + di) = ac + bci + adi + bd i^2 = (ac - bd) + (bc + ad)i$

Division of complex numbers can also be defined (see below). Thus, the set of complex numbers forms a field which, in contrast to the real numbers, is algebraically closed.

In mathematics, the adjective "complex" means that the field of complex numbers is the underlying number field considered, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra.

### The complex number field

Formally, the complex numbers can be defined as ordered pairs of real numbers (a, b) together with the operations:

$(a,b) + (c,d) = (a + c,b + d) \,$
$(a,b) \cdot (c,d) = (ac - bd,bc + ad). \,$

So defined, the complex numbers form a field, the complex number field, denoted by C.

Since a complex number a + bi is uniquely specified by an ordered pair (a, b) of real numbers, the complex numbers are in one-to-one correspondence with points on a plane, called the complex plane.

We identify the real number a with the complex number (a, 0), and in this way the field of real numbers R becomes a subfield of C. The imaginary unit i is the complex number (0, 1).

In C, we have:

• additive identity ("zero"): (0, 0)
• multiplicative identity ("one"): (1, 0)
• additive inverse of (a,b): (−a, −b)
• multiplicative inverse (reciprocal) of non-zero (a, b): $\left({a\over a^2+b^2},{-b\over a^2+b^2}\right).$

C can also be defined as the topological closure of the algebraic numbers or as the algebraic closure of R, both of which are described below.

### The complex plane

A complex number can be viewed as a point or a position vector on a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (named after Jean-Robert Argand).

The Cartesian coordinates of the complex number are the real part x and the imaginary part y, while the polar coordinates are r = |z|, called the absolute value or modulus, and φ = arg(z), called the complex argument of z (mod-arg form). Together with Euler's formula we have

$z = x + iy = r (\cos \varphi + i\sin \varphi ) = r e^{i \varphi}. \,$

The notation cis φ is sometimes used for cos φ + i sin φ.

The complex argument of 0 is not defined by the equations above. There are two possible approaches for this case. The first is to consider arg(0) an undefined form, just like 0/0. The other is to choose some fixed value and to define arg(0) to have that value. For this approach, a conventional choice is to set arg(0) = 0.

Note that for a non-zero complex number the complex argument is unique modulo 2π, that is, if any two values of the complex argument exactly differ by an integer multiple of 2π, they are considered equivalent.

By simple trigonometric identities, we see that

$r_1 e^{i\varphi_1} \cdot r_2 e^{i\varphi_2} = r_1 r_2 e^{i(\varphi_1 + \varphi_2)} \,$

and that

$\frac{r_1 e^{i\varphi_1}} {r_2 e^{i\varphi_2}} = \frac{r_1}{r_2} e^{i (\varphi_1 - \varphi_2)}. \,$

Now the addition of two complex numbers is just the vector addition of two vectors, and the multiplication with a fixed complex number can be seen as a simultaneous rotation and stretching.

Multiplication with i corresponds to a counter clockwise rotation by 90 degrees (π / 2 radians). The geometric content of the equation i2 = −1 is that a sequence of two 90 degree rotations results in a 180 degree (π radians) rotation. Even the fact (−1) · (−1) = +1 from arithmetic can be understood geometrically as the combination of two 180 degree turns.

### Absolute value, conjugation and distance

The absolute value (or modulus or magnitude) of a complex number z = r eiφ is defined as |z| = r. Algebraically, if z = a + ib, then $| z | = \sqrt{a^2+b^2}.$

One can check readily that the absolute value has three important properties:

$| z | = 0 \,$ iff $z = 0 \,$
$| z + w | \leq | z | + | w | \,$ (triangle inequality)
$| z w | = | z | \; | w | \,$

for all complex numbers z and w. It then follows, for example, that | 1 | = 1 and | z / w | = | z | / | w | . By defining the distance function d(z, w) = |zw| we turn the complex numbers into a metric space and we can therefore talk about limits and continuity. The addition, subtraction, multiplication and division of complex numbers are then continuous operations. Unless anything else is said, this is always the metric being used on the complex numbers.

The complex conjugate of the complex number z = a + ib is defined to be a - ib, written as $\bar{z}$ or $z^*\,$. As seen in the figure, $\bar{z}$ is the "reflection" of z about the real axis. The following can be checked:

$\overline{z+w} = \bar{z} + \bar{w}$
$\overline{zw} = \bar{z}\bar{w}$
$\overline{(z/w)} = \bar{z}/\bar{w}$
$\bar{\bar{z}}=z$
$\bar{z}=z$   if and only if z is real
$|z|=|\bar{z}|$
$|z|^2 = z\bar{z}$
$z^{-1} = \bar{z}|z|^{-2}$   if z is non-zero.

The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.

That conjugation commutes with all the algebraic operations (and many functions; e.g. $\sin\bar z=\overline{\sin z}$) is rooted in the ambiguity in choice of i (−1 has two square roots); note, however, that conjugation is not differentiable (see holomorphic).

### Complex fractions

Given a complex number (a + bi) which is to be divided by another complex number (c + di) whose magnitude is non-zero, there are two ways to do this; in either case it is the same as multiplying the first by the multiplicative inverse of the second. The first way has already been implied: to convert both complex numbers into exponential form, from which their quotient is easy to derive. The second way is to express the division as a fraction, then to multiply both numerator and denominator by the complex conjugate of the denominator. This causes the denominator to simplify into a real number:

${a + bi \over c + di} = {(a + bi) (c - di) \over (c + di) (c - di)} = {(ac + bd) + (bc - ad) i \over c^2 + d^2}$
$= \left({ac + bd \over c^2 + d^2}\right) + i\left( {bc - ad \over c^2 + d^2} \right).$

### Matrix representation of complex numbers

While usually not useful, alternative representations of complex fields can give some insight into their nature. One particularly elegant representation interprets every complex number as 2×2 matrix with real entries which stretches and rotates the points of the plane. Every such matrix has the form

$\begin{bmatrix} a & -b \\ b & \;\; a \end{bmatrix}$

with real numbers a and b. The sum and product of two such matrices is again of this form. Every non-zero such matrix is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a field. In fact, this is exactly the field of complex numbers. Every such matrix can be written as

$\begin{bmatrix} a & -b \\ b & \;\; a \end{bmatrix} = a \begin{bmatrix} 1 & \;\; 0 \\ 0 & \;\; 1 \end{bmatrix} + b \begin{bmatrix} 0 & -1 \\ 1 & \;\; 0 \end{bmatrix}$

which suggests that we should identify the real number 1 with the matrix

$\begin{bmatrix} 1 & \;\; 0 \\ 0 & \;\; 1 \end{bmatrix}$

and the imaginary unit i with

$\begin{bmatrix} 0 & -1 \\ 1 & \;\; 0 \end{bmatrix}$

a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeed equal to −1.

The absolute value of a complex number expressed as a matrix is equal to the square root of the determinant of that matrix. If the matrix is viewed as a transformation of a plane, then the transformation rotates points through an angle equal to the argument of the complex number and scales by a factor equal to the complex number's absolute value. The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be described by the transpose of the matrix corresponding to z.

If the matrix elements are themselves complex numbers, then the resulting algebra is that of the quaternions. In this way, the matrix representation can be seen as a way of expressing the Cayley-Dickson construction of algebras.