Complex Numbers

A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the "imaginary unit" that satisfies i² = −1.

In mathematics, a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a symbol, called the imaginary unit, that satisfies the equation i² = −1. Because no real number satisfies this equation, i was called an imaginary number by René Descartes. For the complex number a + bi, a is called the real part and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols $\mathbb {C}$ or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation $(x+1)^{2}=-9$ has no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions −1 + 3i and −1 − 3i.

Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule i² = −1 combined with the associative, commutative and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field that has the real numbers as a subfield. The complex numbers form also a real vector space of dimension two, with {1, i as a standard basis.

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 expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the real line which is identified to the horizontal axis of the complex plane. The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with respect to the real axis. The complex absolute value is a Euclidean norm.

In summary, 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

An illustration of the complex number z = x + iy on the complex plane. The real part is x, and its imaginary part is y.

A complex number is a number of the form a + bi, where a and b are real numbers, and i is an indeterminate satisfying i² = −1. For example, 2 + 3i is a complex number.

This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation {{{1}}} is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials. The relation {{{1}}} induces the equalities {{{1}}} and {{{1}}} which hold for all integers k; these allow the reduction of any polynomial that results from the addition and multiplication of complex numbers to a linear polynomial in i, again of the form a + bi with real coefficients a, b.

The real number a is called the real part of the complex number a + bi; the real number b is called its imaginary part. To emphasize, the imaginary part does not include a factor i; that is, the imaginary part is b, not bi.

Formally, the complex numbers are defined as the quotient ring of the polynomial ring in the indeterminate i, by the ideal generated by the polynomial i² + 1.

Notation

A real number a can be regarded as a complex number a + 0i, whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi, whose real part is zero. As with polynomials, it is common to write a for a + 0i and bi for 0 + bi. Moreover, when the imaginary part is negative, that is, b = −|b| < 0, it is common to write a − |b|i instead of a + (−|b|)i; for example, for b = −4, 3 − 4i can be written instead of 3 + (−4)i.

Since the multiplication of the indeterminate i and a real is commutative in polynomials with real coefficients, the polynomial a + bi may be written as a + ib. This is often expedient for imaginary parts denoted by expressions, for example, when b is a radical.

The real part of a complex number z is denoted by Re(z), ${\mathcal {Re}}(z)$, or ${\mathfrak {R}}(z)$; the imaginary part of a complex number z is denoted by Im(z), ${\mathcal {Im}}(z)$, or ${\mathfrak {I}}(z).$ For example,

$$\operatorname {Re} (2+3i)=2\quad {\text{ and }}\quad \operatorname {Im} (2+3i)=3~.$$

The set of all complex numbers is denoted by $\mathbb {C}$ (blackboard bold) or C (upright bold).

In some disciplines, particularly in electromagnetism and electrical engineering, j is used instead of i as i is frequently used to represent electric current.

Visualization

A complex number z, as a point (black) and its position vector (blue)

A complex number z can thus be identified with an ordered pair $(\Re (z),\Im (z))$ of real numbers, which in turn may be interpreted as coordinates of a point in a two-dimensional space. The most immediate space is the Euclidean plane with suitable coordinates, which is then called complex plane or Argand diagram, named after Jean-Robert Argand. Another prominent space on which the coordinates may be projected is the two-dimensional surface of a sphere, which is then called Riemann sphere.

Cartesian complex plane

The definition of the complex numbers involving two arbitrary real values immediately suggests the use of Cartesian coordinates in the complex plane. The horizontal (real) axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical (imaginary) axis, with increasing values upwards.

A charted number may be viewed either as the coordinatized point or as a position vector from the origin to this point. The coordinate values of a complex number z can hence be expressed in its Cartesian, rectangular, or algebraic form.

Notably, the operations of addition and multiplication take on a very natural geometric character, when complex numbers are viewed as position vectors: addition corresponds to vector addition, while multiplication (see below) corresponds to multiplying their magnitudes and adding the angles they make with the real axis. Viewed in this way, the multiplication of a complex number by i corresponds to rotating the position vector counterclockwise by a quarter turn (90°) about the origin—a fact which can be expressed algebraically as follows:

$$(a+bi)\cdot i=ai+b(i)^{2}=-b+ai.$$

Polar complex plane

Argument φ and modulus r locate a point in the complex plane.

Modulus and argument

An alternative option for coordinates in the complex plane is the polar coordinate system that uses the distance of the point z from the origin (O), and the angle subtended between the positive real axis and the line segment Oz in a counterclockwise sense. This leads to the polar form of complex numbers.

The absolute value (or modulus or magnitude) of a complex number z = x + yi is

$$r=|z|={\sqrt {x^{2}+y^{2}}}.$$

If z is a real number (that is, if y = 0), then r = |x|. That is, the absolute value of a real number equals its absolute value as a complex number.

By Pythagoras' theorem, the absolute value of a complex number is the distance to the origin of the point representing the complex number in the complex plane.

The argument of z (in many applications referred to as the "phase" φ) is the angle of the radius Oz with the positive real axis, and is written as arg z. As with the modulus, the argument can be found from the rectangular form x + yi by applying the inverse tangent to the quotient of imaginary-by-real parts. By using a half-angle identity, a single branch of the arctan suffices to cover the range of the arg-function, (−π, π], and avoids a more subtle case-by-case analysis

$$\varphi =\arg(x+yi)={\begin{cases}2\arctan \left({\dfrac {y}{{\sqrt {x^{2}+y^{2}}}+x}}\right)&{\text{if }}x>0{\text{ or }}y\neq 0,\\\pi &{\text{if }}x<0{\text{ and }}y=0,\\{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}$$

Normally, as given above, the principal value in the interval (−π, π] is chosen. If the arg value is negative, values in the range (−π, π] or [0, 2π) can be obtained by adding 2π. The value of φ is expressed in radians in this article. It can increase by any integer multiple of 2π and still give the same angle, viewed as subtended by the rays of the positive real axis and from the origin through z. Hence, the arg function is sometimes considered as multivalued. The polar angle for the complex number 0 is indeterminate, but arbitrary choice of the polar angle 0 is common.

The value of φ equals the result of atan2:

$$\varphi =\operatorname {atan2} \left(\operatorname {Im} (z),\operatorname {Re} (z)\right).$$

Together, r and φ give another way of representing complex numbers, the polar form, as the combination of modulus and argument fully specify the position of a point on the plane. Recovering the original rectangular co-ordinates from the polar form is done by the formula called trigonometric form

$$z=r(\cos \varphi +i\sin \varphi ).$$

Using Euler's formula this can be written as

$$z=re^{i\varphi }{\text{ or }}z=r\exp i\varphi .$$

Using the cis function, this is sometimes abbreviated to

$$z=r\operatorname {\mathrm {cis} } \varphi .$$

In angle notation, often used in electronics to represent a phasor with amplitude r and phase φ, it is written as

$$z=r\angle \varphi .$$

Complex graphs

A color wheel graph of the expression ${\frac {(z^{2}-1)(z-2-i)^{2}}{z^{2}+2+2i}}$

When visualizing complex functions, both a complex input and output are needed. Because each complex number is represented in two dimensions, visually graphing a complex function would require the perception of a four dimensional space, which is possible only in projections. Because of this, other ways of visualizing complex functions have been designed.

In domain coloring the output dimensions are represented by color and brightness, respectively. Each point in the complex plane as domain is ornated, typically with color representing the argument of the complex number, and brightness representing the magnitude. Dark spots mark moduli near zero, brighter spots are farther away from the origin, the gradation may be discontinuous, but is assumed as monotonous. The colors often vary in steps of ${\frac {\pi }{3}}$ for 0 to $2\pi$ from red, yellow, green, cyan, blue, to magenta. These plots are called color wheel graphs. This provides a simple way to visualize the functions without losing information. The picture shows zeros for ±1, (2 + i) and poles at ${\sqrt {-2-2{\text{i}}}}$

Riemann surfaces are another way to visualize complex functions. Riemann surfaces can be thought of as deformations of the complex plane; while the horizontal axes represent the real and imaginary inputs, the single vertical axis only represents either the real or imaginary output. However, Riemann surfaces are built in such a way that rotating them 180 degrees shows the imaginary output, and vice versa. Unlike domain coloring, Riemann surfaces can represent multivalued functions like ${\sqrt {z}}$.

Relations and operations

Equality

Complex numbers have a similar definition of equality to real numbers; two complex numbers a₁ + b₁i and a₂ + b₂i are equal if and only if both their real and imaginary parts are equal, that is, if a₁ = a₂ and b₁ = b₂. Nonzero complex numbers written in polar form are equal if and only if they have the same magnitude and their arguments differ by an integer multiple of 2π.

Ordering

Unlike the real numbers, there is no natural ordering of the complex numbers. In particular, there is no linear ordering on the complex numbers that is compatible with addition and multiplication – the complex numbers cannot have the structure of an ordered field. This is e.g. because every non-trivial sum of squares in an ordered field is ≠ 0, and i² + 1² = 0 is a non-trivial sum of squares. Thus, complex numbers are naturally thought of as existing on a two-dimensional plane.

Conjugate

Geometric representation of z and its conjugate ${\overline {z}}$ in the complex plane

The complex conjugate of the complex number z = x + yi is given by x − yi. It is denoted by either ${\overline {z}}$ or z*. This unary operation on complex numbers cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division.

Geometrically, ${\overline {z}}$ is the "reflection" of z about the real axis. Conjugating twice gives the original complex number

$${\overline {\overline {z}}}=z,$$

which makes this operation an involution. The reflection leaves both the real part and the magnitude of z unchanged, that is

$$\operatorname {Re} ({\overline {z}})=\operatorname {Re} (z)\quad$$

and $\quad |{\overline {z}}|=|z|.$

The imaginary part and the argument of a complex number z change their sign under conjugation

$$\operatorname {Im} ({\overline {z}})=-\operatorname {Im} (z)\quad {\text{ and }}\quad \operatorname {arg} {\overline {z}}\equiv -\operatorname {arg} z{\pmod {2\pi }}.$$

For details on argument and magnitude, see the section on Polar form.

The product of a complex number {{{1}}} and its conjugate is known as the absolute square. It is always a non-negative real number and equals the square of the magnitude of each:

$$z\cdot {\overline {z}}=x^{2}+y^{2}=|z|^{2}=|{\overline {z}}|^{2}.$$

This property can be used to convert a fraction with a complex denominator to an equivalent fraction with a real denominator by expanding both numerator and denominator of the fraction by the conjugate of the given denominator. This process is sometimes called "rationalization" of the denominator (although the denominator in the final expression might be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.

The real and imaginary parts of a complex number z can be extracted using the conjugation:

$$\operatorname {Re} (z)={\dfrac {z+{\overline {z}}}{2}},\quad {\text{ and }}\quad \operatorname {Im} (z)={\dfrac {z-{\overline {z}}}{2i}}.$$

Moreover, a complex number is real if and only if it equals its own conjugate.

Conjugation distributes over the basic complex arithmetic operations:

$${\overline {z\pm w}}={\overline {z}}\pm {\overline {w}},$$

$${\overline {z\cdot w}}={\overline {z}}\cdot {\overline {w}},\quad {\overline {z/w}}={\overline {z}}/{\overline {w}}.$$

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.

Addition and subtraction

Addition of two complex numbers can be done geometrically by constructing a parallelogram.

Two complex numbers a and b are most easily added by separately adding their real and imaginary parts of the summands. That is to say:

$$a+b=(x+yi)+(u+vi)=(x+u)+(y+v)i.$$

Similarly, subtraction can be performed as

$$a-b=(x+yi)-(u+vi)=(x-u)+(y-v)i.$$

Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers a and b, interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices O, and the points of the arrows labeled a and b (provided that they are not on a line). Equivalently, calling these points A, B, respectively and the fourth point of the parallelogram X the triangles OAB and XBA are congruent. A visualization of the subtraction can be achieved by considering addition of the negative subtrahend.

Multiplication and square

The rules of the distributive property, the commutative properties (of addition and multiplication), and the defining property i² = −1 apply to complex numbers. It follows that

$$(x+yi)\,(u+vi)=(xu-yv)+(xv+yu)i.$$

In particular,

$$(x+yi)^{2}=x^{2}-y^{2}+2xyi.$$

Reciprocal and division

Using the conjugation, the reciprocal of a nonzero complex number z = x + yi can always be broken down to

$${\frac {1}{z}}={\frac {\overline {z}}{z{\overline {z}}}}={\frac {\overline {z}}{|z|^{2}}}={\frac {\overline {z}}{x^{2}+y^{2}}}={\frac {x}{x^{2}+y^{2}}}-{\frac {y}{x^{2}+y^{2}}}i,$$

since non-zero implies that x² + y² is greater than zero.

This can be used to express a division of an arbitrary complex number {{{1}}} by a non-zero complex number z as

$${\frac {w}{z}}=w\cdot {\frac {1}{z}}=(u+vi)\cdot \left({\frac {x}{x^{2}+y^{2}}}-{\frac {y}{x^{2}+y^{2}}}i\right)={\frac {(ux+vy)+(vx-uy)i}{x^{2}+y^{2}}}.$$

Multiplication and division in polar form

Multiplication of 2 + i (blue triangle) and 3 + i (red triangle). The red triangle is rotated to match the vertex of the blue one and stretched by √5, the length of the hypotenuse of the blue triangle.

Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates. Given two complex numbers z₁ = r₁(cos φ₁ + i sin φ₁) and z₂ = r₂(cos φ₂ + i sin φ₂), because of the trigonometric identities

$${\begin{alignedat}{4}\cos a\cos b&-\sin a\sin b\,&=\,&\cos(a+b){}\\\cos a\sin b&+\sin a\cos b\,&=\,&\sin(a+b).\end{alignedat}}$$

we may derive

$$z_{1}z_{2}=r_{1}r_{2}(\cos(\varphi _{1}+\varphi _{2})+i\sin(\varphi _{1}+\varphi _{2})).$$

In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. For example, multiplying by i corresponds to a quarter-turn counter-clockwise, which gives back i² = −1. The picture at the right illustrates the multiplication of

$$(2+i)(3+i)=5+5i.$$

Since the real and imaginary part of 5 + 5i are equal, the argument of that number is 45 degrees, or π/4 (in radians). 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 $\pi$.

Similarly, division is given by

$${\frac {z_{1}}{z_{2}}}={\frac {r_{1}}{r_{2}}}\left(\cos(\varphi _{1}-\varphi _{2})+i\sin(\varphi _{1}-\varphi _{2})\right).$$

Square root

The square roots of a + bi (with b ≠ 0) are $\pm (\gamma +\delta i)$, where

$$\gamma ={\sqrt {\frac {a+{\sqrt {a^{2}+b^{2}}}}{2}}}$$

and

$$\delta =(\operatorname {sgn} b){\sqrt {\frac {-a+{\sqrt {a^{2}+b^{2}}}}{2}}},$$

where sgn is the signum function. This can be seen by squaring $\pm (\gamma +\delta i)$ to obtain a + bi.

Exponential function

The exponential function $\exp \colon \mathbb {C} \to \mathbb {C} ;z\mapsto \exp z$ can be defined for every complex number z by the power series

$$\exp z=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}},$$

which has an infinite radius of convergence.

The value at 1 of the exponential function is Euler's number

$$e=\exp 1=\sum _{n=0}^{\infty }{\frac {1}{n!}}\approx 2.71828.$$

If z is real, one has $\exp z=e^{z}.$ Analytic continuation allows extending this equality for every complex value of z, and thus to define the complex exponentiation with base e as

$$e^{z}=\exp z.$$

Functional equation

The exponential function satisfies the functional equation $e^{z+t}=e^{z}e^{t}.$ This can be proved either by comparing the power series expansion of both members or by applying analytic continuation from the restriction of the equation to real arguments.

Euler's formula

Euler's formula states that, for any real number y,

$$e^{iy}=\cos y+i\sin y.$$

The functional equation implies thus that, if x and y are real, one has

$$e^{x+iy}=e^{x}(\cos y+i\sin y)=e^{x}\cos y+ie^{x}\sin y,$$

which is the decomposition of the exponential function into its real and imaginary parts.

Complex logarithm

In the real case, the natural logarithm can be defined as the inverse $\ln \colon \mathbb {R} ^{+}\to \mathbb {R} ;x\mapsto \ln x$ of the exponential function. For extending this to the complex domain, one can start from Euler's formula. It implies that, if a complex number $z\in \mathbb {C} ^{\times }$ is written in polar form

$$z=r(\cos \varphi +i\sin \varphi )$$

with $r,\varphi \in \mathbb {R} ,$ then with

$$\ln z=\ln r+i\varphi$$

as complex logarithm one has a proper inverse:

$$\exp \ln z=\exp(\ln r+i\varphi )=r\exp i\varphi =r(\cos \varphi +i\sin \varphi )=z.$$

However, because cosine and sine are periodic functions, the addition of an integer multiple of 2π to φ does not change z. For example, e^iπ = e^3iπ = −1 , so both iπ and 3iπ are possible values for the natural logarithm of −1.

Therefore, if the complex logarithm is not to be defined as a multivalued function

$$\ln z=\left\{\ln r+i(\varphi +2\pi k)\mid k\in \mathbb {Z} \right\},$$

one has to use a branch cut and to restrict the codomain, resulting in the bijective function

$$\ln \colon \;\mathbb {C} ^{\times }\;\to \;\;\;\mathbb {R} ^{+}+\;i\,\left(-\pi ,\pi \right].$$

If $z\in \mathbb {C} \setminus \left(-\mathbb {R} _{\geq 0}\right)$ 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 −π < φ < π. 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 $z\in -\mathbb {R} ^{+}$, where the principal value is ln z = ln(−z) + iπ.

Exponentiation

If x > 0 is real and z complex, the exponentiation is defined as

$$x^{z}=e^{z\ln x},$$

where ln denotes the natural logarithm.

It seems natural to extend this formula to complex values of x, but there are some difficulties resulting from the fact that the complex logarithm is not really a function, but a multivalued function.

It follows that if z is as above, and if t is another complex number, then the exponentiation is the multivalued function

$$z^{t}=\left\{e^{t\ln r}\,(\cos(\varphi t+2\pi kt)+i\sin(\varphi t+2\pi kt))\}\mid k\in \mathbb {Z} \right\}$$

Integer and fractional exponents

If, in the preceding formula, t is an integer, then the sine and the cosine are independent of k. Thus, if the exponent n is an integer, then $z^{n}$ is well defined, and the exponentiation formula simplifies to de Moivre's formula:

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

The n nth roots of a complex number z 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 0 ≤ k ≤ n − 1. (Here ${\sqrt[{n}]{r}}$ is the usual (positive) nth root of the positive real number r.) Because sine and cosine are periodic, other integer values of k do not give other values.

While the nth root of a positive real number r is chosen to be the positive real number c satisfying cⁿ = r, there is no natural way of distinguishing one particular complex nth root of a complex number. Therefore, the nth root is a n-valued function of z. This implies that, contrary to the case of positive real numbers, one has

$$(z^{n})^{1/n}\neq z,$$

since the left-hand side consists of n values, and the right-hand side is a single value.

Licensing

Content obtained and/or adapted from:

• Complex number, Wikipedia under a CC BY-SA license