Complex number

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:

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

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[15]

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

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)[14] 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[16]—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

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

Normally, as given above, the principal value in the interval (−π, π] is chosen. Values in the range [0, 2π) are obtained by adding 2π — if the value is negative. 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:

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

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

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

Using Euler's formula this can be written as

${\displaystyle z=re^{i\varphi }.}$

Using the cis function, this is sometimes abbreviated to

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

In angle notation, often used in electronics to represent a phasor with amplitude r and phase φ, it is written as[17]

${\displaystyle z=r\angle \varphi .}$

A color wheel graph of the expression (z² − 1)(z − 2 − i)²/z² + 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 π/3 for 0 to 2π 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 ±√−2 −2 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 √z.

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.

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.

Geometric representation of z and its conjugate 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 z or z*.[42] This unary operation on complex numbers cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division.

Geometrically, z is the "reflection" of z about the real axis. Conjugating twice gives the original complex number

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

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

${\displaystyle \operatorname {\mathcal {Re}} ({\overline {z}})=\operatorname {\mathcal {Re}} (z)\quad }$ and ${\displaystyle \quad |{\overline {z}}|=|z|.}$

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

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

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

The product of a complex number z = x + yi and its conjugate is known as the absolute square. It is always a positive real number and equals the square of the magnitude of each:

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

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:

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

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

Conjugation distributes over the basic complex arithmetic operations:

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

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

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 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:

${\displaystyle a+b=(x+yi)+(u+vi)=(x+u)+(y+v)i.}$

Similarly, subtraction can be performed as

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

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.

Since the real part, the imaginary part, and the indeterminate i in a complex number are all considered as numbers in themselves, two complex numbers, given as z = x + yi and w = u + vi are multiplied under the rules of the distributive property, the commutative properties and the defining property i² = −1 in the following way

${\displaystyle {\begin{aligned}z\cdot w&=(x+yi)\cdot (u+vi)&\\&=x(u+vi)+yi(u+vi)&&{\text{by the (right) distributive law}}\\&=xu+xvi+yiu+yivi&&{\text{by the (left) distributive law}}\\&=xu+yivi+xvi+yiu&&{\text{by the commutativity of addition}}\\&=xu+yvi^{2}+xvi+yui&&{\text{by the commutativity of multiplication}}\\&=(xu+yvi^{2})+(xvi+yui)&&{\text{by the associativity of addition}}\\&=(xu-yv)+(xvi+yui)&&{\text{by the defining property of }}i\\&=(xu-yv)+(xv+yu)i&&{\text{by the distributive law}}.\end{aligned}}}$

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

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

since non-zero implies that ${\displaystyle x^{2}+y^{2}}$ is greater than zero.

This can be used to express a division of an arbitrary complex number w = u + vi by a non-zero complex number z as

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

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

${\displaystyle \cos(a)\cos(b)-\sin(a)\sin(b)=\cos(a+b)}$

${\displaystyle \cos(a)\sin(b)+\sin(a)\cos(b)=\sin(a+b)}$

we may derive

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

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

${\displaystyle (2+i)(3+i)=5+5i.}$

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

${\displaystyle {\frac {\pi }{4}}=\arctan \left({\frac {1}{2}}\right)+\arctan \left({\frac {1}{3}}\right)}$

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 π.

Similarly, division is given by

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

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

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

and

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

where sgn is the signum function. This can be seen by squaring ${\displaystyle \pm (\gamma +\delta i)}$ to obtain a + bi.[43][44] Here ${\displaystyle {\sqrt {a^{2}+b^{2}}}}$ is called the modulus of a + bi, and the square root sign indicates the square root with non-negative real part, called the principal square root; also ${\displaystyle {\sqrt {a^{2}+b^{2}}}={\sqrt {z{\overline {z}}}},}$ where z = a + bi.[45]

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

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

which has an infinite radius of convergence.

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

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

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

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

The exponential function satisfies the functional equation ${\displaystyle e^{z+t}=e^{z}e^{t}.}$ 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 states that, for any real number y,

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

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

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

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

In the real case, the natural logarithm can be defined as the inverse ${\displaystyle \ln \colon \mathbb {R} ^{+}\to \mathbb {R} ;x\mapsto \ln x}$ 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 ${\displaystyle z\in \mathbb {C} ^{\times }}$ is written in polar form

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

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

${\displaystyle \ln z=\ln r+i\varphi }$

as complex logarithm one has a proper inverse:

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

However, because cosine and sine are periodic functions, the addition of an integer multiple of 2π to φ does not change z. For example, ${\displaystyle e^{i\pi }=e^{3i\pi }=-1}$, 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

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

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

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

If ${\displaystyle z\in \mathbb {C} \setminus {\bigl (}\!\!-\!\mathbb {R} _{\geq 0}{\bigr )}}$ 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 ${\displaystyle -\pi <\varphi <\pi .}$ 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 ${\displaystyle z\in -\mathbb {R} ^{+}}$, where the principal value is ${\displaystyle \ln z=\ln(-z)+i\pi .}$[lower-alpha 9]

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

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

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

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

Geometric representation of the 2nd to 6th roots of a complex number z, in polar form re^iφ  where r = |z | and φ = arg z. If z is real, φ = 0 or π. Principal roots are shown in black.

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 zⁿ is well defined, and the exponentiation formula simplifies to de Moivre's formula:

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

The n nth roots of a complex number z are given by

${\displaystyle z^{1/n}={\sqrt[{n}]{r}}\left(\cos \left({\frac {\varphi +2k\pi }{n}}\right)+i\sin \left({\frac {\varphi +2k\pi }{n}}\right)\right)}$

for 0 ≤ k ≤ n − 1. (Here ${\displaystyle {\sqrt[{n}]{r}}}$ 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

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

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

The set ℂ of complex numbers is a field.[46] Briefly, this means that the following facts hold: first, any two complex numbers can be added and multiplied to yield another complex number. Second, for any complex number z, its additive inverse −z is also a complex number; and third, every nonzero complex number has a reciprocal complex number. Moreover, these operations satisfy a number of laws, for example the law of commutativity of addition and multiplication for any two complex numbers z₁ and z₂:

${\displaystyle z_{1}+z_{2}=z_{2}+z_{1},}$

${\displaystyle z_{1}z_{2}=z_{2}z_{1}.}$

These two laws and the other requirements on a field can be proven by the formulas given above, using the fact that the real numbers themselves form a field.

Unlike the reals, ℂ is not an ordered field, that is to say, it is not possible to define a relation z₁ < z₂ that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so i² = −1 precludes the existence of an ordering on ℂ.[47]

When the underlying field for a mathematical topic or construct is the field of complex numbers, the topic's name is usually modified to reflect that fact. For example: complex analysis, complex matrix, complex polynomial, and complex Lie algebra.

Given any complex numbers (called coefficients) a₀, ..., a_n, the equation

${\displaystyle a_{n}z^{n}+\dotsb +a_{1}z+a_{0}=0}$

has at least one complex solution z, provided that at least one of the higher coefficients a₁, ..., a_n is nonzero.[7]^(VIII.1) This is the statement of the fundamental theorem of algebra, of Carl Friedrich Gauss and Jean le Rond d'Alembert. Because of this fact, ℂ is called an algebraically closed field. This property does not hold for the field of rational numbers ℚ (the polynomial x² − 2 does not have a rational root, since √2 is not a rational number) nor the real numbers ℝ (the polynomial x² + a does not have a real root for a > 0, since the square of x is positive for any real number x).

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.

Because of this fact, theorems that hold for any algebraically closed field apply to ℂ. For example, any non-empty complex square matrix has at least one (complex) eigenvalue.

The field ℂ has the following three properties:

• First, it has characteristic 0. This means that 1 + 1 + ⋯ + 1 ≠ 0 for any number of summands (all of which equal one).
• Second, its transcendence degree over ℚ, the prime field of ℂ, 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 ℂ. For example, the algebraic closure of ℚ_p also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields).[48] Also, ℂ 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 ℂ contains many proper subfields that are isomorphic to ℂ.

The preceding characterization of ℂ describes only the algebraic aspects of ℂ. 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 ℂ 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. ℂ contains a subset P (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:

• P is closed under addition, multiplication and taking inverses.
• If x and y are distinct elements of P, then either x − y or y − x is in P.
• If S is any nonempty subset of P, then S + P = x + P for some x in ℂ.

Moreover, ℂ has a nontrivial involutive automorphism x ↦ x* (namely the complex conjugation), such that x x* is in P for any nonzero x in ℂ.

Any field F with these properties can be endowed with a topology by taking the sets B(x, p) = { y | p − (y − x)(y − x)* ∈ P }  as a base, where x ranges over the field and p ranges over P. With this topology F is isomorphic as a topological field to ℂ.

The only connected locally compact topological fields are ℝ and ℂ. This gives another characterization of ℂ as a topological field, since ℂ can be distinguished from ℝ because the nonzero complex numbers are connected, while the nonzero real numbers are not.[7]^(§VIII.4)

William Rowan Hamilton introduced the approach to define the set ℂ of complex numbers[49] as the set ℝ² of ordered pairs (a, b) of real numbers, in which the following rules for addition and multiplication are imposed:[46]

${\displaystyle {\begin{aligned}(a,b)+(c,d)&=(a+c,b+d)\\(a,b)\cdot (c,d)&=(ac-bd,bc+ad).\end{aligned}}}$

It is then just a matter of notation to express (a, b) as a + bi.

Though this low-level construction does accurately describe the structure of the complex numbers, the following equivalent definition reveals the algebraic nature of ℂ more immediately. This characterization relies on the notion of fields and polynomials. A field is a set endowed with addition, subtraction, multiplication and division operations that behave as is familiar from, say, rational numbers. For example, the distributive law

${\displaystyle (x+y)z=xz+yz}$

must hold for any three elements x, y and z of a field. The set ℝ of real numbers does form a field. A polynomial p(X) with real coefficients is an expression of the form

${\displaystyle a_{n}X^{n}+\dotsb +a_{1}X+a_{0},}$

where the a₀, ..., a_n are real numbers. The usual addition and multiplication of polynomials endows the set ℝ[X] of all such polynomials with a ring structure. This ring is called the polynomial ring over the real numbers.

The set of complex numbers is defined as the quotient ring ℝ[X]/(X ² + 1).[7]^(§VIII.1) This extension field contains two square roots of −1, namely (the cosets of) X and −X, respectively. (The cosets of) 1 and X form a basis of ℝ[X]/(X ² + 1) as a real vector space, which means that each element of the extension field can be uniquely written as a linear combination in these two elements. Equivalently, elements of the extension field can be written as ordered pairs (a, b) of real numbers. The quotient ring is a field, because X² + 1 is irreducible over ℝ, so the ideal it generates is maximal.

The formulas for addition and multiplication in the ring ℝ[X], modulo the relation X² = −1, correspond to the formulas for addition and multiplication of complex numbers defined as ordered pairs. So the two definitions of the field ℂ are isomorphic (as fields).

Accepting that ℂ is algebraically closed, since it is an algebraic extension of ℝ in this approach, ℂ is therefore the algebraic closure of ℝ.

Complex numbers a + bi can also be represented by 2 × 2 matrices that have the following form:

${\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}}$

Here the entries a and b are real numbers. The sum and product of two such matrices is again of this form, and the sum and product of complex numbers corresponds to the sum and product of such matrices, the product being:

${\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}{\begin{pmatrix}c&-d\\d&\;\;c\end{pmatrix}}={\begin{pmatrix}ac-bd&-ad-bc\\bc+ad&\;\;-bd+ac\end{pmatrix}}}$

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. Moreover, the square of the absolute value of a complex number expressed as a matrix is equal to the determinant of that matrix:

${\displaystyle |z|^{2}={\begin{vmatrix}a&-b\\b&a\end{vmatrix}}=a^{2}+b^{2}.}$

The conjugate z corresponds to the transpose of the matrix.

Though this representation of complex numbers with matrices is the most common, many other representations arise from matrices other than ${\displaystyle {\bigl (}{\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}}{\bigr )}}$ that square to the negative of the identity matrix. See the article on 2 × 2 real matrices for other representations of complex numbers.

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, ℂ, endowed with the metric

${\displaystyle \operatorname {d} (z_{1},z_{2})=|z_{1}-z_{2}|}$

is a complete metric space, which notably includes the triangle inequality

${\displaystyle |z_{1}+z_{2}|\leq |z_{1}|+|z_{2}|}$

for any two complex numbers z₁ and z₂.

Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the exponential function exp z, also written e^z, is defined as the infinite series

${\displaystyle \exp z:=1+z+{\frac {z^{2}}{2\cdot 1}}+{\frac {z^{3}}{3\cdot 2\cdot 1}}+\cdots =\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.}$

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.

Euler's formula states:

${\displaystyle \exp(i\varphi )=\cos \varphi +i\sin \varphi }$

for any real number φ, in particular

${\displaystyle \exp(i\pi )=-1}$

Unlike in the situation of real numbers, there is an infinitude of complex solutions z of the equation

${\displaystyle \exp z=w}$

for any complex number w ≠ 0. It can be shown that any such solution z – called complex logarithm of w – satisfies

${\displaystyle \log w=\ln |w|+i\arg w,}$

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 2π, log is also multivalued. The principal value of log is often taken by restricting the imaginary part to the interval (−π, π].

Complex exponentiation z^ω is defined as

${\displaystyle z^{\omega }=\exp(\omega \log z),}$

and is multi-valued, except when ω is an integer. For ω = 1 / n, for some natural number n, this recovers the non-uniqueness of nth roots mentioned above.

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

${\displaystyle a^{bc}=\left(a^{b}\right)^{c}.}$

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.

A function f : ℂ → ℂ is called holomorphic if it satisfies the Cauchy–Riemann equations. For example, any ℝ-linear map ℂ → ℂ can be written in the form

${\displaystyle f(z)=az+b{\overline {z}}}$

with complex coefficients a and b. This map is holomorphic if and only if b = 0. The second summand ${\displaystyle b{\overline {z}}}$ is real-differentiable, but does not satisfy the Cauchy–Riemann equations.

Complex analysis shows some features not apparent in real analysis. For example, any two holomorphic functions f and g that agree on an arbitrarily small open subset of ℂ necessarily agree everywhere. Meromorphic functions, functions that can locally be written as f(z)/(z − z₀)ⁿ with a holomorphic function f, still share some of the features of holomorphic functions. Other functions have essential singularities, such as sin(1/z) at z = 0.

Three non-collinear points ${\displaystyle u,v,w}$ in the plane determine the shape of the triangle ${\displaystyle \{u,v,w\}}$. Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic as

${\displaystyle S(u,v,w)={\frac {u-w}{u-v}}.}$

The shape ${\displaystyle S}$ 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 ${\displaystyle \{u,v,w\}}$ is in a similarity class of triangles with the same shape.[50]

The Mandelbrot set with the real and imaginary axes labeled.

The Mandelbrot set is a popular example of a fractal formed on the complex plane. It is defined by plotting every location ${\displaystyle c}$ where iterating the sequence ${\displaystyle f_{c}(z)=z^{2}+c}$ does not diverge when iterated infinitely. Similarly, Julia sets have the same rules, except where ${\displaystyle c}$ remains constant.

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:[51][52] Denote the triangle's vertices in the complex plane as a = x_A + y_Ai, b = x_B + y_Bi, and c = x_C + y_Ci. Write the cubic equation ${\displaystyle (x-a)(x-b)(x-c)=0}$, 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.

Construction of a regular pentagon using straightedge and compass.

As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in 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. Compared to ℚ, the algebraic closure of ℚ, which also contains all algebraic numbers, 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 are Gaussian integers, that is, numbers of the form x + iy, where x and y are integers, which can be used to classify sums of squares.

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 ζ(s) is related to the distribution of prime numbers.

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.

In differential equations, it is common to first find all complex roots r 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 f(t) = e^rt. Likewise, in difference equations, the complex roots r 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 f(t) = r^t.

In control theory, systems are often transformed from the time domain to the 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. 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.

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 |z| of the corresponding z is the amplitude and the argument arg z 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

${\displaystyle x(t)=\operatorname {\mathcal {Re}} \{X(t)\}}$

and

${\displaystyle X(t)=Ae^{i\omega t}=ae^{i\phi }e^{i\omega t}=ae^{i(\omega t+\phi )}}$

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 utilize 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:

${\displaystyle {\begin{aligned}\cos((\omega +\alpha )t)+\cos \left((\omega -\alpha )t\right)&=\operatorname {\mathcal {Re}} \left(e^{i(\omega +\alpha )t}+e^{i(\omega -\alpha )t}\right)\\&=\operatorname {\mathcal {Re}} \left(\left(e^{i\alpha t}+e^{-i\alpha t}\right)\cdot e^{i\omega t}\right)\\&=\operatorname {\mathcal {Re}} \left(2\cos(\alpha t)\cdot e^{i\omega t}\right)\\&=2\cos(\alpha t)\cdot \operatorname {\mathcal {Re}} \left(e^{i\omega t}\right)\\&=2\cos(\alpha t)\cdot \cos \left(\omega t\right).\end{aligned}}}$

In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. 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 j, to avoid confusion with I, which is generally in use to denote electric current, or, more particularly, i, which is generally in use to denote instantaneous electric current.

Since the voltage in an AC circuit is oscillating, it can be represented as

${\displaystyle V(t)=V_{0}e^{j\omega t}=V_{0}\left(\cos \omega t+j\sin \omega t\right),}$

To obtain the measurable quantity, the real part is taken:

${\displaystyle v(t)=\operatorname {\mathcal {Re}} (V)=\operatorname {\mathcal {Re}} \left[V_{0}e^{j\omega t}\right]=V_{0}\cos \omega t.}$

The complex-valued signal V(t) is called the analytic representation of the real-valued, measurable signal v(t). [53]

In fluid dynamics, complex functions are used to describe potential flow in two dimensions.

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.

In special 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.