Complex number

Addition and subtraction

Two complex numbers ${\displaystyle a=x+yi}$ and ${\displaystyle b=u+vi}$ are added by separately adding their real and imaginary parts.^[13] That is to say:

$${\displaystyle a+b=(x+yi)+(u+vi)=(x+u)+(y+v)i.}$$ Similarly, subtraction can be performed as $${\displaystyle a-b=(x+yi)-(u+vi)=(x-u)+(y-v)i.}$$

The addition can be geometrically visualized as follows: 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.

Multiplication

The product of two complex numbers is computed as follows:^[13]

${\displaystyle (a+bi)\cdot (c+di)=ac-bd+(ad+bc)i.}$

For example, ${\displaystyle (2-i)(3+4i)=2\cdot 3-((-1)\cdot 4)+(2\cdot 4+(-1)\cdot 3)i=10+5i.}$ In particular, this includes as a special case the fundamental formula

${\displaystyle i^{2}=i\cdot i=-1.}$

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.^[14]

Complex conjugate, absolute value, argument and division

The complex conjugate of the complex number z = x + yi is defined as ${\displaystyle {\overline {z}}=x-yi.}$^[15] It is also denoted by some authors by ${\displaystyle z^{*}}$. Geometrically, z is the "reflection" of z about the real axis. Conjugating twice gives the original complex number: ${\displaystyle {\overline {\overline {z}}}=z.}$ 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 z = x + yi , the product

${\displaystyle z\cdot {\overline {z}}=(x+iy)(x-iy)=x^{2}+y^{2}}$

is a non-negative real number. This allows to define the absolute value (or modulus or magnitude) of z to be the square root^[16] $${\displaystyle |z|={\sqrt {x^{2}+y^{2}}}.}$$ By Pythagoras' theorem, ${\displaystyle |z|}$ 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 ${\displaystyle |z|=1}$, known as the unit complex numbers. If ${\displaystyle z=x=x+0i}$ is a real number, then ${\displaystyle |z|=|x|}$: its absolute value as a complex number and as a real number are equal.

Using the conjugate, the reciprocal of a nonzero complex number ${\displaystyle z=x+yi}$ can be computed to be^[13]

$${\displaystyle {\frac {1}{z}}={\frac {\bar {z}}{z{\bar {z}}}}={\frac {\bar {z}}{|z|^{2}}}={\frac {x-yi}{x^{2}+y^{2}}}={\frac {x}{x^{2}+y^{2}}}-{\frac {y}{x^{2}+y^{2}}}i.}$$ More generally, the division of an arbitrary complex number ${\displaystyle w=u+vi}$ by a non-zero complex number ${\displaystyle z=x+yi}$ equals $${\displaystyle {\frac {w}{z}}={\frac {w{\bar {z}}}{|z|^{2}}}={\frac {(u+vi)(x-iy)}{x^{2}+y^{2}}}={\frac {ux+vy}{x^{2}+y^{2}}}+{\frac {vx-uy}{x^{2}+y^{2}}}i.}$$ 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.^[17][18]

The argument of z (sometimes called the "phase" φ)^[10] is the angle of the radius Oz with the positive real axis, and is written as arg z, expressed in radians in this article. The angle is defined only up to adding integer multiples of ${\displaystyle 2\pi }$, since a rotation by ${\displaystyle 2\pi }$ (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 ${\displaystyle (-\pi ,\pi ]}$, which is referred to as the principal value.^[19] The argument can be computed from the rectangular form x + yi by means of the arctan (inverse tangent) function.^[20]

Polar form

For any complex number z, with absolute value ${\displaystyle r=|z|}$ and argument ${\displaystyle \varphi }$, the equation^[13]

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

holds. This identity is referred to as the polar form of z. It is sometimes abbreviated as ${\textstyle z=r\operatorname {\mathrm {cis} } \varphi }$. In electronics, one represents a phasor with amplitude r and phase φ in angle notation:^[21]$${\displaystyle z=r\angle \varphi .}$$

If two complex numbers are given in polar form, i.e., z₁ = r₁(cos φ₁ + i sin φ₁) and z₂ = r₂(cos φ₂ + i sin φ₂), the product and division can be computed as $${\displaystyle z_{1}z_{2}=r_{1}r_{2}(\cos(\varphi _{1}+\varphi _{2})+i\sin(\varphi _{1}+\varphi _{2})).}$$ $${\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {r_{1}}{r_{2}}}\left(\cos(\varphi _{1}-\varphi _{2})+i\sin(\varphi _{1}-\varphi _{2})\right),{\text{if }}z_{2}\neq 0.}$$ (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 $${\displaystyle (2+i)(3+i)=5+5i.}$$ Because 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:^[22] $${\displaystyle {\frac {\pi }{4}}=4\arctan \left({\frac {1}{5}}\right)-\arctan \left({\frac {1}{239}}\right)}$$

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:^[23] $${\displaystyle z^{n}=\underbrace {z\cdot \dots \cdot z} _{n{\text{ factors}}}=(r(\cos \varphi +i\sin \varphi ))^{n}=r^{n}\,(\cos n\varphi +i\sin n\varphi ).}$$ For example, the first few powers of the imaginary unit i are ${\displaystyle i,i^{2}=-1,i^{3}=-i,i^{4}=1,i^{5}=i,\dots }$.

The n nth roots of a complex number z are given by^[24] $${\displaystyle z^{1/n}={\sqrt[{n}]{r}}\left(\cos \left({\frac {\varphi +2k\pi }{n}}\right)+i\sin \left({\frac {\varphi +2k\pi }{n}}\right)\right)}$$ 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. For any ${\displaystyle z\neq 0}$, there are, in particular n distinct complex n-th roots. For example, there are 4 fourth roots of 1, namely

${\displaystyle z_{1}=1,z_{2}=i,z_{3}=-1,z_{4}=-i.}$

In general there is no natural way of distinguishing one particular complex nth 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 nth root is an n-valued function of z.

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) 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.^[25] This property does not hold for the field of rational numbers ${\displaystyle \mathbb {Q} }$ (the polynomial x² − 2 does not have a rational root, because √2 is not a rational number) nor the real numbers ${\displaystyle \mathbb {R} }$ (the polynomial x² + 4 does not have a real root, because the square of x is positive for any real number x).

Because of this fact, ${\displaystyle \mathbb {C} }$ 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.

The field of complex numbers is defined as the (unique) algebraic extension field of the real numbers later in #Abstract algebraic definitions.

Construction as a quotient ring

One approach to ${\displaystyle \mathbb {C} }$ is via polynomials, i.e., expressions of the form $${\displaystyle p(X)=a_{n}X^{n}+\dotsb +a_{1}X+a_{0},}$$ where the coefficients a₀, ..., a_n are real numbers. The set of all such polynomials is denoted by ${\displaystyle \mathbb {R} [X]}$. Since sums and products of polynomials are again polynomials, this set ${\displaystyle \mathbb {R} [X]}$ forms a commutative ring, called the polynomial ring (over the reals). To every such polynomial p, one may assign the complex number ${\displaystyle p(i)=a_{n}i^{n}+\dotsb +a_{1}i+a_{0}}$, i.e., the value obtained by setting ${\displaystyle X=i}$. This defines a function

${\displaystyle \mathbb {R} [X]\to \mathbb {C} }$

This function is surjective since every complex number can be obtained in such a way: the evaluation of a linear polynomial ${\displaystyle a+bX}$ at ${\displaystyle X=i}$ is ${\displaystyle a+bi}$. However, the evaluation of polynomial ${\displaystyle X^{2}+1}$ at i is 0, since ${\displaystyle i^{2}+1=0.}$ 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

${\displaystyle \mathbb {R} [X]/(X^{2}+1){\stackrel {\cong }{\to }}\mathbb {C} }$

between the quotient ring and ${\displaystyle \mathbb {C} }$. Some authors take this as the definition of ${\displaystyle \mathbb {C} }$.^[66][67] This definition expresses ${\displaystyle \mathbb {C} }$ as a quadratic algebra.

The field ${\displaystyle \mathbb {C} }$ is algebraically closed by the fundamental theorem of algebra, and is therefore the algebraic closure of ${\displaystyle \mathbb {R} .}$

Matrix representation of complex numbers

Complex numbers a + bi can also be represented by 2 × 2 matrices that have the form^[68] $${\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}.}$$ Here the entries a and b 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 2 × 2 matrices.

A simple computation shows that the map $${\displaystyle a+ib\mapsto {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}}$$ 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 polar form representation of complex numbers explicitly gives these matrices as scaled rotation matrices. $${\displaystyle r(\cos \theta +i\sin \theta )\mapsto {\begin{pmatrix}r\cos \theta &-r\sin \theta \\r\sin \theta &\;\;r\cos \theta \end{pmatrix}}}$$ In particular, the case of r = 1, which is ${\displaystyle |a+ib|={\sqrt {a^{2}+b^{2}}}=1}$, gives (unscaled) rotation matrices.

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, ${\displaystyle \mathbb {C} }$, 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.

Complex exponential

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, which can be shown to converge for any z: $${\displaystyle \exp z:=1+z+{\frac {z^{2}}{2\cdot 1}}+{\frac {z^{3}}{3\cdot 2\cdot 1}}+\cdots =\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.}$$ For example, ${\displaystyle \exp(1)}$ is Euler's number ${\displaystyle e\approx 2.718}$. Euler's formula states: $${\displaystyle \exp(i\varphi )=\cos \varphi +i\sin \varphi }$$ for any real number φ. 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 $${\displaystyle \exp(i\pi )=-1.}$$

Complex logarithm

For any positive real number t, there is a unique real number x such that ${\displaystyle \exp(x)=t}$. This leads to the definition of the natural logarithm as the inverse ${\displaystyle \ln \colon \mathbb {R} ^{+}\to \mathbb {R} ;x\mapsto \ln x}$ of the exponential function. The situation is different for complex numbers, since

${\displaystyle \exp(z+2\pi i)=\exp z\exp(2\pi i)=\exp z}$

by the functional equation and Euler's identity. For example, e^iπ = e^3iπ = −1 , so both iπ and 3iπ are possible values for the complex logarithm of −1.

In general, given any non-zero complex number w, any number z solving the equation

${\displaystyle \exp z=w}$

is called a complex logarithm of w, denoted ${\displaystyle \log w}$. It can be shown that these numbers satisfy^[69] $${\displaystyle z=\log w=\ln |w|+i\arg w,}$$ where ${\displaystyle \arg }$ is the argument defined above, and ${\displaystyle \ln }$ 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]. This leads to the complex logarithm being a bijective function taking values in the strip ${\displaystyle \mathbb {R} ^{+}+\;i\,\left(-\pi ,\pi \right]}$ (that is denoted ${\displaystyle S_{0}}$ in the above illustration) $${\displaystyle \ln \colon \;\mathbb {C} ^{\times }\;\to \;\;\;\mathbb {R} ^{+}+\;i\,\left(-\pi ,\pi \right].}$$

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

Complex exponentiation z^ω is defined as $${\displaystyle z^{\omega }=\exp(\omega \ln z),}$$ 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. If z > 0 is real (and ω an arbitrary complex number), one has a preferred choice of ${\displaystyle \ln x}$, 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 $${\displaystyle a^{bc}=\left(a^{b}\right)^{c}.}$$ 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.

Complex sine and cosine

The series defining the real trigonometric functions sin and cos, 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 tan, 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.

The value of a trigonometric or hyperbolic function of a complex number can be expressed in terms of those functions evaluated on real numbers, via angle-addition formulas. For z = x + iy,

$${\displaystyle \sin {z}=\sin {x}\cosh {y}+i\cos {x}\sinh {y}}$$

$${\displaystyle \cos {z}=\cos {x}\cosh {y}-i\sin {x}\sinh {y}}$$

$${\displaystyle \tan {z}={\frac {\tan {x}+i\tanh {y}}{1-i\tan {x}\tanh {y}}}}$$

$${\displaystyle \cot {z}=-{\frac {1+i\cot {x}\coth {y}}{\cot {x}-i\coth {y}}}}$$

$${\displaystyle \sinh {z}=\sinh {x}\cos {y}+i\cosh {x}\sin {y}}$$

$${\displaystyle \cosh {z}=\cosh {x}\cos {y}+i\sinh {x}\sin {y}}$$

$${\displaystyle \tanh {z}={\frac {\tanh {x}+i\tan {y}}{1+i\tanh {x}\tan {y}}}}$$

$${\displaystyle \coth {z}={\frac {1-i\coth {x}\cot {y}}{\coth {x}-i\cot {y}}}}$$

Where these expressions are not well defined, because a trigonometric or hyperbolic function evaluates to infinity or there is division by zero, they are nonetheless correct as limits.

Holomorphic functions

A function ${\displaystyle f:\mathbb {C} }$ → ${\displaystyle \mathbb {C} }$ is called holomorphic or complex differentiable at a point ${\displaystyle z_{0}}$ if the limit

${\displaystyle \lim _{z\to z_{0}}{f(z)-f(z_{0}) \over z-z_{0}}}$

exists (in which case it is denoted by ${\displaystyle f'(z_{0})}$). This mimics the definition for real differentiable functions, except that all quantities are complex numbers. Loosely speaking, the freedom of approaching ${\displaystyle z_{0}}$ in different directions imposes a much stronger condition than being (real) differentiable. For example, the function

${\displaystyle f(z)={\overline {z}}}$

is differentiable as a function ${\displaystyle \mathbb {R} ^{2}\to \mathbb {R} ^{2}}$, 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

${\displaystyle {\frac {\partial f}{\partial {\overline {z}}}}=0.}$

Complex analysis shows some features not apparent in real analysis. For example, the identity theorem asserts that two holomorphic functions f and g agree if they agree on an arbitrarily small open subset of ${\displaystyle \mathbb {C} }$. 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.

Geometry

Fractal geometry

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.

Algebraic number theory

As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in ${\displaystyle \mathbb {C} }$. 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. Compared to ${\displaystyle {\overline {\mathbb {Q} }}}$, the algebraic closure of ${\displaystyle \mathbb {Q} }$, which also contains all algebraic numbers, ${\displaystyle \mathbb {C} }$ 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 x + iy, where x and y 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 ζ(s) 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 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.

Linear algebra

Since ${\displaystyle \mathbb {C} }$ 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

In physics

Algebraic characterization

The field ${\displaystyle \mathbb {C} }$ 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 ${\displaystyle \mathbb {Q} }$, the prime field of ${\displaystyle \mathbb {C} ,}$ 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 ${\displaystyle \mathbb {C} .}$ For example, the algebraic closure of the field ${\displaystyle \mathbb {Q} _{p}}$ of the p-adic number also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields).^[76] Also, ${\displaystyle \mathbb {C} }$ 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 ${\displaystyle \mathbb {C} }$ contains many proper subfields that are isomorphic to ${\displaystyle \mathbb {C} }$.

Characterization as a topological field

The preceding characterization of ${\displaystyle \mathbb {C} }$ describes only the algebraic aspects of ${\displaystyle \mathbb {C} .}$ 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 ${\displaystyle \mathbb {C} }$ 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. ${\displaystyle \mathbb {C} }$ 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 ${\displaystyle \mathbb {C} .}$

Moreover, ${\displaystyle \mathbb {C} }$ has a nontrivial involutive automorphism x ↦ x* (namely complex conjugation), such that x x* is in P for any nonzero x in ${\displaystyle \mathbb {C} .}$

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 ${\displaystyle \mathbb {C} .}$

The only connected locally compact topological fields are ${\displaystyle \mathbb {R} }$ and ${\displaystyle \mathbb {C} .}$ This gives another characterization of ${\displaystyle \mathbb {C} }$ as a topological field, because ${\displaystyle \mathbb {C} }$ can be distinguished from ${\displaystyle \mathbb {R} }$ because the nonzero complex numbers are connected, while the nonzero real numbers are not.^[77]

Other number systems

Number systems

	rational numbers ${\displaystyle \mathbb {Q} }$	real numbers ${\displaystyle \mathbb {R} }$	complex numbers ${\displaystyle \mathbb {C} }$	quaternions ${\displaystyle \mathbb {H} }$	octonions ${\displaystyle \mathbb {O} }$	sedenions ${\displaystyle \mathbb {S} }$
complete	No	Yes	Yes	Yes	Yes	Yes
dimension as an ${\displaystyle \mathbb {R} }$-vector space	[does not apply]	1	2	4	8	16
ordered	Yes	Yes	No	No	No	No
multiplication commutative (${\displaystyle xy=yx}$)	Yes	Yes	Yes	No	No	No
multiplication associative (${\displaystyle (xy)z=x(yz)}$)	Yes	Yes	Yes	Yes	No	No
normed division algebra (over ${\displaystyle \mathbb {R} }$)	[does not apply]	Yes	Yes	Yes	Yes	No

The process of extending the field ${\displaystyle \mathbb {R} }$ of reals to ${\displaystyle \mathbb {C} }$ is an instance of the Cayley–Dickson construction. Applying this construction iteratively to ${\displaystyle \mathbb {C} }$ then yields the quaternions, the octonions,^[78] the sedenions, and the trigintaduonions. This construction turns out to diminish the structural properties of the involved number systems.

Unlike the reals, ${\displaystyle \mathbb {C} }$ 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 ${\displaystyle \mathbb {C} .}$^[79] Passing from ${\displaystyle \mathbb {C} }$ to the quaternions ${\displaystyle \mathbb {H} }$ 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 ${\displaystyle \mathbb {R} }$. 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 ${\displaystyle \mathbb {C} ,}$ thought of as an ${\displaystyle \mathbb {R} }$-algebra (an ${\displaystyle \mathbb {R} }$-vector space with a multiplication), with respect to the basis (1, i). This means the following: the ${\displaystyle \mathbb {R} }$-linear map $${\displaystyle {\begin{aligned}\mathbb {C} &\rightarrow \mathbb {C} \\z&\mapsto wz\end{aligned}}}$$ for some fixed complex number w can be represented by a 2 × 2 matrix (once a basis has been chosen). With respect to the basis (1, i), this matrix is $${\displaystyle {\begin{pmatrix}\operatorname {Re} (w)&-\operatorname {Im} (w)\\\operatorname {Im} (w)&\operatorname {Re} (w)\end{pmatrix}},}$$ that is, the one mentioned in the section on matrix representation of complex numbers above. While this is a linear representation of ${\displaystyle \mathbb {C} }$ in the 2 × 2 real matrices, it is not the only one. Any matrix $${\displaystyle J={\begin{pmatrix}p&q\\r&-p\end{pmatrix}},\quad p^{2}+qr+1=0}$$ has the property that its square is the negative of the identity matrix: J² = −I. Then $${\displaystyle \{z=aI+bJ:a,b\in \mathbb {R} \}}$$ is also isomorphic to the field ${\displaystyle \mathbb {C} ,}$ and gives an alternative complex structure on ${\displaystyle \mathbb {R} ^{2}.}$ This is generalized by the notion of a linear complex structure.

Hypercomplex numbers also generalize ${\displaystyle \mathbb {R} ,}$ ${\displaystyle \mathbb {C} ,}$ ${\displaystyle \mathbb {H} ,}$ and ${\displaystyle \mathbb {O} .}$ For example, this notion contains the split-complex numbers, which are elements of the ring ${\displaystyle \mathbb {R} [x]/(x^{2}-1)}$ (as opposed to ${\displaystyle \mathbb {R} [x]/(x^{2}+1)}$ for complex numbers). In this ring, the equation a² = 1 has four solutions.

The field ${\displaystyle \mathbb {R} }$ is the completion of ${\displaystyle \mathbb {Q} ,}$ the field of rational numbers, with respect to the usual absolute value metric. Other choices of metrics on ${\displaystyle \mathbb {Q} }$ lead to the fields ${\displaystyle \mathbb {Q} _{p}}$ of p-adic numbers (for any prime number p), which are thereby analogous to ${\displaystyle \mathbb {R} }$. There are no other nontrivial ways of completing ${\displaystyle \mathbb {Q} }$ than ${\displaystyle \mathbb {R} }$ and ${\displaystyle \mathbb {Q} _{p},}$ by Ostrowski's theorem. The algebraic closures ${\displaystyle {\overline {\mathbb {Q} _{p}}}}$ of ${\displaystyle \mathbb {Q} _{p}}$ still carry a norm, but (unlike ${\displaystyle \mathbb {C} }$) are not complete with respect to it. The completion ${\displaystyle \mathbb {C} _{p}}$ of ${\displaystyle {\overline {\mathbb {Q} _{p}}}}$ turns out to be algebraically closed. By analogy, ${\displaystyle \mathbb {C} _{p}}$ is called the p-adic complex numbers.

The fields ${\displaystyle \mathbb {R} ,}$ ${\displaystyle \mathbb {Q} _{p},}$ and their finite field extensions, including ${\displaystyle \mathbb {C} ,}$ are called local fields.