nLab complex number

Definition

A complex number is a number of the form $a + \mathrm{i} b$, where $a$ and $b$ are real numbers and $\mathrm{i}^2 = - 1$ is an imaginary unit. The set of complex numbers (in fact a field and topological vector space) is denoted $\mathbf{C}$ or $\mathbb{C}$.

This can be thought of as:

• the vector space $\mathbb{R}^2$ made into an algebra by the rule
  $$(a, b) \cdot (c, d) = (a c - b d, a d + b c) ;$$
• the subalgebra of those $2$-by-$2$ real matrices of the form
  $$\left(\array { a & b \\ - b & a } \right);$$
• the polynomial ring $\mathbb{R}[\mathrm{x}]$ modulo $\mathrm{x}^2 + 1$;
• the result of applying the Cayley–Dickson construction to $\mathbb{R}$;
• the $2$-dimensional normed division algebra over $\mathbb{R}$;
• the Clifford algebra $Cl_{0,1}(\mathbb{R})$;
• the elliptic $2$-dimensional algebra of hypercomplex numbers;
• the complexification of $\mathbb{R}$;
• In mathematics with weak countable choice, the algebraic closure of $\mathbb{R}$ as a field;

We think of $\mathbb{R}$ as a subset (in fact ${\mathbb{R}}$-vector subspace) of $\mathbb{C}$ by identifying $a$ with $a + \mathrm{i} 0$. $\mathbb{C}$ is equipped with a $\mathbb{R}$-linear involution , called complex conjugation, that maps $\mathrm{i}$ to $\bar{\mathrm{i}} = -\mathrm{i}$. Concretely, $\overline{a + \mathrm{i} b} = a - \mathrm{i} b$.

Complex conjugation is the nontrivial field automorphism of $\mathbb{C}$ which leaves ${\mathbb{R}}$ invariant. In other words, the Galois group $Gal({\mathbb{C}}/\mathbb{R})$ is cyclic of order two and generated by complex conjugation. $\mathbb{C}$ also has an absolute value:

$$|{a + \mathrm{i} b}| = \sqrt{a^2 + b^2} ;$$

notice that the absolute value of a complex number is a nonnegative real number, with

$$|z|^2 = z \bar{z} .$$

Most concepts in analysis can be extended from $\mathbb{R}$ to $\mathbb{C}$, as long as they do not rely on the order in $\mathbb{R}$. Sometimes $\mathbb{C}$ even works better, either because it is algebraically closed or because of Goursat's theorem. Even when the order in $\mathbb{R}$ is important, often it is enough to order the absolute values of complex numbers. See ground field for some of the concepts whose precise definition may vary with the choice of $\mathbb{R}$ or $\mathbb{C}$ (or even other possibilities).

Properties

Automorphisms

Proposition

The automorphism group of the complex numbers, as an associative algebra over the real numbers, is Z/2, acting by complex conjugation.

See also at normed division algebra – automorphism.

Over other subfields, the automorphism group may be considerably larger. Over the rational numbers, for instance, $\mathbb{C}$ has transcendence degree equal to the cardinality of the continuum, i.e., there is an algebraic extension $\mathbb{Q}(X) \hookrightarrow \mathbb{C}$ with ${|X|} = \mathfrak{c} = 2^{\aleph_0}$. Any bijection $X \to X$ induces a field automorphism $\mathbb{Q}(X) \to \mathbb{Q}(X)$ which may be extended to an automorphism of $\mathbb{C}$ over $\mathbb{Q}$. Therefore the number of automorphisms of $\mathbb{C}$ is at least $2^\mathfrak{c}$ (and in fact at most this as well, since the number of functions $\mathbb{C} \to \mathbb{C}$ is also $2^\mathfrak{c}$).

See also at automorphism of the complex numbers.

Geometry of complex numbers

The complex numbers form a plane, the complex plane. Indeed, a map $\mathbb{C} \to \mathbb{R}^2$ given by sending $\mathrm{x} + \mathrm{i}\mathrm{y}$ to the standard real-valued coordinates $(\mathrm{x},\mathrm{y})$ on this plane is a bijection. Much of complex analysis can be understood through differential topology by identifying $\mathbb{C}$ with $\mathbb{R}^2$, using either $\mathrm{x}$ and $\mathrm{y}$ or $\mathrm{z}$ and $\bar{\mathrm{z}}$. (For example, Cauchy's integral theorem is Green's/Stokes's theorem.)

It is often convenient to use the Alexandroff compactification of $\mathbb{C}$, the Riemann sphere $\mathbb{C}P^1$. One may think of $\mathbb{C}P^1$ as $\mathbb{C} \cup \{\infty\}$; functions valued in $\mathbb{C}$ but containing ‘poles’ may be taken to be valued in $\overline{\mathbb{C}}$, with $f(\zeta) = \infty$ whenever $\zeta$ is a pole of $f$.

Related concepts

• automorphism of the complex numbers

• absolute value, phase

• real part, imaginary part

• complex plane, complex projective space, Riemann sphere

• complex vector space

  • complex line

• conjugate transpose matrix

• complex vector bundle

  • complex line bundle

• contour integration

• complex geometry

• polydisc

• p-adic complex number

exceptional spinors and real normed division algebras

Lorentzian spacetime dimension	$\phantom{AA}$spin group	normed division algebra	$\,\,$ brane scan entry
$3 = 2+1$	$Spin(2,1) \simeq SL(2,\mathbb{R})$	$\phantom{A}$ $\mathbb{R}$ the real numbers	super 1-brane in 3d
$4 = 3+1$	$Spin(3,1) \simeq SL(2, \mathbb{C})$	$\phantom{A}$ $\mathbb{C}$ the complex numbers	super 2-brane in 4d
$6 = 5+1$	$Spin(5,1) \simeq$ SL(2,H)	$\phantom{A}$ $\mathbb{H}$ the quaternions	little string
$10 = 9+1$	Spin(9,1) ${\simeq}$ “SL(2,O)”	$\phantom{A}$ $\mathbb{O}$ the octonions	heterotic/type II string

References

On the history of the notion of complex numbers:

• Jean-Pierre Tignol, p. 19 of: Galois’ Theory of Algebraic Equations, World Scientific (2001) [doi:10.1142/4628]

• Orlando Merino, A Short History of Complex Numbers (2006) [pdf, pdf]

• Leo Corry, A Brief History of Numbers, Oxford University Press (2015) [ISBN:9780198702597]

See also:

• Wikipedia, Complex numbers

• Tom Leinster, Objects of categories as complex numbers, arXiv:math/0212377