nLab
complex number

Definition

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

This can be thought of as:

• the vector space ${ℝ}^2$ made into an algebra by the rule

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

  $$\left(\begin{array}{cc}a& b\\ -b& a\end{array}\right);$$\left(\array { a & b \\ - b & a } \right);

• the polynomial ring $ℝ[\mathrm{x}]$ modulo ${\mathrm{x}}^2+1$;

• the algebraic closure of $ℝ$ as a field;

• the result of applying the Cayley–Dickson construction to $ℝ$;

• the $2$-dimensional normed division algebra;

• the Clifford algebra ${\mathrm{Cl}}_{0,1}(ℝ)$;

• the elliptic $2$-dimensional algebra of hypercomplex numbers;

• the complexification of $ℝ$.

We think of $ℝ$ as a subset of $ℂ$ by identifying $a$ with $a+0\mathrm{i}$. $ℂ$ is equipped with an involution, called complex conjugation, that maps $\mathrm{i}$ to $\overline{\mathrm{i}}=-\mathrm{i}$:

$$\overline{a+\mathrm{i}b}=a-\mathrm{i}b.$$\overline{a + \mathrm{i} b} = a - \mathrm{i} b .

$ℂ$ also has an absolute value:

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

notice that the absolute value of a complex number is a non-negative real number, with

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

Most concepts in analysis can be extended from $ℝ$ to $ℂ$, as long as they do not rely on the order in $ℝ$. Sometimes $ℂ$ even works better, either because it is algebraically closed or because of Goursat's theorem. Even when the order in $ℝ$ 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 $ℝ$ or $ℂ$ (or even other possibilities).

Properties

Geometry of complex numbers

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

It is often convenient to use the Alexandroff compactification of $ℂ$, the Riemann sphere? $\overline{ℂ}$ (note that the bar here does not indicate complex conjugation). One may think of $\overline{ℂ}$ as $ℂ\cup \{\mathrm{\infty }\}$; functions valued in $ℂ$ but containing ‘poles’ may be taken to be valued in $\overline{ℂ}$, with $f(\zeta )=\mathrm{\infty }$ whenever $\zeta$ is a pole of $f$.

Related concepts

• absolute value, phase

• complex vector space

  • complex line

• complex vector bundle

  • complex line bundle

• contour integration

References

• Tom Leinster, Objects of categories as complex numbers, arXiv:math/0212377v1