Complex numbers

Numbers of the form a+bi with i^2=-1, forming a field extending the reals.

Complex numbers

The complex numbers are

\mathbb{C}:=\{a+bi : a,b\in\mathbb{R},\ i^2=-1\}.

Addition and multiplication are defined by

(a+bi)+(c+di)=(a+c)+(b+d)i,

(a+bi)(c+di)=(ac-bd)+(ad+bc)i.

The field $\mathbb{C}$ is a two-dimensional real vector space and can be identified with $\mathbb{R}^2$ via $a+bi\leftrightarrow(a,b)$ . Complex numbers are the natural setting for Fourier analysis, power series, and many aspects of analysis and geometry.

Examples:

$z=3-2i$ has real part $3$ and imaginary part $-2$ .
$(1+i)^2 = 1+2i+i^2 = 2i$ .
The real numbers embed into $\mathbb{C}$ via $a\mapsto a+0i$ .