complex number

The ring of complex numbers $\mathbb{C}$ is defined to be the quotient ring of the polynomial ring $\mathbb{R}[X]$ in one variable over the reals by the principal ideal $(X^{2}+1)$ . For $a,b\in\mathbb{R}$ , the equivalence class of $a+bX$ in $\mathbb{C}$ is usually denoted $a+bi$ , and one has $i^{2}=-1$ .

The complex numbers form an algebraically closed field. There is a standard metric on the complex numbers, defined by

$d(a_{1}+b_{1}i,a_{2}+b_{2}i):=\sqrt{(a_{2}-a_{1})^{2}+(b_{2}-b_{1})^{2}}.$