# 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

 $d(a_{1}+b_{1}i,a_{2}+b_{2}i):=\sqrt{(a_{2}-a_{1})^{2}+(b_{2}-b_{1})^{2}}.$
Title complex number ComplexNumber 2013-03-22 11:52:35 2013-03-22 11:52:35 djao (24) djao (24) 9 djao (24) Definition msc 12D99 msc 30-00 msc 32-00 msc 46L05 msc 18B40 msc 46M20 msc 17B37 msc 22A22 msc 81R50 msc 22D25 $\mathbb{C}$ Complex