complex number

The ring of complex numbers $ℂ$ is defined to be the quotient ring of the polynomial ring $ℝ[X]$ in one variable over the reals by the principal ideal $(X^2+1)$. For $a,b\in ℝ$, the equivalence class of $a+bX$ in $ℂ$ 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

$$d(a_1+b_{1}i,a_2+b_{2}i):=\sqrt{{(a_2-a_1)}^2+{(b_2-b_1)}^2}.$$