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