Modulus of a Complex Number

Let $\mathbb{C}$ be the field of complex numbers, let $z\in\mathbb{C}$, and let $a=\operatorname{Re}z$ and $b=\operatorname{Im}z$ be its real and imaginary parts.

The modulus of $z$, written $|z|$, is the unique real number $r$ with $0\le r$ and $r^{2}=a^{2}+b^{2}$; such an $r$ exists and is unique by Existence and Uniqueness of the Square Root of a Sum of Two Squares.