Complex Modulus

complex-numbers

Definition

Complex Modulus

The modulus of a complex number $c=a+bi$ is

$$|c|=\sqrt{c\overline{c}}=\sqrt{a^2+b^2}.$$

Geometrically, $|c|$ is the distance from the origin to $(a,b)$ in the complex plane.$

Example

Computing the Modulus

For $c=3+4i$:

$$|c|=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5.$$