Complex Modulus

mathematics complex-numbers

Definition

Complex Modulus

The modulus (or absolute value) of a complex number $c=a+bi$, where $a,b\in \mathbb{R}$, is the non-negative real number

$$|c|=\sqrt{(a+bi)(a-bi)}=\sqrt{a^2+b^2}.$$

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

Properties

For complex numbers $c,c_1,c_2\in \mathbb{C}$:

1. Non-negativity: $|c|\ge 0$ with $|c|=0⟺c=0$
2. Multiplicativity: $|c_1\cdot c_2|=|c_1|\cdot |c_2|$
3. Triangle inequality: $|c_1+c_2|\le |c_1|+|c_2|$
4. Conjugate relation: $|c{|}^2=c\cdot \overline{c}$

Example

Computing the Modulus

For $c=3+4i$:

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