Modulus (absolute value) on ℂ

For $z=a+bi\in\mathbb{C}$ , writing $\overline z$ for its complex conjugate, the modulus (or absolute value) of $z$ is

|z|:=\sqrt{a^2+b^2}=\sqrt{z\overline{z}}.

The square root is the nonnegative square root.

Remarks

The modulus makes $\mathbb{C}$ into a normed space and induces the standard metric $d(z,w)=|z-w|$ . It is the complex analogue of absolute value and is crucial for convergence of complex sequences and series.

Examples

If $z=3-4i$ , then $|z|=\sqrt{3^2+(-4)^2}=5$ .
If $z\in\mathbb{R}\subseteq\mathbb{C}$ , then this definition agrees with the real absolute value.
$|e^{i\theta}|=1$ for all $\theta\in\mathbb{R}$ (Euler's formula context).