Modulus (absolute value) on ℂ

For $z=a+bi\in\mathbb{C}$, the modulus (or absolute value) of $z$ is

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).