Knowlpedia

Modulus (absolute value) on ℂ

The nonnegative magnitude |z| of a complex number z, equal to its distance from 0.

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

∣z∣:=a2+b2=zz‾.|z|:=\sqrt{a^2+b^2}=\sqrt{z\overline{z}}.
Remarks

The modulus makes C\mathbb{C} into a normed space and induces the standard metric d(z,w)=∣z−w∣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=3−4iz=3-4i, then ∣z∣=32+(−4)2=5|z|=\sqrt{3^2+(-4)^2}=5.
  • If z∈R⊆Cz\in\mathbb{R}\subseteq\mathbb{C}, then this definition agrees with the real absolute value.
  • ∣eiθ∣=1|e^{i\theta}|=1 for all θ∈R\theta\in\mathbb{R} (Euler's formula context).