Polar form

Statement

Let $z$ be a complex number with $z \neq 0$, let $r = |z|$ be its modulus and let $\theta = \operatorname{Arg} z$ be its principal argument. Then $$z = r (\cos \theta + i \sin \theta) = r e^{i \theta}.$$

Remarks

The second equality is Euler's formula. The pair $(r, \theta)$ gives the polar coordinates of the point $z$ in the complex plane. Polar form makes products easy: $r e^{i \theta} \cdot s e^{i \varphi} = r s \, e^{i (\theta + \varphi)}$.

Explanations

L1 human

You can describe a point of the plane by its distance from $0$ and its direction. For example, $1 + i$ has distance $\sqrt{2}$ and direction $\pi/4$, so $1 + i = \sqrt{2} \, e^{i \pi/4}$.