Polar form of a complex number ($z=r{e}^{i\theta }$)

A nonzero complex number has the form $z=r{e}^{i\theta }$ with $r=|z|>0$ and an argument determined modulo $2\pi$. Multiplication multiplies moduli and adds arguments. This makes powers and roots of a complex number particularly transparent.

For $z=r{e}^{i\theta }\ne 0$ and a positive integer $n$, all solutions of $w^n=z$ are $r^{1/n}{e}^{i(\theta +2\pi k)/n}$ for $k=0,\dots ,n-1$. They differ by multiplication by roots of unity and are distinct. If $z=0$, the sole root is zero.