Roots of complex numbers

The equation $w^n=z$ has exactly $n$ distinct complex solutions for any nonzero $z\in \mathbb{C}$ and positive integer $n$. They sit evenly spaced around a circle of radius $|z{|}^{1/n}$ centered at the origin.

Derivation

Write $z$ in polar form as $z=r{e}^{i\theta }$. Let $w=\rho e^{i\varphi }$. Then $w^n={\rho }^n{e}^{in\varphi }$, and $w^n=z$ requires

${\rho }^n=r,n\varphi =\theta +2\pi k,k\in \mathbb{Z}.$

So $\rho =r^{1/n}$ (the unique positive real $n$-th root) and

${\varphi }_k=\frac{\theta +2\pi k}{n},k=0,1,\dots ,n-1.$

Values of $k$ outside $\{0,1,\dots ,n-1\}$ repeat the same $n$ angles modulo $2\pi$, giving the same $n$ distinct roots.

The $n$ roots

$w_k=r^{1/n}\exp \left(i\frac{\theta +2\pi k}{n}\right),k=0,1,\dots ,n-1.$

Geometrically, the roots are the vertices of a regular $n$-gon inscribed in the circle $|w|=r^{1/n}$. Consecutive roots differ by an angle $2\pi /n$.

Square roots of $-4$

$-4$ has modulus $4$, argument $\pi$. So $w^2=-4$ gives $\rho =2$, $\varphi =\pi /2+\pi k$. Two distinct angles in $[0,2\pi )$: $\varphi =\pi /2$ and $\varphi =3\pi /2$. The square roots are $2i$ and $-2i$.

$n$-th roots of unity

The roots of $w^n=1$ are

${\omega }_k=e^{2\pi ik/n},k=0,1,\dots ,n-1.$

They lie on the unit circle, one at $1$, the others spaced by $2\pi /n$. The primitive root ${\omega }_1=e^{2\pi i/n}$ generates all the others by multiplication.

The roots of unity show up in the discrete Fourier transform: its kernel is built entirely from ${\omega }_N^{kn}$ for various $k,n$.

Connection to the fundamental theorem of algebra

The polynomial $w^n-z$ has degree $n$, and we just found $n$ distinct roots (when $z\ne 0$). Consistent with the Fundamental theorem of algebra: every degree-$n$ polynomial with complex coefficients has exactly $n$ complex roots (counted with multiplicity).

Branch consideration

Writing "$z^{1/n}$" as a single number requires choosing one of the $n$ values, usually the principal $n$-th root, with argument in $(-\pi /n,\pi /n]$ (this follows from the principal argument convention). For non-integer powers the multivaluedness propagates further; see Complex power.