The $n$th roots of unity are the unique solution to the equation $\zeta^{n} = 1$. In the [[Real Numbers]], the solution is obviously $1$ (and $-1$ when $n$ is even), but the roots of unity includes both these and members in the [[Complex Plane]]. These numbers all lie on the [[Unit Circle]], and can be derived by this formula using the [[Complex Exponential|complex exponential]]: $\huge \begin{align} 1 &= \zeta^{n} \\ \zeta &= \set{ 1, e^{ {2\pi i}/{n} }, e^{ {(2)2\pi i}/{n} }, e^{ {(3)2\pi i}/{n} }, \cdots, e^{ {(n-1)2\pi i}/{n} } } \\ \zeta &= \setbuild{k \in [0, n)}{e^{k2\pi i/n}} \end{align} $