:PROPERTIES: :ID: 5d435f4a-eb4d-4db4-be42-e0e804955e6f :mtime: 20231024010428 20231021014909 :ctime: 20231021014908 :END: #+title: roots of unity #+filetags: :public:project: * Roots of Unity The roots of unity are all [[id:2553e0fb-12c1-42cc-8e47-54937a36e2c7][complex number]] solutions to the equation \[z^{n} = 1\] where $n$ is a natural number. * Solutions to the roots of unity if we use [[id:9a9960ad-ea53-4d94-ac3b-3156da5d213e][polar representation of complex numbers]], then the solutions are \[e^{2\pi i\frac{k}{n}}\] for $k \in \mathbb{Z}$ and $0 < k \le n$ There are exactly $n$ distinct [[id:2553e0fb-12c1-42cc-8e47-54937a36e2c7][complex number]] to the equation \[z^{n} = 1\]