section_path: Complex roots/Roots of unity
section_name: Roots of unity
paragraph_text: Roots of unity
  
  The number 1 has n different n th roots in the complex plane, namely
  
  1, ω, ω 2, …, ω n − 1, {\displaystyle 1,\;\omega,\;\omega ^{2},\;\ldots,\;\omega ^{n-1},}
  
  where
  
  ω = e 2 π i n = cos (2 π n) + i sin (2 π n). {\displaystyle \omega =e^{\frac {2\pi i}{n}}=\cos \left({\frac {2\pi }{n}}\right)+i\sin \left({\frac {2\pi }{n}}\right).}
  
  These roots are evenly spaced around the unit circle in the complex plane, at angles which are multiples of 2 π / n {\displaystyle 2\pi /n}. For example, the square roots of unity are 1 and −1, and the fourth roots of unity are 1, i {\displaystyle i}, −1, and − i {\displaystyle -i}.