section_path: Non-integer powers of complex numbers/n th roots of a complex number/Roots of unity
section_name: Roots of unity
paragraph_text: Roots of unity
  
  The n th roots of unity are the n complex numbers such that w = 1, where n is a positive integer. They arise in various areas of mathematics, such as in discrete Fourier transform or algebraic solutions of algebraic equations (Lagrange resolvent).
  
  The n n th roots of unity are the n first powers of ω = e 2 π i n {\displaystyle \omega =e^{\frac {2\pi i}{n}}}, that is 1 = ω 0 = ω n, ω = ω 1, ω 2, ω n − 1. {\displaystyle 1=\omega ^{0}=\omega ^{n},\omega =\omega ^{1},\omega ^{2},\omega ^{n-1}.} The n th roots of unity that have this generating property are called primitive n th roots of unity; they have the form ω k = e 2 k π i n, {\displaystyle \omega ^{k}=e^{\frac {2k\pi i}{n}},} with k coprime with n. The unique primitive square root of unity is − 1; {\displaystyle -1;} the primitive fourth roots of unity are i {\displaystyle i} and − i. {\displaystyle -i.}
  
  The n th roots of unity allow expressing all n th roots of a complex number z as the n products of a given n th roots of z with a n th root of unity.
  
  Geometrically, the n th roots of unity lie on the unit circle of the complex plane at the vertices of a regular n -gon with one vertex on the real number 1.
  
  As the number e 2 k π i n {\displaystyle e^{\frac {2k\pi i}{n}}} is the primitive n th root of unity with the smallest positive argument, it is called the principal primitive n th root of unity, sometimes shortened as principal n th root of unity, although this terminology can be confused with the principal value of 1 1 / n {\displaystyle 1^{1/n}}, which is 1.