Roots of unity
zn = 1 has n solutions, equally spaced on the unit circle.
The n nth roots of unity are e^(2πik/n) for k = 0, 1, ..., n−1. They form a regular n-gon on the unit circle.
Worked examples
3rd roots of 1: 1, e^(2πi/3), e^(4πi/3) (i.e. 1 and the two complex cube roots).
Sum of all nth roots = 0 (n > 1).
4th roots: 1, i, −1, −i.
Frequently asked questions
Why equally spaced?
Argument increments by 2π/n; modulus is always 1.
Useful where?
Roots of unity appear in DFTs, polynomial factorisation, and de Moivre identity derivations.