University⏱ 4 min

Roots of Unity

n solutions of zⁿ = 1

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1-1z0z1z2z3z4z5

Number of roots n

6
zk=e2πik/n=cos⁡\ rac2πkn+isin⁡\ rac2πknz_k = e^{2\pi i k/n} = \cos\ rac{2\pi k}{n} + i\sin\ rac{2\pi k}{n}

Click a root for detail

The nth roots of unity are evenly spaced on the unit circle – they form a regular n-gon. Each root is a rotation of 360°/n from the previous.