Roots of Unity Calculator
The nth roots of unity are the n complex numbers that satisfy z to the power n equals 1. They lie evenly spaced around the unit circle, separated by an angle of 2 pi divided by n, and each has modulus 1. Enter an index n and this calculator lists every root as a real and imaginary pair together with its angle in radians, plus the angular spacing and the count. The roots underpin the discrete Fourier transform and the factorisation of z to the n minus 1.
Roots of unity formula
z_k = cos(2 pi k / n) + i sin(2 pi k / n), for k = 0, 1, ..., n - 1
Number of roots = n
Angular spacing = 2 pi / n
All roots have modulus 1
Each root sits on the unit circle at an angle that is a whole-number multiple of 2 pi over n. Together they divide the circle into n equal arcs.
About the roots of unity
- The root at k = 0 is always 1, since 1 to any power equals 1.
- For even n, minus 1 is always among the roots.
- The roots are the vertices of a regular n-sided polygon inscribed in the unit circle.
- They form a cyclic group under multiplication.
- The discrete Fourier transform evaluates a sequence at these roots.
Roots of unity: frequently asked questions
What are the roots of unity?
The nth roots of unity are the n complex solutions of z to the power n equals 1. They sit evenly spaced around the unit circle at angles 2 pi k over n, for k from 0 up to n minus 1.
How are the roots of unity calculated?
Each root is cos(2 pi k / n) + i sin(2 pi k / n) for k = 0, 1, up to n - 1. They all have modulus 1 and their arguments are equally spaced by 2 pi divided by n.
How many roots of unity are there?
There are exactly n distinct nth roots of unity. For example there are 3 cube roots of unity and 4 fourth roots of unity, always equal in count to the index n.
What is a primitive root of unity?
A primitive nth root of unity is one whose powers generate all n roots. The root at k = 1, namely cos(2 pi / n) + i sin(2 pi / n), is always primitive, and there are phi(n) primitive roots in total.
Why do the roots of unity matter?
They are central to the discrete Fourier transform, to factoring polynomials like z^n - 1, and to cyclic group structure. Their even spacing on the unit circle also makes them a natural set of evenly spaced directions.
Official sources
- NIST Digital Library of Mathematical Functions: Complex Variables.
- NIST Digital Library of Mathematical Functions: Zeros of Polynomials.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.