Roots of Unity

The n nth-roots of unity on the unit circle

zⁿ = 1 → z_k = e^(2πik/n) = cos(2πk/n) + i·sin(2πk/n)

Number of roots

Primitive roots φ(n)

k	Rectangular form	Angle	Primitive
0	1	0°	—
1	0.5 + 0.866i	60°	✓
2	-0.5 + 0.866i	120°	—
3	-1	180°	—
4	-0.5 − 0.866i	240°	—
5	0.5 − 0.866i	300°	✓

About this calculator

The Roots of Unity Calculator finds all n complex solutions of zⁿ = 1. It plots the roots as equally spaced points on the unit circle, lists each root in rectangular form e^(2πik/n) = cos(2πk/n) + i·sin(2πk/n) with its angle, and counts the primitive roots φ(n). Use it for complex analysis, abstract algebra, and signal-processing study.

Common examples

n = 1 → z = 1
n = 2 → z = 1, −1
n = 3 → z = 1, −0.5 ± 0.8660i (equilateral triangle)
n = 4 → z = 1, i, −1, −i (square); primitive roots i and −i
n = 6 → six roots forming a regular hexagon; φ(6) = 2 primitive roots