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
6
Primitive roots φ(n)
2
kRectangular formAnglePrimitive
010°—
10.5 + 0.866i60°✓
2‎-0.5 + 0.866i120°—
3‎-1180°—
4‎-0.5 − 0.866i240°—
50.5 − 0.866i300°✓
Roots of UnityUnit circle showing the 6 nth-roots of unity as equally spaced points forming a regular polygon, with 2 primitive roots highlighted.ReIm2π/6012345

حول هذه الآلة الحاسبة

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.

أمثلة شائعة

  • 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