Roots of unity

The solutions of zⁿ = 1 are e^2πik/n for k = 0, …, n−1 — n points evenly spaced on the unit circle, forming a regular n-gon with one vertex at z = 1.

root primitive root ω = e^2πi/n target z = 1 w (click to set)

Controls

n 3

Show labels (hover any root for details)

Default opens at n = 3 — the equilateral triangle 1, −½ + (√3/2)i, −½ − (√3/2)i you derived by hand.

The angle derivation, live

z = e^iθ, zⁿ = e^inθ = 1 ⟺ nθ ≡ 0 (mod 2π)

Press “Animate zⁿ → 1” to watch each root rotate to z = 1 while nθ climbs to a multiple of 2π.

Showing the n roots of unity.

General case zⁿ = w

Click anywhere in the plane to set w. Roots become a copy of the n-gon scaled to radius |w|^1/n and rotated by arg(w)/n.

w = 1 (the zⁿ = 1 case)

Roots of unity

Controls

The angle derivation, live

General case zn = w

General case zⁿ = w