Roots of Unity

Polygons from equations

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• $z^n = 1$ has $n$ solutions, evenly spaced around the unit circle.
• They form a regular n-gon. Their sum is 0 (for n ≥ 2).
• Equations in complex numbers have pictures: algebra and geometry merge.

Steps

1. How many cube roots?
2. The nth roots of 1
3. Fourth roots
4. Evenly spaced
5. Shared roots
6. They cancel