Tutorial 1 · About 8 minutes
Roots in the complex plane
Relate the algebraic equation wⁿ = z to angles on the unit circle.
z = exp(iθ), wⁿ = z, |z| = 1
Prediction
Predict the result
Experiment
Vary the parameters
Change the argument and the root order. The green point is z; the blue points are all the solutions of wⁿ = z.
- Number of distinct roots
- 3
- Angular separation
- 120°
- Argument of the first root
- 30°
Questions to consider
- Set n = 4 and check the angular separation of consecutive roots.
- Raise each root to the power n. Each result should be the same point z.
Explanation
Explain your result
Read the mathematical explanation
If z = exp(iθ), its nth roots are wₖ = exp(i(θ + 2πk)/n), for k = 0,…,n−1. All lie on the unit circle, with angular separation 2π/n. Arguments are considered modulo 2π.