The equation zⁿ = 1 has n solutions:
z_k = cos(2kπ/n) + i sin(2kπ/n) (k = 0, 1, …, n−1)
On the complex plane, these points are arranged as the vertices of a regular n-gon with center at the origin and radius 1.
Example ① Find all solutions of z³ = 1.
Solution:z = 1, −1/2 + (√3/2)i, −1/2 − (√3/2)i
Solution:z = 1, −1/2 + (√3/2)i, −1/2 − (√3/2)i
Key Point
- The nth roots of unity are the vertices of a regular n-gon.
Practice Problems
- Find all solutions of z⁴ = 1.
Solution and Explanation
- Answer:1, i, −1, −i