Question

Let ω=cos 2π/n + i sin 2π/n where n is a positive integer. Show that 1, ω, ω², ω³, ....., ωⁿ⁻¹ are the n distinct n th roots of 1

Answer 1

The complex number ω=cos 2π/n + i sin 2π/n, where n is a positive integer, represents points on the unit circle in the complex plane. Multiplying ω repeatedly gives n distinct nth roots of 1.

Step-by-step explanation:

The complex number ω=cos 2π/n + i sin 2π/n, where n is a positive integer, represents a point on the unit circle in the complex plane. The angle 2π/n corresponds to dividing the circle into n equal parts. Starting from the point (1, 0) on the unit circle, we can multiply by ω repeatedly to obtain n distinct nth roots of 1.

For example, if n = 4, ω = cos(2π/4) + i sin(2π/4) = (0, 1). Multiplying ω by itself gives ω² = (0, -1), ω³ = (-1, 0), and finally ω⁴ = (1, 0), which is 1.

This pattern continues for any value of n, where ω raised to the power of n equals 1. Therefore, 1, ω, ω², ω³, ..., ωⁿ⁻¹ are the n distinct nth roots of 1.