Roots of Complex Number

A number ω is called an n^th root of a complex number z, if ωⁿ = z.

Working rule to find the nth roots of a complex number:

Step 1: Write the given number in polar form.

Step 2: Add 2kπ to the argument.

Step 3: Apply De Moivre’s theorem and bring the power to inside.

Step 4: Put k = 0, 1 … up to n−1.

n^th Roots of Unity

The n^th roots of unity are 1, ω, ω² … ωⁿ⁻¹.

ωⁿ = 1

Sum of the roots is 0.

1 + ω + ω² + … + ω^n–1 = 0

The roots are in G.P with common ratio ω.

Product of the roots = (−1)ⁿ⁺¹