Roots of Complex Number

A number ω is called an nth 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.

nth Roots of Unity

The nth roots of unity are 1, ω, ω2 … ωn−1.

ωn = 1

Sum of the roots is 0.

1 + ω + ω2 + … + ωn–1 = 0

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

Product of the roots = (−1)n+1