. . The n th roots of unity The solutions of the equation z n = , for positive values of integer n , are the n roots of the unity. In polar form the equation z n = can be written as n = ) + π = e i k p , k = , , , .
Using De Moivre’s theorem, we find the n th roots of unity from the equation given below: e i k + = , , , , . … ( ) Given a positive integer n , a complex number z is called an n th root of unity if and only if z n = . If we denote the complex number by ω , then ω = e ⇒ ω n = e e . Fig.
. Re Im O P ω m ω ω − ω n th root of a complex number n r Therefore ω is an n th root of unity. From equation ( ), the complex numbers , , n are n th roots of unity. The complex numbers , , n are the points in the complex plane and are the vertices of a regular polygon of n sides inscribed in a unit circle as shown in Fig .
. Note that because the n th roots all have the same modulus , they will lie on a circle of radius with centre at the origin. Furthermore, the n roots are equally spaced along the circle, because successive n th roots have arguments that differ by p n . The n th roots of unity , , n are in geometric progression with common ratio ω .
Therefore since n and ω ¹ . The sum of all the n th roots of unity is The product of n n th roots of unit