Properties of Roots of Unity
Roots of unity have many special properties and applications. These are just some of them:
1) If x x x is an n t h n^{th} nth root of unity, then so is x k x^k xk where k k k is any integer. 2) If x x x is an n t h n^{th} nth root of unity, then x n = 1 x^n = 1 xn=1. 3) The sum of all n t h n^{th} nth roots of unity is always zero for n ≠ 1 n \neq 1 n̸=1. 4) The product of all n t h n^{th} nth roots of unity is always ( − 1 ) n + 1 (-1)^{n+1} (−1)n+1. 5) 1 1 1 and − 1 -1 −1 are the only real roots of unity. 6) If a number is a root of unity, then so is its complex conjugate. 7) The sum of all the k t h k^{th} kth power of the n t h n^{th} nth roots of unity is 0 0 0 for all integers k k k such that k k k is not divisible by n n n. 8) The sum of the absolute values of all the n t h n^{th} nth roots of unity is n n n. 9) If x x x is an n t h n^{th} nth root of unity not equal to 1 1 1, then ∑ k = 0 n − 1 x k = 0 \sum_{k=0}^{n-1}x^k = 0 ∑k=0n−1xk=0.
