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.
