Positive Real Complex Root of Unity

Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $U_n = \set {z \in \C: z^n = 1}$ be the set of complex $n$th roots of unity.

The only $x \in U_n$ such that $x \in \R_{>0}$ is:
 * $x = 1$

That is, $1$ is the only complex $n$th root of unity which is a positive real number.

Proof
We have that $1$ is a positive real number.

The result follows from Existence and Uniqueness of Positive Root of Positive Real Number.