Positive Real Complex Root of Unity
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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.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity