Modulus of Complex Root of Unity equals 1

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Theorem

Let $n \in \Z_{>0}$ be a (strictly) positive integer such that $n$ is even.

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

Let $z \in U_n$.


Then:

$\cmod z = 1$

where $\cmod z$ denotes the modulus of $z$.


Proof

\(\displaystyle z^n\) \(=\) \(\displaystyle 1\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \cmod {z^n}\) \(=\) \(\displaystyle \cmod 1\)
\(\displaystyle \) \(=\) \(\displaystyle 1\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \cmod z^n\) \(=\) \(\displaystyle 1\) Power of Complex Modulus equals Complex Modulus of Power
\(\displaystyle \leadsto \ \ \) \(\displaystyle \cmod z\) \(=\) \(\displaystyle 1\) Positive Real Complex Root of Unity

$\blacksquare$


Sources