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
\(\ds z^n\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cmod {z^n}\) | \(=\) | \(\ds \cmod 1\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cmod z^n\) | \(=\) | \(\ds 1\) | Power of Complex Modulus equals Complex Modulus of Power | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cmod z\) | \(=\) | \(\ds 1\) | Positive Real Complex Root of Unity |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 44$