Modulus of Complex Root of Unity equals 1

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: $\S 44$