Complex Roots of Unity are Vertices of Regular Polygon Inscribed in Circle

Theorem

Let $n \in \Z$ be an integer such that $n \ge 3$.

Let $z \in \C$ be a complex number such that $z^n = 1$.

Let $U_n = \set {e^{2 i k \pi / n}: k \in \N_n}$ be the set of $n$th roots of unity.

Let $U_n$ be plotted on the complex plane.

Then the elements of $U_n$ are located at the vertices of a regular $n$-sided polygon $P$, such that:

$(1):\quad$ $P$ is circumscribed by a unit circle whose center is at $\tuple {0, 0}$

$(2):\quad$ one of those vertices is at $\tuple {1, 0}$.

Proof

The above diagram illustrates the $7$th roots of unity.

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Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: The $n$th Roots of Unity