Complex Roots of Unity/Examples/7th Roots
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Example of Complex Roots of Unity
The complex $7$th roots of unity are the elements of the set:
- $U_n = \set {z \in \C: z^7 = 1}$
They are:
\(\ds e^{0 \pi / 7}\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds e^{2 \pi / 7}\) | \(=\) | \(\ds \cis \dfrac {2 \pi} 7\) | ||||||||||||
\(\ds e^{4 \pi / 7}\) | \(=\) | \(\ds \cis \dfrac {4 \pi} 7\) | ||||||||||||
\(\ds e^{6 \pi / 7}\) | \(=\) | \(\ds \cis \dfrac {6 \pi} 7\) | ||||||||||||
\(\ds e^{8 \pi / 7}\) | \(=\) | \(\ds \cis \dfrac {8 \pi} 7\) | ||||||||||||
\(\ds e^{10 \pi / 7}\) | \(=\) | \(\ds \cis \dfrac {10 \pi} 7\) | ||||||||||||
\(\ds e^{12 \pi / 7}\) | \(=\) | \(\ds \cis \dfrac {12 \pi} 7\) |
Proof
By definition, the first complex $7$th root of unity $\alpha$ is given by:
\(\ds \alpha\) | \(=\) | \(\ds e^{2 i \pi / 7}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cos \frac {2 \pi} 7 + i \sin \frac {2 \pi} 7\) |
Hence the result:
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: The $n$th Roots of Unity: $105 \ \text {(b)}$