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:

Complex 7th Roots of 1.png

$\blacksquare$


Sources