Complex Roots of Unity/Examples/5th Roots

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Example of Complex Roots of Unity

The complex $5$th roots of unity are the elements of the set:

$U_n = \set {z \in \C: z^5 = 1}$


They are:

\(\ds e^{0 \pi / 5}\) \(=\) \(\ds 1\)
\(\ds e^{2 \pi / 5}\) \(=\) \(\ds \dfrac {\sqrt 5 - 1} 4 + i \dfrac {\sqrt {10 + 2 \sqrt 5} } 4\)
\(\ds e^{4 \pi / 5}\) \(=\) \(\ds -\dfrac {1 + \sqrt 5} 4 + i \sqrt {\dfrac 5 8 - \dfrac {\sqrt 5} 8}\)
\(\ds e^{6 \pi / 5}\) \(=\) \(\ds -\dfrac {1 + \sqrt 5} 4 - i \sqrt {\dfrac 5 8 - \dfrac {\sqrt 5} 8}\)
\(\ds e^{8 \pi / 5}\) \(=\) \(\ds \dfrac {\sqrt 5 - 1} 4 - i \dfrac {\sqrt {10 + 2 \sqrt 5} } 4\)


Proof

By definition, the first complex $5$th root of unity $\alpha$ is given by:

\(\ds \alpha\) \(=\) \(\ds e^{2 i \pi / 5}\)
\(\ds \) \(=\) \(\ds \cos \frac {2 \pi} 5 + i \sin \frac {2 \pi} 5\)
\(\ds \) \(=\) \(\ds \dfrac {\sqrt 5 - 1} 4 + i \dfrac {\sqrt {10 + 2 \sqrt 5} } 4\) Cosine of $\dfrac {2 \pi} 5$, Sine of $\dfrac {2 \pi} 5$


We have that:

$e^{0 i \pi / 5} = e^0 = 1$

which gives us, as always, the zeroth complex $n$th root of unity for all $n$.


The remaining complex $5$th roots of unity follow:

\(\ds \alpha^2\) \(=\) \(\ds e^{4 i \pi / 5}\)
\(\ds \) \(=\) \(\ds -\dfrac {1 + \sqrt 5} 4 + i \sqrt {\dfrac 5 8 - \dfrac {\sqrt 5} 8}\)
\(\ds \) \(=\) \(\ds \dfrac {\sqrt 5 - 1} 4 + i \dfrac {\sqrt {10 + 2 \sqrt 5} } 4\) Cosine of $\dfrac {4 \pi} 5$, Sine of $\dfrac {4 \pi} 5$


\(\ds \alpha^3\) \(=\) \(\ds \overline {\alpha^{5 - 3} }\) Complex Roots of Unity occur in Conjugate Pairs
\(\ds \) \(=\) \(\ds \overline {\alpha^2}\)
\(\ds \) \(=\) \(\ds \overline {-\dfrac {1 + \sqrt 5} 4 + i \sqrt {\dfrac 5 8 - \dfrac {\sqrt 5} 8} }\)
\(\ds \) \(=\) \(\ds -\dfrac {1 + \sqrt 5} 4 - i \sqrt {\dfrac 5 8 - \dfrac {\sqrt 5} 8}\) Definition of Complex Conjugate


\(\ds \alpha^4\) \(=\) \(\ds \overline {\alpha^{4 - 1} }\) Complex Roots of Unity occur in Conjugate Pairs
\(\ds \) \(=\) \(\ds \overline \alpha\)
\(\ds \) \(=\) \(\ds \overline {\dfrac {\sqrt 5 - 1} 4 + i \dfrac {\sqrt {10 + 2 \sqrt 5} } 4}\)
\(\ds \) \(=\) \(\ds \dfrac {\sqrt 5 - 1} 4 - i \dfrac {\sqrt {10 + 2 \sqrt 5} } 4\) Definition of Complex Conjugate

$\blacksquare$


Sources

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