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
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity: Example $2$.
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: The $n$th Roots of Unity: $37$