Complex Roots of Unity/Examples/6th Roots
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Example of Complex Roots of Unity
The complex $6$th roots of unity are the elements of the set:
- $U_n = \set {z \in \C: z^6 = 1}$
They are:
| \(\ds e^{0 i \pi / 6}\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds e^{i \pi / 3}\) | \(=\) | \(\ds \frac 1 2 + \frac {i \sqrt 3} 2\) | ||||||||||||
| \(\ds e^{2 i \pi / 3}\) | \(=\) | \(\ds -\frac 1 2 + \frac {i \sqrt 3} 2\) | ||||||||||||
| \(\ds e^{i \pi}\) | \(=\) | \(\ds -1\) | ||||||||||||
| \(\ds e^{4 i \pi / 3}\) | \(=\) | \(\ds -\frac 1 2 - \frac {i \sqrt 3} 2\) | ||||||||||||
| \(\ds e^{5 i \pi / 3}\) | \(=\) | \(\ds \frac 1 2 - \frac {i \sqrt 3} 2\) |
Proof
By definition, the first complex $6$th root of unity $\alpha$ is given by:
| \(\ds \alpha\) | \(=\) | \(\ds e^{2 i \pi / 6}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds e^{i \pi / 3}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \cos \frac \pi 3 + i \sin \frac \pi 3\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 + i \frac {\sqrt 3} 2\) | Cosine of $\dfrac \pi 3$, Sine of $\dfrac \pi 3$ |
We have that:
- $e^{0 i \pi / 6} = e^0 = 1$
which gives us, as always, the zeroth complex $n$th root of unity for all $n$.
The remaining complex $6$th roots of unity follow:
| \(\ds \alpha^2\) | \(=\) | \(\ds \paren {\frac 1 2 + i \frac {\sqrt 3} 2}^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\frac 1 2}^2 + 2 i \cdot \frac 1 2 \cdot \frac {\sqrt 3} 2 + \paren {i \frac {\sqrt 3} 2}^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 4 + i \frac {\sqrt 3} 2 - \frac 3 4\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -\frac 1 2 + i \frac {\sqrt 3} 2\) |
| \(\ds \alpha^3\) | \(=\) | \(\ds e^{6 i \pi / 6}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds e^{i \pi}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -1\) | Euler's Identity |
| \(\ds \alpha^4\) | \(=\) | \(\ds \overline {\alpha^{6 - 4} }\) | Complex Roots of Unity occur in Conjugate Pairs | |||||||||||
| \(\ds \) | \(=\) | \(\ds \overline {\alpha^2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \overline {-\frac 1 2 + i \frac {\sqrt 3} 2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -\frac 1 2 - i \frac {\sqrt 3} 2\) | Definition of Complex Conjugate |
| \(\ds \alpha^5\) | \(=\) | \(\ds \overline {\alpha^{6 - 5} }\) | Complex Roots of Unity occur in Conjugate Pairs | |||||||||||
| \(\ds \) | \(=\) | \(\ds \overline {\alpha}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \overline {\frac 1 2 + i \frac {\sqrt 3} 2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 - i \frac {\sqrt 3} 2\) | Definition of Complex Conjugate |
$\blacksquare$
Illustration
From Complex Roots of Unity are Vertices of Regular Polygon Inscribed in Circle the complete set of complex $6$th roots of unity can be depicted as the vertices of the following regular hexagon:
where $\alpha$ is used to denote the first complex $6$th root of unity.
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity: Exercise $1$.