Complex Roots of Unity/Examples
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Examples of Complex Roots of Unity
Complex Square Roots of Unity
The complex square roots of unity are the elements of the set:
- $U_2 = \set {z \in \C: z^2 = 1}$
They are:
\(\ds e^{0 i \pi / 2}\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds e^{2 i \pi / 2}\) | \(=\) | \(\ds -1\) |
Complex Cube Roots of Unity
The complex cube roots of unity are the elements of the set:
- $U_3 = \set {z \in \C: z^3 = 1}$
They are:
\(\ds \) | \(\) | \(\, \ds e^{0 i \pi / 3} \, \) | \(\, \ds = \, \) | \(\ds 1\) | ||||||||||
\(\ds \omega\) | \(=\) | \(\, \ds e^{2 i \pi / 3} \, \) | \(\, \ds = \, \) | \(\ds -\frac 1 2 + \frac {i \sqrt 3} 2\) | ||||||||||
\(\ds \omega^2\) | \(=\) | \(\, \ds e^{4 i \pi / 3} \, \) | \(\, \ds = \, \) | \(\ds -\frac 1 2 - \frac {i \sqrt 3} 2\) |
The notation $\omega$ for, specifically, the complex cube roots of unity is conventional.
Complex $4$th Roots of Unity
The complex $4$th roots of unity are the elements of the set:
- $U_n = \set {z \in \C: z^4 = 1}$
They are:
\(\ds e^{0 i \pi / 4}\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds e^{i \pi / 2}\) | \(=\) | \(\ds i\) | ||||||||||||
\(\ds e^{i \pi}\) | \(=\) | \(\ds -1\) | ||||||||||||
\(\ds e^{3 i \pi / 2}\) | \(=\) | \(\ds -i\) |
Complex $5$th 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:
\(\ds e^{0 \pi / 5}\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds e^{2 \pi / 5}\) | \(=\) | \(\ds \dfrac {\sqrt 5 - 1} 4 + i \dfrac {\sqrt {10 + 2 \sqrt 5} } 4\) | ||||||||||||
\(\ds e^{4 \pi / 5}\) | \(=\) | \(\ds -\dfrac {1 + \sqrt 5} 4 + i \sqrt {\dfrac 5 8 - \dfrac {\sqrt 5} 8}\) | ||||||||||||
\(\ds e^{6 \pi / 5}\) | \(=\) | \(\ds -\dfrac {1 + \sqrt 5} 4 - i \sqrt {\dfrac 5 8 - \dfrac {\sqrt 5} 8}\) | ||||||||||||
\(\ds e^{8 \pi / 5}\) | \(=\) | \(\ds \dfrac {\sqrt 5 - 1} 4 - i \dfrac {\sqrt {10 + 2 \sqrt 5} } 4\) |
Complex $6$th 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:
\(\ds e^{0 i \pi / 6}\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds e^{i \pi / 3}\) | \(=\) | \(\ds \frac 1 2 + \frac {i \sqrt 3} 2\) | ||||||||||||
\(\ds e^{2 i \pi / 3}\) | \(=\) | \(\ds -\frac 1 2 + \frac {i \sqrt 3} 2\) | ||||||||||||
\(\ds e^{i \pi}\) | \(=\) | \(\ds -1\) | ||||||||||||
\(\ds e^{4 i \pi / 3}\) | \(=\) | \(\ds -\frac 1 2 - \frac {i \sqrt 3} 2\) | ||||||||||||
\(\ds e^{5 i \pi / 3}\) | \(=\) | \(\ds \frac 1 2 - \frac {i \sqrt 3} 2\) |
Complex $7$th 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:
\(\ds e^{0 \pi / 7}\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds e^{2 \pi / 7}\) | \(=\) | \(\ds \cis \dfrac {2 \pi} 7\) | ||||||||||||
\(\ds e^{4 \pi / 7}\) | \(=\) | \(\ds \cis \dfrac {4 \pi} 7\) | ||||||||||||
\(\ds e^{6 \pi / 7}\) | \(=\) | \(\ds \cis \dfrac {6 \pi} 7\) | ||||||||||||
\(\ds e^{8 \pi / 7}\) | \(=\) | \(\ds \cis \dfrac {8 \pi} 7\) | ||||||||||||
\(\ds e^{10 \pi / 7}\) | \(=\) | \(\ds \cis \dfrac {10 \pi} 7\) | ||||||||||||
\(\ds e^{12 \pi / 7}\) | \(=\) | \(\ds \cis \dfrac {12 \pi} 7\) |