Complex Roots of Unity/Examples

From ProofWiki
Jump to navigation Jump to search

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:

\(\ds e^{0 \pi / 7}\) \(=\) \(\ds 1\)
\(\ds e^{2 \pi / 7}\) \(=\) \(\ds \cis \dfrac {2 \pi} 7\)
\(\ds e^{4 \pi / 7}\) \(=\) \(\ds \cis \dfrac {4 \pi} 7\)
\(\ds e^{6 \pi / 7}\) \(=\) \(\ds \cis \dfrac {6 \pi} 7\)
\(\ds e^{8 \pi / 7}\) \(=\) \(\ds \cis \dfrac {8 \pi} 7\)
\(\ds e^{10 \pi / 7}\) \(=\) \(\ds \cis \dfrac {10 \pi} 7\)
\(\ds e^{12 \pi / 7}\) \(=\) \(\ds \cis \dfrac {12 \pi} 7\)
Retrieved from "https://proofwiki.org/w/index.php?title=Complex_Roots_of_Unity/Examples&oldid=520329"