Complex Roots of Unity/Examples/4th Roots

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Example of Complex 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:

\(\displaystyle e^{0 i \pi / 4}\) \(=\) \(\displaystyle 1\)
\(\displaystyle e^{i \pi / 2}\) \(=\) \(\displaystyle i\)
\(\displaystyle e^{i \pi}\) \(=\) \(\displaystyle -1\)
\(\displaystyle e^{3 i \pi / 2}\) \(=\) \(\displaystyle -i\)


Proof

By definition, the first complex $4$th root of unity $\alpha$ is given by:

\(\displaystyle \alpha\) \(=\) \(\displaystyle e^{2 i \pi / 4}\)
\(\displaystyle \) \(=\) \(\displaystyle e^{i \pi / 2}\)
\(\displaystyle \) \(=\) \(\displaystyle \cos \frac \pi 2 + i \sin \frac \pi 2\)
\(\displaystyle \) \(=\) \(\displaystyle 0 + i \times 1\) Cosine of $\dfrac \pi 2$, Sine of $\dfrac \pi 2$
\(\displaystyle \) \(=\) \(\displaystyle i\)


We have that:

$e^{0 i \pi / 4} = e^0 = 1$

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


The remaining complex $4$th roots of unity can be expressed as $e^{4 i \pi / 4} = e^{i \pi}$ and $e^{6 i \pi / 4} = e^{3 i \pi / 2}$, but it is simpler to calculate them as follows:

\(\displaystyle \alpha^2\) \(=\) \(\displaystyle i^2\)
\(\displaystyle \) \(=\) \(\displaystyle -1\) Definition of Imaginary Unit


\(\displaystyle \alpha^3\) \(=\) \(\displaystyle \alpha^2 \times \alpha\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {-1} \times i\)
\(\displaystyle \) \(=\) \(\displaystyle -i\)


Complex 4th Roots of 1.png

$\blacksquare$


Sources