Roots of Complex Number/Examples/4th Roots of i

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Example of Roots of Complex Number: Corollary

The complex $4$th roots of $i$ are given by:

$i^{1/4} = \set {b, bi, -b, -bi}$

where:

$b = \paren {\cos \dfrac \pi 8 + i \sin \dfrac \pi 8}$


Proof

Let $z = i$.

Then:

\(\ds \cmod z\) \(=\) \(\ds \sqrt {0 + \paren 1^2}\)
\(\ds \) \(=\) \(\ds \sqrt 1\)
\(\ds \) \(=\) \(\ds 1\)


\(\ds \cos \paren {\arg z}\) \(=\) \(\ds \frac 0 1\)
\(\ds \) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds \arg z\) \(=\) \(\ds \pm \frac \pi 2\) Cosine of Right Angle


\(\ds \sin \paren {\arg z}\) \(=\) \(\ds \frac 1 1\)
\(\ds \) \(=\) \(\ds 1\)
\(\ds \leadsto \ \ \) \(\ds \arg z\) \(=\) \(\ds \frac \pi 2\) Sine of Right Angle

and so:

$\arg z = \frac \pi 2$


Let $b$ be defined as:

\(\ds b\) \(=\) \(\ds \sqrt [4] 1 \paren {\cos \paren {\dfrac 1 4 \frac \pi 2} + i \sin \paren {\dfrac 1 4 \frac \pi 2} }\)
\(\ds \) \(=\) \(\ds \cos \dfrac \pi 8 + i \sin \dfrac \pi 8\)


Then we have that the complex $4$th roots of unity are:

$1, i, -1, -i$

The result follows from Roots of Complex Number: Corollary.

$\blacksquare$


Sources

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