Roots of Complex Number/Corollary/Examples/Fourth Roots of 2-2i

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Example of Roots of Complex Number: Corollary

The complex $4$th roots of $2 - 2 i$ are given by:

$\paren {2 - 2 i}^{1/4} = \set {b, bi, -b, -bi}$

where:

$b = \sqrt [8] 8 \paren {\cos \dfrac \pi {16} + i \sin \dfrac \pi {16} }$


Proof

Let $z = 2 - 2 i$.

Then:

\(\ds \cmod z\) \(=\) \(\ds \sqrt {2^2 + \paren {-2}^2}\)
\(\ds \) \(=\) \(\ds \sqrt 8\)
\(\ds \) \(=\) \(\ds 2 \sqrt 2\)


\(\ds \cos \paren {\arg z}\) \(=\) \(\ds \frac 2 {2 \sqrt 2}\)
\(\ds \) \(=\) \(\ds \frac {\sqrt 2} 2\)
\(\ds \leadsto \ \ \) \(\ds \arg z\) \(=\) \(\ds \pm \frac \pi 4\) Cosine of $45 \degrees$


\(\ds \sin \paren {\arg z}\) \(=\) \(\ds \frac {-2} {2 \sqrt 2}\)
\(\ds \) \(=\) \(\ds -\frac {\sqrt 2} 2\)
\(\ds \leadsto \ \ \) \(\ds \arg z\) \(=\) \(\ds -\frac \pi 4 \text { or } -\frac {3 \pi} 4\) Sine of $315 \degrees$

and so:

$\arg z = -\dfrac \pi 4$


Let $b$ be defined as:

\(\ds b\) \(=\) \(\ds \sqrt [4] {2 \sqrt 2} \paren {\cos \paren {-\dfrac 1 4 \dfrac \pi 4} + i \sin \paren {-\dfrac 1 4 \dfrac \pi 4} }\)
\(\ds \) \(=\) \(\ds \sqrt [8] 8 \paren {\cos \dfrac \pi {16} - i \sin \dfrac \pi {16} }\)


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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