Roots of Complex Number/Examples/Cube Roots of 2+2i

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Example of Roots of Complex Number: Corollary

The complex cube roots of $2 + 2 i$ are given by:

$\paren {2 + 2 i}^{1/3} = \set {\sqrt 2 \paren {\cos \dfrac \pi {12} + i \sin \dfrac \pi {12} }, -1 + i, -\sqrt 2 \paren {\cos \dfrac {5 \pi} {12} + i \sin \dfrac {5 \pi} {12} }}$


Proof

Let $z = 2 + 2 i$.

Then: Then:

\(\ds \cmod z\) \(=\) \(\ds \sqrt {2^2 + \paren 2^2}\)
\(\ds \) \(=\) \(\ds \sqrt 8\)
\(\ds \) \(=\) \(\ds 2 \sqrt 2\)
\(\ds \) \(=\) \(\ds \sqrt {2^3}\)


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


\(\ds \map \sin {\arg z}\) \(=\) \(\ds \frac 2 {2 \sqrt 2}\)
\(\ds \) \(=\) \(\ds \frac {\sqrt 2} 2\)
\(\ds \leadsto \ \ \) \(\ds \arg z\) \(=\) \(\ds \frac \pi 4 \text { or } \frac {3 \pi} 4\) Sine of $\dfrac \pi 4$

and so:

$\arg z = \dfrac \pi 4$


Let $b$ be defined as:

\(\ds b\) \(=\) \(\ds \sqrt [3] {\sqrt {2^3} } \map \exp {\dfrac 1 3 \dfrac {i \pi} 4}\)
\(\ds \) \(=\) \(\ds \sqrt 2 \map \exp {\dfrac {i \pi} {12} }\)


Then we have that the complex cube roots of unity are:

$1, \exp {\dfrac {2 i \pi} 3}, \exp {\dfrac {-2 i \pi} 3}$


Thus from Roots of Complex Number: Corollary:

\(\ds b\) \(=\) \(\ds \sqrt 2 \map \exp {\dfrac {i \pi} {12} }\)
\(\ds \) \(=\) \(\ds 2 \map \exp {\dfrac {i \pi} 6}\)
\(\ds \) \(=\) \(\ds \sqrt 2 \paren {\cos \dfrac \pi {12} + i \sin \dfrac \pi {12} }\)


\(\ds b \exp {\dfrac {2 i \pi} 3}\) \(=\) \(\ds \sqrt 2 \map \exp {\dfrac {i \pi} {12} } \exp {\dfrac {2 i \pi} 3}\)
\(\ds \) \(=\) \(\ds \sqrt 2 \map \exp {\dfrac {i \pi} {12} + \dfrac {2 i \pi} 3}\)
\(\ds \) \(=\) \(\ds \sqrt 2 \map \exp {\dfrac {9 i \pi} {12} }\)
\(\ds \) \(=\) \(\ds \sqrt 2 \map \exp {\dfrac {3 i \pi} 4}\)
\(\ds \) \(=\) \(\ds \sqrt 2 \paren {\cos \dfrac {3 i \pi} 4 + i \sin \dfrac {3 i \pi} 4}\)
\(\ds \) \(=\) \(\ds \sqrt 2 \paren {-\dfrac {\sqrt 2} 2 + i -\dfrac {\sqrt 2} 2}\) Cosine of $\dfrac {3 \pi} 4$, Sine of $\dfrac {3 \pi} 4$
\(\ds \) \(=\) \(\ds -1 + i\)


\(\ds b \exp {\dfrac {-2 i \pi} 3}\) \(=\) \(\ds \sqrt 2 \map \exp {\dfrac {i \pi} {12} } \exp {\dfrac {-2 i \pi} 3}\)
\(\ds \) \(=\) \(\ds \sqrt 2 \map \exp {\dfrac {i \pi} {12} - \dfrac {2 i \pi} 3}\)
\(\ds \) \(=\) \(\ds \sqrt 2 \map \exp {\dfrac {-7 i \pi} 2}\)
\(\ds \) \(=\) \(\ds -\sqrt 2 \paren {\cos \dfrac {5 \pi} {12} + i \sin \dfrac {5 \pi} {12} }\)

$\blacksquare$


Sources

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