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
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity: Exercise $3$.