Roots of Complex Number/Examples/Cube Roots of 2 + 2 root 3 i
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Example of Roots of Complex Number
The complex cube roots of $2 + 2 \sqrt 3 i$ are given by:
- $\paren {2 + 2 \sqrt 3 i}^{1/3} = \set {\sqrt [3] 4 \, \map \cis {20 + 120 k} \degrees}$
for $k = 0, 1, 2$.
That is:
\(\ds k = 0: \ \ \) | \(\ds z = z_1\) | \(=\) | \(\ds \sqrt [3] 4 \cis 20 \degrees\) | |||||||||||
\(\ds k = 1: \ \ \) | \(\ds z = z_2\) | \(=\) | \(\ds \sqrt [3] 4 \cis 140 \degrees\) | |||||||||||
\(\ds k = 2: \ \ \) | \(\ds z = z_3\) | \(=\) | \(\ds \sqrt [3] 4 \cis 260 \degrees\) |
Proof
Let $z^3 = 2 + 2 \sqrt 3 i$.
We have that:
- $z^3 = 4 \, \map \cis {\dfrac \pi 3 + 2 k \pi} = 4 \, \map \cis {60 \degrees + k \times 360 \degrees}$
Let $z = r \cis \theta$.
Then:
\(\ds z^3\) | \(=\) | \(\ds r^3 \cis 3 \theta\) | De Moivre's Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \, \map \cis {\dfrac \pi 3 + 2 k \pi}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds r^3\) | \(=\) | \(\ds 4\) | |||||||||||
\(\ds 3 \theta\) | \(=\) | \(\ds \dfrac \pi 3 + 2 k \pi\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds r\) | \(=\) | \(\ds 4^{1/3}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt [3] 4\) | ||||||||||||
\(\ds \theta\) | \(=\) | \(\ds \dfrac \pi 9 + \dfrac {2 k \pi} 3\) | for $k = 0, 1, 2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 20 \degrees + 120 k \degrees\) | for $k = 0, 1, 2$ |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Roots of Complex Numbers: $95 \ \text{(c)}$