Roots of Complex Number/Examples/5th Roots of -4 + 4i
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Example of Roots of Complex Number
The complex $5$th roots of $-4 + 4i$ are given by:
- $\paren {-4 + 4i}^{1/5} = \set {\sqrt 2 \, \map \cis {27 + 72 k} \degrees}$
for $k = 0, 1, 2, 3, 4$.
That is:
\(\ds k = 0: \ \ \) | \(\ds z = z_1\) | \(=\) | \(\ds \sqrt 2 \cis 27 \degrees\) | |||||||||||
\(\ds k = 1: \ \ \) | \(\ds z = z_2\) | \(=\) | \(\ds \sqrt 2 \cis 99 \degrees\) | |||||||||||
\(\ds k = 2: \ \ \) | \(\ds z = z_3\) | \(=\) | \(\ds \sqrt 2 \cis 171 \degrees\) | |||||||||||
\(\ds k = 3: \ \ \) | \(\ds z = z_4\) | \(=\) | \(\ds \sqrt 2 \cis 243 \degrees\) | |||||||||||
\(\ds k = 4: \ \ \) | \(\ds z = z_5\) | \(=\) | \(\ds \sqrt 2 \cis 315 \degrees\) |
Proof
Let $z^5 = -4 + 4 i$.
We have that:
- $z^5 = 4 \sqrt 2 \, \map \cis {\dfrac {3 \pi} 4 + 2 k \pi} = 4 \sqrt 2 \, \map \cis {135 \degrees + k \times 360 \degrees}$
Let $z = r \cis \theta$.
Then:
\(\ds z^5\) | \(=\) | \(\ds r^5 \cis 5 \theta\) | De Moivre's Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \sqrt 2 \, \map \cis {\dfrac {3 \pi} 4 + 2 k \pi}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds r^5\) | \(=\) | \(\ds 4 \sqrt 2\) | |||||||||||
\(\ds 5 \theta\) | \(=\) | \(\ds \dfrac {3 \pi} 4 + 2 k \pi\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds r\) | \(=\) | \(\ds \paren {4 \sqrt 2} ^{1/5}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt 2\) | ||||||||||||
\(\ds \theta\) | \(=\) | \(\ds \dfrac {3 \pi} {20} + \dfrac {2 k \pi} 5\) | for $k = 0, 1, 2, 3, 4$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 27 \degrees + 72 k \degrees\) | for $k = 0, 1, 2, 3, 4$ |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Roots of Complex Numbers: $95 \ \text{(b)}$