Complex Arithmetic/Examples/((1 + root 3 i)(1 - root 3 i)^-1)^10/Proof 2
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Example of Complex Arithmetic
- $\paren {\dfrac {1 + \sqrt 3 i} {1 - \sqrt 3 i} }^{10} = -\dfrac 1 2 + \dfrac {\sqrt 3} 2 i$
Proof
| \(\ds \paren {\dfrac {1 + \sqrt 3 i} {1 - \sqrt 3 i} }^{10}\) | \(=\) | \(\ds \paren {\dfrac {2 e^{\pi i / 3} } {2 e^{-\pi i / 3} } }^{10}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {e^{2 \pi i / 3} }^{10}\) | Division of Complex Numbers in Polar Form | |||||||||||
| \(\ds \) | \(=\) | \(\ds e^{20 \pi i / 3}\) | De Moivre's Theorem | |||||||||||
| \(\ds \) | \(=\) | \(\ds e^{6 \pi i} e^{2 \pi i / 3}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 \times \paren {\cos \dfrac {2 \pi} 3 + i \sin \dfrac {2 \pi} 3}\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\dfrac 1 2 + \dfrac {\sqrt 3} 2 i\) | Cosine of $120 \degrees$, Sine of $120 \degrees$ |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: De Moivre's Theorem: $26 \ \text {(c)}$