# Complex Arithmetic/Examples/((1 + root 3 i)(1 - root 3 i)^-1)^10

## Example of Complex Arithmetic

$\paren {\dfrac {1 + \sqrt 3 i} {1 - \sqrt 3 i} }^{10} = -\dfrac 1 2 + \dfrac {\sqrt 3} 2 i$

## Proof 1

 $\ds \paren {\dfrac {1 + \sqrt 3 i} {1 - \sqrt 3 i} }^{10}$ $=$ $\ds \paren {\dfrac {2 \, \map \cis {60 \degrees} } {2 \, \map \cis {-60 \degrees} } }^{10}$ $\ds$ $=$ $\ds \paren {\cis 120 \degrees}^{10}$ Division of Complex Numbers in Polar Form $\ds$ $=$ $\ds \cis 1200 \degrees$ De Moivre's Theorem $\ds$ $=$ $\ds \map \cis {3 \times 360 \degrees + 120 \degrees}$ $\ds$ $=$ $\ds \cis 120 \degrees$ simplifying $\ds$ $=$ $\ds -\dfrac 1 2 + \dfrac {\sqrt 3} 2 i$ Cosine of $120 \degrees$, Sine of $120 \degrees$

$\blacksquare$

## Proof 2

 $\ds \paren {\dfrac {1 + \sqrt 3 i} {1 - \sqrt 3 i} }^{10}$ $=$ $\ds \paren {\dfrac {2 e^{\pi i / 3} } {2 e^{-\pi i / 3} } }^{10}$ $\ds$ $=$ $\ds \paren {e^{2 \pi i / 3} }^{10}$ 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 $120 \degrees$

$\blacksquare$