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

Example of Complex Arithmetic

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

Proof

\(\ds \paren {\dfrac {1 + \sqrt 3 i} {1 - \sqrt 3 i} }^{10}\)

\(=\)

\(\ds \paren {\dfrac {2 \, \map \cis {60 \degrees} } {2 \, \map \cis {-60 \degrees} } }^{10}\)

\(\ds \)

\(=\)

\(\ds \paren {\cis 120 \degrees}^{10}\)

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 $120 \degrees$

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: De Moivre's Theorem: $26 \ \text {(c)}$