Complex Arithmetic/Examples/Half (z 3 over conj z 3 + conj z 3 over z 3)

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Example of Complex Arithmetic

Let $z_3 = \sqrt 3 - 2 i$.

Then:

$\dfrac 1 2 \paren {\dfrac {z_3} {\overline z_3} + \dfrac {\overline z_3} {z_3} } = -\dfrac 1 7$


Proof

\(\ds \dfrac 1 2 \paren {\dfrac {z_3} {\overline z_3} + \dfrac {\overline z_3} {z_3} }\) \(=\) \(\ds \dfrac 1 2 \paren {\dfrac {\sqrt 3 - 2 i} {\sqrt 3 + 2 i} + \dfrac {\sqrt 3 + 2 i} {\sqrt 3 - 2 i} }\)
\(\ds \) \(=\) \(\ds \dfrac 1 2 \paren {\dfrac {\paren {\sqrt 3 - 2 i}^2} {\paren {\sqrt 3 + 2 i} \paren {\sqrt 3 - 2 i} } + \dfrac {\paren {\sqrt 3 + 2 i}^2 } {\paren {\sqrt 3 - 2 i} \paren {\sqrt 3 + 2 i} } }\)
\(\ds \) \(=\) \(\ds \dfrac 1 2 \paren {\dfrac {3 - 4 \sqrt3 i + 4 i^2} {3 + 2^2} + \dfrac {3 + 4 \sqrt3 i + 4 i^2} {3 + 2^2} }\)
\(\ds \) \(=\) \(\ds \dfrac 1 2 \paren {\dfrac {3 + 3 - 4 - 4} 7}\)
\(\ds \) \(=\) \(\ds \dfrac 1 2 \paren {\dfrac {-2} 7}\)
\(\ds \) \(=\) \(\ds -\dfrac 1 7\)

$\blacksquare$


Sources

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