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
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(f)}$