Complex Arithmetic/Examples/Modulus of (z 1 conj z 2 + z 2 conj z 1)

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Example of Complex Arithmetic

Let $z_1 = 1 - i$ and $z_2 = -2 + 4 i$.

Then:

$\cmod {z_1 \overline {z_2} + z_2 \overline {z_1} } = 12$


Proof

\(\ds \cmod {z_1 \overline {z_2} + z_2 \overline {z_1} }\) \(=\) \(\ds \cmod {\paren {1 - i} \paren {-2 - 4 i} + \paren {-2 + 4 i} \paren {1 + i} }\)
\(\ds \) \(=\) \(\ds \cmod {\paren {-2 - 4 i + 2 i + 4 i^2} + \paren {-2 - 2 i + 4 i + 4 i^2} }\)
\(\ds \) \(=\) \(\ds \cmod {\paren {-6 - 2 i} + \paren {-6 + 2 i} }\)
\(\ds \) \(=\) \(\ds \cmod {12}\)
\(\ds \) \(=\) \(\ds 12\)

$\blacksquare$


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