Complex Arithmetic/Examples/Modulus of (z 1 conj z 2 + z 2 conj z 1)
Jump to navigation
Jump to search
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
\(\ds \cmod {z_1 \overline {z_2} + z_2 \overline {z_1} }\) | \(=\) | \(\ds \cmod {\paren {1 - i} \paren {-2 - 4 i} + \paren {-2 + 4 i} \paren {1 + i} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {\paren {-2 - 4 i + 2 i + 4 i^2} + \paren {-2 - 2 i + 4 i + 4 i^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {\paren {-6 - 2 i} + \paren {-6 + 2 i} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {12}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 12\) |
$\blacksquare$
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: $54 \ \text {(d)}$