Complex Arithmetic/Examples/(Modulus of 2 z 2 - 3 z 1)^2
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Example of Complex Arithmetic
Let $z_1 = 1 - i$ and $z_2 = -2 + 4 i$.
Then:
- $\cmod {2 z_2 - 3 z_1}^2 = 170$
Proof
\(\ds \cmod {2 z_2 - 3 z_1}^2\) | \(=\) | \(\ds \cmod {2 \paren {-2 + 4 i} - 3 \paren {1 - i} }^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {\paren {-4 + 8 i} - \paren {3 - 3 i} }^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {-7 + 11 i}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 7^2 + 11^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 49 + 121\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 170\) |
$\blacksquare$
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: $54 \ \text {(b)}$