Complex Multiplication/Examples/(2 - i) ((-3 + 2i) (5 - 4i))
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Example of Complex Multiplication
- $\paren {2 - i} \paren {\paren {-3 + 2 i} \paren {5 - 4 i} } = 8 + 51 i$
Proof
\(\ds \paren {2 - i} \paren {\paren {-3 + 2 i} \paren {5 - 4 i} }\) | \(=\) | \(\ds \paren {2 - i} \paren {\paren {\paren {-3} \times 5 - 2 \times \paren {-4} } + \paren {2 \times 5 + \paren {-3} \times \paren {-4} } i}\) | Definition of Complex Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 - i} \paren {\paren {-15 + 8} + \paren {10 + 12} i}\) | simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 - i} \paren {-7 + 22 i}\) | simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 \times \paren {-7} - \paren {-1} \times 22} + \paren {\paren {-1} \times \paren {-7} + 2 \times 22} i\) | Definition of Complex Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-14 + 22} + \paren {7 + 44} i\) | simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds 8 + 51 i\) |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $1 \ \text{(h)}$