Complex Multiplication/Examples/((2 - i) (-3 + 2i)) (5 - 4i)
Jump to navigation
Jump to search
Example of Complex Multiplication
- $\paren {\paren {2 - i} \paren {-3 + 2 i} } \paren {5 - 4 i} = 8 + 51 i$
Proof
\(\ds \paren {\paren {2 - i} \paren {-3 + 2 i} } \paren {5 - 4 i}\) | \(=\) | \(\ds \paren {\paren {2 \times \paren {-3} - \paren {-1} \times 2} + \paren {2 \times 2 + \paren {-3} \times \paren {-1} } i} \paren {5 - 4 i}\) | Definition of Complex Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {-6 + 2} + \paren {4 + 3} i} \paren {5 - 4 i}\) | simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-4 + 7 i} \paren {5 - 4 i}\) | simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {-4} \times 5 - 7 \times \paren {-4} } + \paren {\paren {-4} \times \paren {-4} + 7 \times 5} i\) | Definition of Complex Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-20 + 28} + \paren {7 + 44} i\) | 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: $1 \ \text{(i)}$