Complex Multiplication/Examples/((2 - i) (-3 + 2i)) (5 - 4i)

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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