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

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: $1 \ \text{(h)}$