Complex Arithmetic/Examples/(-1 + 2i) (7 - 5i) + (-1 + 2i) (-3 + 4i)

Example of Complex Arithmetic

$\paren {-1 + 2 i} \paren {7 - 5 i} + \paren {-1 + 2 i} \paren {-3 + 4 i} = -2 + 9 i$

Proof

\(\ds \paren {-1 + 2 i} \paren {7 - 5 i} + \paren {-1 + 2 i} \paren {-3 + 4 i}\)

\(=\)

\(\ds \paren {\paren {\paren {-1} \times 7 - 2 \times \paren {-5} } + \paren {-1 \times \paren {-5} + 2 \times 7} i} + \paren {\paren {\paren {-1} \times \paren {-3} - 2 \times 4} + \paren {\paren {-1} \times 4 + 2 \times \paren {-3} } i}\)

Definition of Complex Multiplication

\(\ds \)

\(=\)

\(\ds \paren {\paren {-7 + 10} + \paren {5 + 14 i} } + \paren {\paren {3 - 8} + \paren {-4 - 6 i} }\)

simplification

\(\ds \)

\(=\)

\(\ds \paren {3 + 19 i} + \paren {-5 - 10 i}\)

simplification

\(\ds \)

\(=\)

\(\ds \paren {3 - 5} + \paren {19 - 10} i\)

Definition of Complex Addition

\(\ds \)

\(=\)

\(\ds -2 + 9 i\)

$\blacksquare$

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