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

Example of Complex Arithmetic

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

Proof

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

\(=\)

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

Definition of Complex Addition

\(\ds \)

\(=\)

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

simplification

\(\ds \)

\(=\)

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

Definition of Complex Multiplication

\(\ds \)

\(=\)

\(\ds \paren {-4 + 2} + \paren {1 + 8} i\)

simplification

\(\ds \)

\(=\)

\(\ds -2 + 9 i\)

$\blacksquare$

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