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

Example of Complex Arithmetic

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

Proof

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

\(=\)

\(\ds \paren {i - 2} \paren {\paren {2 + 2 i} - \paren {-3 + 3 i} }\)

\(\ds \)

\(=\)

\(\ds \paren {-2 + i} \paren {5 - i}\)

Definition of Complex Subtraction

\(\ds \)

\(=\)

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

Definition of Complex Multiplication

\(\ds \)

\(=\)

\(\ds \paren {-10 + 1} + \paren {2 + 7} i\)

\(\ds \)

\(=\)

\(\ds -9 + 7 i\)

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $53 \ \text {(d)}$