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

Example of Complex Arithmetic

$\dfrac {\paren {2 + i} \paren {3 - 2 i} \paren {1 + 2 i} } {\paren {1 - i}^2} = -\dfrac {15} 2 + 5 i$

Proof

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

\(=\)

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

Definition of Complex Multiplication

\(\ds \)

\(=\)

\(\ds \dfrac {\paren {2 + i} \paren {7 + 4 i} } {-2 i}\)

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

\(\ds \dfrac {10 i + 15 i^2} 2\)

\(\ds \)

\(=\)

\(\ds -\dfrac {15} 2 + 5 i\)

$\blacksquare$

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: $53 \ \text {(g)}$