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 $\ds$ $=$ $\ds \dfrac {\paren {2 + i} \paren {7 + 4 i} } {-2 i}$ $\ds$ $=$ $\ds \dfrac {i \paren {\paren {2 \times 7 - 1 \times 4} + \paren {2 \times 4 + 1 \times 7} i} } 2$ $\ds$ $=$ $\ds \dfrac {10 i + 15 i^2} 2$ $\ds$ $=$ $\ds -\dfrac {15} 2 + 5 i$

$\blacksquare$