# Complex Arithmetic/Examples/(4+2i)^-1 (2-3i)^-1

## Example of Complex Arithmetic

$\dfrac 1 {\paren {4 + 2 i} \paren {2 - 3 i} } = \dfrac 7 {130} + \dfrac {2 i} {65}$

## Proof

 $\ds \dfrac 1 {4 + 2 i} \times \dfrac 1 {2 - 3 i}$ $=$ $\ds \dfrac {4 - 2 i} {\paren {4 + 2 i} \paren {4 - 2 i} } \times \dfrac {2 + 3 i} {\paren {2 - 3 i} \paren {2 + 3 i} }$ multiplying top and bottom by conjugate of bottom $\ds$ $=$ $\ds \dfrac {4 - 2 i} {4^2 + 2^2} \times \dfrac {2 + 3 i} {2^2 + 3^2}$ simplifying $\ds$ $=$ $\ds \dfrac {\paren {4 - 2 i} \paren {2 + 3 i} } {20 \times 13}$ simplifying $\ds$ $=$ $\ds \dfrac {\paren {4 \times 2 - \paren {-2} \times 3} + \paren {4 \times 3 + \paren {-2} \times 2} i} {20 \times 13}$ Definition of Complex Multiplication $\ds$ $=$ $\ds \dfrac {14 + 8 i} {260}$ simplification $\ds$ $=$ $\ds \dfrac 7 {130} + \dfrac {2 i} {65}$ simplification

$\blacksquare$