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

## Example of Complex Arithmetic

$\dfrac {\paren {1 + 2 i}^2} {1 - i} = -\dfrac 7 2 + \dfrac 1 2 i$

## Proof

 $\displaystyle \dfrac {\paren {1 + 2 i}^2} {1 - i}$ $=$ $\displaystyle \dfrac {1 + 4 i + 4 i^2} {1 - i}$ multiplying out numerator $\displaystyle$ $=$ $\displaystyle \dfrac {1 + 4 i - 4} {1 - i}$ Definition of Imaginary Unit $\displaystyle$ $=$ $\displaystyle \dfrac {-3 + 4 i} {1 - i}$ $\displaystyle$ $=$ $\displaystyle \dfrac {\paren {-3 + 4 i} \left({1 + i}\right)} {\left({1 - i}\right) \left({1 + i}\right)}$ multiplying top and bottom by $1 + i$ $\displaystyle$ $=$ $\displaystyle \dfrac {-3 - 3 i + 4 i + 4 i^2} {1^2 + 1^2}$ simplifying $\displaystyle$ $=$ $\displaystyle \dfrac {-7 + i} 2$ simplifying

$\blacksquare$