# How do you simplify  (2+2i)/(1+2i)  and write in a+bi form?

Dec 11, 2015

Multiply the numerator and denominator by the complex conjugate of the denominator to find

$\frac{2 + 2 i}{1 + 2 i} = \frac{6}{5} - \frac{2}{5} i$

#### Explanation:

Given a complex number $a + b i$ with $a , b \in \mathbb{R}$ we have

$\left(a + b i\right) \left(a - b i\right) = {a}^{2} + {b}^{2}$

$a - b i$ is called the complex conjugate (or conjugate) of $a + b i$. Using this:

$\frac{2 + 2 i}{1 + 2 i} = \frac{2 + 2 i}{1 + 2 i} \cdot \frac{1 - 2 i}{1 - 2 i}$

$= \frac{\left(2 + 2 i\right) \left(1 - 2 i\right)}{{1}^{2} + {2}^{2}}$

$= \frac{6 - 2 i}{5}$

$= \frac{6}{5} - \frac{2}{5} i$