How do you simplify (2+i)/(1+2i)?

Jan 2, 2018

$\frac{4}{5} - \frac{3}{5} i$

Explanation:

$\text{multiply the numerator/denominator by the}$
$\textcolor{b l u e}{\text{complex conjugate"" of the denominator}}$

$\text{the complex conjugate of "1+2i" is } 1 \textcolor{red}{-} 2 i$

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

$\text{expand factors on numerator/denominator using FOIL}$

$= \frac{2 - 3 i - 2 {i}^{2}}{1 - 4 {i}^{2}}$

$\left[{i}^{2} = {\left(\sqrt{- 1}\right)}^{2} = - 1\right]$

$= \frac{4 - 3 i}{5} = \frac{4}{5} - \frac{3}{5} i \leftarrow \textcolor{b l u e}{\text{in standard form}}$

Jan 2, 2018

$\frac{4}{5} - \frac{3 i}{5}$

Explanation:

To divide complex numbers we first remove the complex number from the denominator, by multiplying by the complex conjugate of the denominator:

This is $1 - 2 i$

The product of a complex number and its conjugate is always a real number.

$\therefore$

$\frac{\left(1 - 2 i\right) \left(2 + i\right)}{\left(1 - 2 i\right) \left(1 + 2 i\right)} = \frac{\left(1 - 2 i\right) \left(2 + i\right)}{5} = \frac{4 - 3 i}{5} = \frac{4}{5} - \frac{3 i}{5}$