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

Jan 21, 2017

The answer is $= \frac{2}{5} + \frac{1}{5} i$

#### Explanation:

When you have a complex number in the denominator, multiply the numerator and denominator by the conjugate of the denominator.

${i}^{2} = - 1$

The conjugate of $\left(1 + 2 i\right)$ is $\left(1 - 2 i\right)$

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

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

$= \frac{2 + i}{5}$

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