# How do you divide 3+i div 1-4i?

Dec 6, 2015

$- \frac{3}{17} + \frac{13}{17} i$

#### Explanation:

Multiply by the complex conjugate.

$\frac{3 + i}{1 - 4 i} = \frac{3 + i}{1 - 4 i} \left(\frac{1 + 4 i}{1 + 4 i}\right) = \frac{3 + 12 i + i + 4 {i}^{2}}{1 + 4 i - 4 i - 16 {i}^{2}} = \frac{1 + 13 i + 4 {i}^{2}}{1 - 16 {i}^{2}}$

Recall that $i = \sqrt{- 1}$, so ${i}^{2} = - 1$.

$= \frac{1 + 13 i + 4 \left(- 1\right)}{1 - 16 \left(- 1\right)} = \frac{1 - 4 + 13 i}{1 + 16} = \frac{- 3 + 13 i}{17} = - \frac{3}{17} + \frac{13}{17} i$

Notice how the answer is written in the $a + b i$ form of a complex number.