# How do you divide  (6-i) / (1-i) ?

Jan 19, 2016

Explanation is given below.

#### Explanation:

For dividing complex numbers, you just need to multiply the numerator and denominator by the conjugate of the denominator.

If the complex number is $a + i b$ then the conjugate is given by $a - i b$. $\textcolor{red}{\text{The imaginary term sign has to be changed}}$

In our problem $\frac{6 - i}{1 - i}$

The denominator is $\left(1 - i\right)$ remember for the conjugate we change the sign of the denominator.

The conjugate of $1 - i$ is $1 + i$

So we have

$\frac{6 - i}{1 - i} \times \frac{1 + i}{1 + i}$

$= \frac{\left(6 - i\right) \left(1 + i\right)}{\left(1 - i\right) \left(1 + i\right)}$

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

$= \frac{6 - 7 i - 1}{1 - \left(- 1\right)}$

$= \frac{7 - 7 i}{1 + 1}$

$= \frac{7 - 7 i}{2}$

$= \frac{7}{2} - \frac{7}{2} i$