# Find the complex conjugate of (3+2i)/(1-i) ?

Mar 29, 2018

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

#### Explanation:

The complex conjugate of a complex number is of the form ${\left(a + b i\right)}^{\ast} = a - b i$.

We have: $\frac{3 + 2 i}{1 - i} = \frac{3 + 2 i}{1 + \left(- i\right)}$

To perform division on complex numbers we multiply the numerator and denominator of the fraction by the denominator's complex conjugate.

This turns the denominator into a real number, allowing the fraction to be expressed in the form $a + b i$.

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

$= \frac{3 + 2 i}{1 - i} \cdot \frac{1 + i}{1 + i}$

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

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

$= \frac{3 + 2 + 5 i}{1 + 1}$

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

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

$\therefore {\left(\frac{5}{2} + \frac{5}{2} i\right)}^{\ast} = \frac{5}{2} - \frac{5}{2} i$

Therefore, the complex conjugate of $\frac{3 + 2 i}{1 - i}$ is $\frac{5}{2} - \frac{5}{2} i$.