# How do you divide (5 - i) / (5 + i)?

Feb 2, 2016

$\frac{12}{13} - \frac{5}{13} i$

#### Explanation:

To divide 2 complex numbers , multiply the numerator and

denominator by the' complex conjugate' of the denominator.

If a + bi is a complex number then$\textcolor{red}{\text{ a - bi is the conjugate}}$

This ensures that the denominator is real , as shown below.

multiplying a complex number and it's conjugate.

$\left(a + b i\right) \left(a - b i\right) = {a}^{2} + a b i - a b i - b {i}^{2} = {a}^{2} + {b}^{2}$

which is real . [ remember  i^2 =( sqrt-1)^2 = -1]

hence question becomes: $\frac{\left(5 - i\right) \left(5 - i\right)}{\left(5 + i\right) \left(5 - i\right)}$

$= \frac{25 - 10 i + {i}^{2}}{25 - {i}^{2}} = \frac{24 - 10 i}{26} = \frac{24}{26} - \frac{10}{26} i$

which simplifies to : $\frac{12}{13} - \frac{5}{13} i$