# How do you divide 10/(10+5i)?

Aug 2, 2016

$= \frac{4}{5} - \frac{2}{5} i$

#### Explanation:

you gotta multiply the denominator by its conjugate to make it real...and of course do the same to the denominator, with the result that the numerator will have the real and imaginary parts

BACKGROUND:

the conjugate of $z = x + i y$ is $t i l \mathrm{de} z = x - i y$

and $z \cdot t i l \mathrm{de} z = {x}^{2} + {y}^{2} = {\left\mid z \right\mid}^{2}$

so here we have

$\frac{10}{10 + 5 i} \cdot \frac{10 - 5 i}{10 - 5 i}$

$= \frac{2}{2 + i} \cdot \frac{2 - i}{2 - i}$ .....to make numbers a bit lighter

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

$= \frac{2}{5} \cdot \left(2 - i\right)$

$= \frac{4}{5} - \frac{2}{5} i$