# How do you rationalize imaginary denominators?

Mar 30, 2015

If $a + b i$ is an imaginary number
its conjugate is $a - b i$ (also an imaginary number)
and
the product of an imaginary number and its conjugate it not an imaginary number.
$\left(a + b i\right) \times \left(a - b i\right) = {a}^{2} - {b}^{2}$

If you have a number with an imaginary denominator multiply both the numerator and denominator by the conjugate of the denominator.

For example, suppose you want to rationalize the denominator of
$\frac{10}{3 + 2 i}$

$\frac{10}{3 + 2 i} \cdot \frac{3 - 2 i}{3 - 2 i}$

$= \frac{10 \left(3 - 2 i\right)}{{3}^{2} - {2}^{2}}$

$= \frac{30 - 20 i}{9 - 4}$

$= 6 - 4 i$