# How do you divide imaginary numbers?

Apr 25, 2018

$\frac{a + b i}{c + \mathrm{di}} = \frac{a c + b d}{{c}^{2} + {d}^{2}} + i \frac{b c - a d}{{c}^{2} + {d}^{2}}$

#### Explanation:

Suppose we wanted to determine

$\frac{a + b i}{c + \mathrm{di}}$

We can multiply the numerator and denominator by the complex conjugate of the denominator. In this case the complex conjugate of the denominator is $c - \mathrm{di}$.

$\frac{a + b i}{c + \mathrm{di}} = \frac{\left(a + b i\right) \left(c - \mathrm{di}\right)}{\left(c + \mathrm{di}\right) \left(c - \mathrm{di}\right)}$

$= \frac{a c - a \mathrm{di} + b c i + b d}{{c}^{2} - c \mathrm{di} + c \mathrm{di} + {d}^{2}}$

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

$= \frac{a c + b d}{{c}^{2} + {d}^{2}} + i \frac{b c - a d}{{c}^{2} + {d}^{2}}$
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