A calculus of the absurd

14.1.2 Dividing complex numbers

Another operation we can perform is division. To divide two complex numbers (which we can call \(w\) and \(z\)), we do something similar to rationalising the denominator (except for complex numbers).

For example, in the specific case where \(z = 1 + i\) and \(w = 3 + 4i\), we have can make the denominator "real" through multiplication by the complex conjugate.

\begin{align} \frac {z}{w} &= \frac {1 + i}{3 + 4i} \\ &= \frac {1 + i}{3 + 4i} \cdot \frac {1+i}{3 - 4i} \\ &= \frac {(1+i)(3-4i)}{25} \end{align}