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). Note that \(\sqrt {-1}\) behaves just like a surd (except that it’s not in the set of real numbers), so it shouldn’t come as any great surprise that the same techniques are useful here as for surds.

  • Example 14.1.1 Divide \(1 + i\) by \(3 + 4i\).

We want to make the denominator real, so we just rationalise it,

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