# 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}