To divide complex numbers when the bottom part has a complex number, we multiply both the top and bottom by the conjugate of the bottom number.

Example:

Let’s look at dividing two complex numbers:

$\frac{3 + 4i}{1 - 2i}$

1. Find the Conjugate: The conjugate of $1 - 2i$ is $1 + 2i$.

2. Multiply: $\frac{(3 + 4i)(1 + 2i)}{(1 - 2i)(1 + 2i)}$

3. Simplify the Bottom: $(1 - 2i)(1 + 2i) = 1^2 + (2i)^2 = 1 + 4 = 5$

4. Expand the Top: $(3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i^2 = 3 + 10i - 8 = -5 + 10i$

5. Final Answer: $\frac{-5 + 10i}{5} = -1 + 2i$

So, $\frac{3 + 4i}{1 - 2i} = -1 + 2i$.