Dividing by a complex number looks awkward: what is $\frac{1}{2 + i}$ in the form $a + bi$? The tool is something you saw in Adding and Multiplying Complex Numbers: a number times its conjugate is real. It is the same trick as rationalizing a denominator with a square root.
Key idea
To divide, multiply the numerator and denominator by the conjugate of the denominator (change the sign of its $i$ part). The denominator becomes the real number $c^2 + d^2$, and then you split the fraction into $a + bi$.
Why it works
Multiplying by $\frac{c - di}{c - di}$ is multiplying by 1, so the value does not change. The point is the denominator: $(c + di)(c - di) = c^2 - d^2i^2$ $= c^2 + d^2$, a positive real number. Once the denominator is real, dividing is just dividing each part by an ordinary number.
Worked examples
Worked example: A whole-number answer
Simplify $\frac{7 + i}{1 - i}$.
- Multiply top and bottom by the conjugate $1 + i$.
- Bottom: $(1 - i)(1 + i) = 1 + 1 = 2$.
- Top: $(7 + i)(1 + i) = 7 + 7i + i + i^2$ $= 6 + 8i$.
- Divide each part by 2: $3 + 4i$.
Answer: $3 + 4i$. Check: $(3 + 4i)(1 - i) = 3 - 3i + 4i + 4$ $= 7 + i$.
Worked example: Fractions in the answer
Simplify $\frac{1}{2 + i}$.
- Multiply by $\frac{2 - i}{2 - i}$: bottom $4 + 1 = 5$, top $2 - i$.
- $\frac{2 - i}{5} = \frac{2}{5} - \frac{1}{5}i$.
Answer: $\frac{2}{5} - \frac{1}{5}i$
Worked example: Dividing by a pure imaginary number
Simplify $\frac{6 - 4i}{2i}$.
- The conjugate of $2i$ is $-2i$, but multiplying by $i$ also works: $$\frac{(6 - 4i)i}{2i \cdot i} = \frac{6i - 4i^2}{2i^2} = \frac{4 + 6i}{-2}.$$
- Divide each part: $-2 - 3i$.
Answer: $-2 - 3i$
Common mistake
Multiplying only the denominator by the conjugate. That changes the value of the fraction. You must multiply the top and the bottom by the same thing, so you are multiplying by 1. Check any answer by multiplying it back by the denominator.