# Complex conjugates and dividing complex numbers

## Complex conjugates

In order to divide a complex number by another, we need to understand the concept of a complex conjugate.

Definition: A complex conjugate of a complex number $$z = x+yi$$, where $$x$$ and $$y$$ are real, is defined as $$\overline{z} = x-yi.$$

In other words, the imaginary part changes sign.

Worked exercise: Evaluate $$\overline{-2-i}$$ and $$\overline{9}$$.
Solution: $$\overline{-2-i} = -2 + i$$ and $$\overline{9} = 9$$

Notice that the conjugate of a real number is itself.

## Dividing complex numbers

Complex conjugates are useful because when you multiply a complex number by its conjugate, you get a real number. This allows us to simplify a fraction of complex numbers by multiplying the top and bottom by the conjugate of the denominator, to “realise the denominator”.

Worked exercise: Find the conjugate of $$z = 4+i$$ and show that $$z \overline{z}$$ is a real number. Hence, evaluate $$\frac{3-4i}{4+i}.$$
Solution: The conjugate of $$4 + i$$ is $$4-i$$. We have $$z \overline{z} = (4+i)(4-i) = 4^2-i^2 = 17.$$ To evaluate the quotient, we have $$\frac{3-4i}{4+i} = \frac{3-4i}{4+i} \times \frac{4-i}{4-i} = \frac{(3-4i)(4-i)}{17} = \frac{8-19i}{17}.$$

## Problems

1. Evaluate the following:
1. $$\overline{5+6i}$$
2. $$\overline{i}$$
3. $$\overline{-4+6i}$$
4. $$\overline{8+8i}$$
5. $$\overline{ \overline{1-i} + \overline{4-3i} }$$
2. Simplify the following to $$x+yi$$ form:
1. $$\displaystyle \frac{1+i}{1-i}$$
2. $$\displaystyle\frac{-2-i}{-5-4i}$$
3. $$\displaystyle\frac{9+6i}{1+4i}$$
4. $$\displaystyle\frac{5-7i}{-10-i}$$
5. $$\displaystyle\frac{9+2i}{5+8i}$$