## 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

- Evaluate the following:
- \( \overline{5+6i} \)
- \( \overline{i}\)
- \(\overline{-4+6i} \)
- \( \overline{8+8i} \)
- \( \overline{ \overline{1-i} + \overline{4-3i} } \)

- Simplify the following to \(x+yi\) form:
- \( \displaystyle \frac{1+i}{1-i}\)
- \(\displaystyle\frac{-2-i}{-5-4i}\)
- \(\displaystyle\frac{9+6i}{1+4i}\)
- \(\displaystyle\frac{5-7i}{-10-i}\)
- \(\displaystyle\frac{9+2i}{5+8i}\)