# 2 Complex numbers

## 2.4 Complex conjugate

Many manipulations involving complex numbers, such as division, can be simplified by using the idea of a *complex conjugate*, which we now introduce.

### Definition

The **complex conjugate**
of the complex number *z* = *x* + *iy* is the complex number *x* − *iy*.

For example, if *z* = 1 − 2*i*, then
. In geometric terms,
is the image of *z* under reflection in the real axis.

### Exercise 9

Let *z*_{1} = −2 + 3*i* and *z*_{2} = 3 − *i*.

Write down
and
, and draw a diagram showing *z*_{1}, *z*_{2},
and
in the complex plane.

### Solution

The following properties of complex conjugates are particularly useful.

### Properties of complex conjugates

Let *z*_{1}, *z*_{2} and *z* be any complex numbers. Then:

- ;
- ;
- ;
- .

In order to prove property 1, we consider two arbitrary complex numbers.

Let *z*_{1} = *x*_{1} + *iy*_{1} and *z*_{2} = *x*_{2} + *iy*_{2}. Then

### Exercise 10

Use a similar approach to prove properties 2, 3 and 4.

### Solution

*Property 2*

Let *z*_{1} = *x*_{1} + *iy*_{1} and *z*_{2} = *x*_{2} + *iy*_{2}. Then

so

Also,

Therefore

*Property 3*

Let *z* = *x* + *iy*. Then

*Property 4*

Let *z* = *x* + *iy*. Then