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 − 2i, then . In geometric terms, is the image of z under reflection in the real axis.

Exercise 9

Let z1 = −2 + 3i and z2 = 3 − i.

Write down and , and draw a diagram showing z1, z2, and in the complex plane.

Solution

The following properties of complex conjugates are particularly useful.

Properties of complex conjugates

Let z1, z2 and z be any complex numbers. Then:

  1. ;
  2. ;
  3. ;
  4. .

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

Let z1 = x1 + iy1 and z2 = x2 + iy2. Then

Exercise 10

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

Solution

Property 2

Let z1 = x1 + iy1 and z2 = x2 + iy2. Then

so

Also,

Therefore

Property 3

Let z = x + iy. Then

Property 4

Let z = x + iy. Then