Number systems
Number systems

This free course is available to start right now. Review the full course description and key learning outcomes and create an account and enrol if you want a free statement of participation.

Free course

Number systems

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.

Answer

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.

Answer

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

M208_6

Take your learning further

Making the decision to study can be a big step, which is why you'll want a trusted University. The Open University has nearly 50 years’ experience delivering flexible learning and 170,000 students are studying with us right now. Take a look at all Open University courses.

If you are new to university level study, find out more about the types of qualifications we offer, including our entry level Access courses and Certificates.

Not ready for University study then browse over 900 free courses on OpenLearn and sign up to our newsletter to hear about new free courses as they are released.

Every year, thousands of students decide to study with The Open University. With over 120 qualifications, we’ve got the right course for you.

Request an Open University prospectus