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

2.5 Modulus of a complex number

We also need the idea of the modulus of a complex number. Recall that the modulus of a real number x is defined by

For example, |7| = 7 and |−6| = 6.

In other words, |x| is the distance from the point x on the real line to the origin. We extend this definition to complex numbers as follows.

Definition

The modulus |z| of a complex number z is the distance from the point z in the complex plane to the origin.

Thus the modulus of the complex number z = x + iy is

For example, if z = 3 − 4i, then

.

Exercise 11

Determine the modulus of each of the following complex numbers.

  • (a)  5 + 12i

  • (b)  1 + i

  • (c)  −5

Answer

Solution

  • (a)  

  • (b)  

  • (c)  

The modulus of a complex number has many properties similar to those of the modulus of a real number.

Properties of modulus

  1. |z| ≥ 0 for any z , with equality only when z = 0.

  2. |z1z2| = |z1||z2| for any z1, z2 .

Property 1 is clear from the definition of |z|. Property 2 can be proved in a similar way to property 2 of complex conjugates given in Exercise 10.

The following useful result shows the link between modulus and distance in the complex plane.

Distance Formula

The distance between the points z1 and z2 in the complex plane is |z1z2|.

This is obtained by applying Pythagoras' Theorem to the triangle in the diagram below.

Exercise 12

For each of the following pairs z1, z2 of complex numbers, draw a diagram showing z1 and z2 in the complex plane, and evaluate |z1z2|.

  • (a)  z1 = 3 + i,   z2 = 1 + 2i.

  • (b)  z1 = 1,   z2 = i.

  • (c)  z1 = −5 − 3i,   z2 = 2 − 7i.

Answer

Solution

  • (a)  

  • Here

  • so

  • (b)  

  • Here

  • so

  • (c)  

  • Here

  • so

The following properties describe the relationship between the modulus and the complex conjugate of a complex number.

Conjugate–modulus properties

  1. for all z .

  2. for all z .

To prove these properties, we let z = x + iy. Then

,

so

and

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 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