# 2 Complex numbers

## 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

### 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.

### 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