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
- |z| ≥ 0 for any z , with equality only when z = 0.
- |z_{1}z_{2}| = |z_{1}||z_{2}| for any z_{1}, z_{2} .
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 z_{1} and z_{2} in the complex plane is |z_{1} − z_{2}|.
This is obtained by applying Pythagoras' Theorem to the triangle in the diagram below.
Exercise 12
For each of the following pairs z_{1}, z_{2} of complex numbers, draw a diagram showing z_{1} and z_{2} in the complex plane, and evaluate |z_{1} − z_{2}|.
- (a) z_{1} = 3 + i, z_{2} = 1 + 2i.
- (b) z_{1} = 1, z_{2} = i.
- (c) z_{1} = −5 − 3i, z_{2} = 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
- for all z .
- for all z .
To prove these properties, we let z = x + iy. Then
,
so
and