In this unit you are going to learn about the modulus and argument of a complex number. These
are quantities which can be recognised by looking at an Argand diagram. Recall that any complex
number, z, can be represented by a point in the complex plane as shown in Figure 1.
The complex number z is represented by point P . Its modulus and argument are shown.
We can join point P to the origin with a line segment, as shown. We associate with this line segment
two important quantities. The length of the line segment, that is OP , is called the modulus of the
complex number. The angle from the positive axis to the line segment is called the argument of
the complex number, z.
The modulus and argument are fairly simple to calculate using trigonometry.
Example. Find the modulus and argument of z = 4 + 3i.
Solution. The complex number z = 4 + 3i is shown in Figure 2. It has been represented by the
point Q which has coordinates (4, 3). The modulus of z is the length of the line OQ which we can
find using Pythagoras’ theorem.
(OQ)2 = 42 + 32 = 16 + 9 = 25
and hence OQ = 5.
Hence the modulus of z = 4 + 3i is 5. To find the argument we must calculate the angle between
the x axis and the line segment OQ. We have labelled this θ in Figure 2.
By referring to the right-angled triangle OQN in Figure 2 we see that
tan θ = 3/4
θ = tan−1 3/4 = 36.97◦
To summarise, the modulus of z = 4 + 3i is 5 and its argument is θ = 36.97◦. There is a special
symbol for the modulus of z; this is |z|. So, in this example, |z| = 5. We also have an abbreviation
for argument: we write arg(z) = 36.97◦.
When the complex number lies in the first quadrant, calculation of the modulus and argument is
straightforward. For complex numbers outside the first quadrant we need to be a little bit more
careful. Consider the following example.
Find the modulus and argument of z = 3 − 2i.
Solution. The Argand diagram is shown in Figure 3. The point P with coordinates (3, −2) represents
z = 3 − 2i.
We use Pythagoras’ theorem in triangle ONP to find the modulus of z:
(OP )2 = 32 + 22 = 13
OP = √13
Using the symbol for modulus, we see that in this example |z| = √13.
We must be more careful with the argument. When the angle θ shown in Figure 3 is measured in a
clockwise sense convention dictates that the angle is negative. We can find the size of the angle by
referring to the right-angled triangle shown. In that triangle tan α = 2 3 so that α = tan−1 2 3 = 33.67◦.
This is not the argument of z. The argument of z is θ = −33.67◦. We often write this as
arg(z) = −33.67◦.