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.

### MAKE MILLIONS MONTHLY DOING SIMPLE EMAIL MARKETING TASKS ON FIVERR - CHECK HERE

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.

**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◦.

PDF – Download How To Calculate Modulus And Argument Of Complex Numbers

### MAKE MONEY ONLINE AS A STUDENT - CHECK HERE

## Subscribe To Our Newsletter

Join our mailing list to receive the latest updates from our team. Ranging from scholarships, study abroad opportunities, job tips, career and mentorship opinions, tutorials, past questions and lecture notes for your institution, success tips to loads of other educational goodies to augment your academic buoyancy.