The Modulus and Argument of Complex Numbers

Generally, the complex number is a vast branch of matric that requires careful examination for appropriate practice. Finding the Modulus and Argument of complex numbers is another step in the hierarchy of complex numbers. While these two calculations will be our focus in this article, it is important to draw some inferences.

In our previous article titled The Algebra of Complex Number, which you can check here, we discussed how the complex number works and how it is formed. The quadratic equation was also cited as the main source of the formation of complex numbers. Also, we expressed the complex numbers in the form z = a + bi which is the standard form of all complex numbers. It is also known as the rectangular form of complex numbers.

In another article titled The Argand Diagram, we discussed the argand diagram and how it can be plotted to represent a complex plane.

The Above Two Topics Are Very Important for our discussion of the Modulus and Argument of a complex, and you should check them before continuing. If you already understand them, you can continue with this topic.

Modulus and Argument of Complex Numbers

Z = a + bi id the rectangular form of the complex number. However, when discussing complex numbers and their relation to trigonometry, it is recommended to transform the rectangular presentation into a polar form presentation.

The polar form representation employs trigonometry functions; cosϴ and sin cosϴ and is given as

Z = cosrϴ + isinrϴ

The polar form and the rectangular form are actually similar because a + bi = cosrϴ + isinrϴ

Where

a = cosrϴ

b = sinrϴ

bi = isinrϴ

we can still simplify the polar form 0f the complex number by taking out the common term r such that it becomes

z = r[cosϴ + isinϴ]

now to find this polar form expression of the complex number, r and ϴ must be known

ris called the modulus of the complex number in polar form

while

ϴ is called the argument of the complex number in polar form

The above implies that finding the modulus and argument of a complex number simply means finding the r and ϴ.

So what is the aim of finding the modulus and arguments? Because they make it possible to express complex numbers in a standard form

The modulus of a Complex number is the Absolute Value of Z

Another point to note is that a complex number’s modulus (r) is also known as the absolute value of z, denoted as |z|. Clearly, absolute means positivity. Therefore, it is impossible for your modulus, which is the absolute value of z to be negative.

r = |z|

The Modulus and Arguments Formulas

The formula of the modulus is given as

The a and b are the real values inside the complex number in the expression Z = a + bi. When we consider an example, you will discover that r = |z| is simply the hypotenuse of a right angle in the diagram.

The formula for the argument is given as

ϴ = ϴref when in the first quadrant of the Argand diagram

ϴ = 180 – ϴref when in the second quadrant of the Argand diagram

ϴ = 180 + ϴref when in the third quadrant of the Argand diagram

ϴ = 360 – ϴref when in the third quadrant of the Argand diagram

Where;

To understand these two formulas, we will first consider the argand diagram. It is based on the argand diagram; we will be able to understand the formula.

How to find The Modulus and Argument of the Complex number (|z| or r and  ϴ): The role of the Argand Diagram

The Argand Diagram plays a very important role in Finding the modulus and argument. Now let’s consider the following.

Example 1

Find the modulus and argument of the complex number z = 3 + 3i

Solution

Recall that the modulus and arguments are based on the polar form; through the argand diagram, we will be able to transform the above complex into its polar alternate to find them.

So the first thing to do is to plot the number in an argand diagram

Clearly, since the a and b are positive, only the x and positive y axes will be involved. Also, the x axis is the axis representing the real number while the y axis is the one representing the imaginary number 3i

This, therefore, makes the plotted diagram something in this form

Now consider the line that creates the intersection between the real number 3 and the imaginary number 3i. If we trace them up, we will have something like this

Now the above clearly looks like a right angle. We can calculate the modulus or absolute value already since we know the a and b

Therefore

Now recall that the Pythagoras theorem states that to get the hypotenuse of a triangle

h2 = a2 + b2

This means that for the modulus of a complex number is the same as the hypotenuse of the right tangle formed when plotting a complex number in an argand diagram. This rule is consistent as all plotted numbers will always form a right angle where the real and imaginary numbers will form the adjacent and opposite. And the modulus or absolute value will be the hypotenuse.

Now we will have to find the argument of the complex number, which is ϴ

Now recall we wrote

ϴ = ϴref when in the first quadrant of the Argand diagram

ϴ = 180 – ϴref when in the second quadrant of the Argand diagram

ϴ = 180 + ϴref when in the third quadrant of the Argand diagram

ϴ = 360 – ϴref when in the third quadrant of the Argand diagram

So to find ϴ, we have to understand what quadrants and ϴref are

Now Quadrants are the four different partitions in an X and Y axis. Moving from the top right-hand side an anticlockwise, of an X,Y axis the partition is labelled 1 to 4. That means for our XY axis, the quadrants are labelled as

Since we have found the reference angle as 450 it is now to determine the argument, which is the real angle.

Now lets us consider this diagram again

From it, we can see that the a and b that formed the reference angle were in the first quadrant. Also, recall that;

ϴ = ϴref when in the first quadrant of the Argand diagram

So this implies that

ϴ = 450 as well.

So the Arguments and modulus for the complex number z = 3 + 3i are

|z| = 4.24 and ϴ = 450

Example 2

Find the argument and modulus of the complex number z = 4 – 3i

Solution

We just have to follow the same steps as the first

Since a = 4, the x axis will be at the positive, while since b = -3, then the negative y axis will be employed

This will make the argand diagram this way

The next step will be to find the argument

From the diagram, the reference angel was formed from the fourth quadrant and as stated in the formula,

ϴ = 360 – ϴref when in the third quadrant of the Argand diagram

Therefore

ϴ = 360 – 76.39

ϴ = 283.61

As such, all the absolute values and arguments of the complex number are 5 and 283.10

Conclusion

Knowing how to find the modulus and argument of a standard complex number is very important in creating the polar alternative. This article has descriptively outlined all that you need to know to fully understand everything about the modulus and argument of the complex number. Two examples were considered to make the expression totally easy and consistent to understand.

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