Modulus and argument of complex numbers. Hello friends, today it’s all about the modulus and argument of complex numbers. Have a look!!

### Modulus and argument of complex numbers

If you’re looking for more in complex numbers, do check in:

**Addition and subtraction of complex numbers**

**Multiplication and division of complex numbers**

**Polar form of a complex number**

**Functions of complex variables**

Let is a complex number. Then the modulus of the number will be

Now, let’s choose as the argument of the complex number . Thus the value of will be

Now I’ll give some examples of the modulus and argument of complex numbers.

#### Examples of the modulus and argument of complex numbers

Disclaimer: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the first example.

##### Example 1

According to Stroud and Booth (2003)*¹, “Find the modulus of .”

##### Solution

Now here I have the complex number as .

First of all, I’ll simplify it.

###### Step 1

So it will be

As I already know, . Thus it will be

Now I’ll simplify it a bit more. Therefore it will look like

Next, I’ll remove the imaginary part from the denominator.

###### Step 2

Therefore I’ll multiply both numerator and denominator with .

And that gives

So this means

Again I’ll substitute . Thus it will be

Next, I’ll simplify it to get

Hence I can say that

Now I’ll get the modulus of .

###### Step 3

Thus the modulus of will be

If I simplify it, will be

Hence I can conclude that is the modulus of the complex number .

And this is the solution to this example.

Now I’ll give another example.

##### Example 2

According to Kreyszig (2005)*², “Determine the principal value of the argument. .”

##### Solution

Now here I have the complex number as . So this means .

Next, I’ll choose as the principal argument of .

Thus the value of will be

So this means

Now this gives the value of as

Hence I can conclude that is the principal value of the argument of . And this is the solution to this given example.

Next, I’ll solve another problem.

##### Example 3

According to Kreyszig (2005)*², “Determine the principal value of the argument. .”

##### Solution

Now here I have the complex number as .

Next, I’ll choose as the principal argument of .

Thus the value of will be

So this gives

Also we all know that . Thus the value of is .

Therefore the value of will be

Hence I can conclude that this is the answer to the given example.

Now comes my next example.

##### Example 4

According to Kreyszig (2005)*², “Determine the principal value of the argument. .”

##### Solution

So here I have the complex number as .

First of all, I’ll get the polar form of .

###### Step 1

So let’s give it a name, say, .

Thus I can say that .

So the modulus of is

Now I’ll simplify it to get

Next, I’ll get the argument of . Let is the argument of the complex number .

Thus it will be

So this means

Therefore the argument of is

Thus the polar form of the complex number is

Next, I’ll get the polar form of .

###### Step 2

As I know, the complex number is .

Now is .

Also, I have already got the polar form of . And that is .

Thus the polar form of will be

As per de Movire’s theorem,

Hence the polar from of will be

Therefore I can say that is the principal argument of .

Hence I can conclude that this is the solution to this example.

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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