Polar form of a complex number. Hello friends, today it’s all about the polar form of a complex number. Have a look!!

### The polar form of a complex number

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

**Addition and subtraction of complex numbers**

**Multiplication and division of complex numbers**

**Modulus and argument of a complex number**

**Functions of complex variables**

Let’s say is a complex number. And my task is to write it in polar form.

So first of all, I’ll get the modulus of . And this is

Next, I’ll get the argument of . Let’s say is the argument of .

Thus it will be

So the polar form of is

Now I’ll give some examples of the polar form of a complex number.

#### Solved examples of the polar form of a complex number

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 Kreyszig (2005)* “Represent in polar form. .”

##### Solution

Now here the complex number is .

First of all, I’ll give it a name, say, . So it will be .

As a first step, I’ll get the modulus of .

###### Step 1

So it will be

Next, I’ll simplify it to get

Then I’ll simplify it a bit more to get

Now this gives

As a next step, I’ll get the argument of .

###### Step 2

So let me choose the argument of as .

Hence will be

So this means

Thus the polar form of will be

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

Now I’ll give another example.

##### Example 2

According to Kreyszig (2005)* “Represent in polar form. .”

##### Solution

Now here the complex number is .

First of all, I’ll give it a name, say, . So it will be .

As a first step, I’ll get the modulus of .

###### Step 1

So it will be

Next, I’ll simplify it to get

As a next step, I’ll get the argument of .

###### Step 2

So let me choose the argument of as .

Hence will be

So this means

Thus the polar form of will be

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

Now I’ll give another example.

##### Example 3

According to Kreyszig (2005)* “Represent in polar form. .”

##### Solution

Now here the complex number is .

First of all, I’ll give it a name, say, . So it will be .

As a first step, I’ll remove the imaginary part from the bottom (denominator) of .

###### Step 1

So, for that, I’ll multiply both top and bottom of with .

Thus it will be

Next, I’ll simplify it to get

Now I’ll expand the top. Also, I’ll substitute at the bottom.

So it becomes

which gives

Therefore the ultimate simplified form of is

As a next step, I’ll get the modulus of .

###### Step 2

So it will be

Now this gives

As a next step, I’ll get the argument of .

###### Step 3

So let me choose the argument of as .

Hence will be

So this means

Thus the polar form of will be

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

Now I’ll give another example.

##### Example 4

According to Kreyszig (2005)* “Represent in polar form. .”

##### Solution

Now here the complex number is .

First of all, I’ll give it a name, say, . So it will be .

As a first step, I’ll remove the imaginary part from the bottom (denominator) of .

###### Step 1

So, for that, I’ll multiply both top and bottom of with .

Thus it will be

Next, I’ll simplify it to get

Now I’ll substitute to get

So this gives

As a next step, I’ll get the modulus of .

###### Step 2

So it will be

Next, I’ll simplify it to get

Then I’ll simplify it a bit more to get

Now this gives

As a next step, I’ll get the argument of .

###### Step 3

So let me choose the argument of as .

Hence will be

So this means

Thus the polar form of will be

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

Dear friends, this is the end of my 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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