Multiplication and division of complex numbers. Hello friends, today it’s all about the multiplication and division of complex numbers. Have a look!!

If you are looking for more in complex numbers, do check-in:

**Addition and subtraction of complex numbers**

**Polar form of a complex number**

**Modulus and argument of a complex number**

**Functions of complex variables**

Multiplication and division of complex numbers_compressed

**Multiplication and division of complex numbers**

Suppose I have two complex numbers and . Also, the number is and the number is . Now here shows the imaginary part of the number.

**Multiplication of two complex numbers**

Now I’ll multiply the two complex numbers and . So it will be

And this gives

Since , I can say

Now I’ll separate the real and imaginary parts to get

Next, I’ll show the division of complex numbers, that is .

**Division of two complex numbers**

So becomes

First of all, I’ll get rid of the imaginary part of the denominator. Therefore I can write it as

(1)

Now I’ll simplify .

Thus it will be

(2)

Next, I’ll simplify .

Thus it will be

Since , I can say that

Now I’ll separate the real and imaginary parts to get

(3)

Thus I’ll substitute equations (2) and (3) to equation (1) to get the value of as

Now I’ll give some examples on the multiplication and division of complex numbers.

**Solved examples of multiplication and division 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 Kreyszig (2005)* “Let and . Showing the details of your work, find (in the form of ): . ”

**Solution**

Now here I have to find out the values of . Also, I already know that the complex number is . So, let’s start now.

**First part**

First of all, I’ll find out the value of .

So the value of is

Now I’ll expand the right-hand side.

Thus it will be

Next, I’ll simplify it.

So it becomes

Since , it will look like

Thus it will be

Hence I can say that .

Now I’ll do the second part.

**Second part**

So here I have to find out the value of .

As I already know that the real part of the complex number is . So I can say that .

Hence the value of is

Therefore I can conclude that and are the answers to this example.

Now I’ll give another example.

**Example 2**

According to Kreyszig (2005)* “Let and . Showing the details of your work, find (in the form of ): . ”

**Solution**

Now here the two complex numbers are and . And I have to find out the value of .

Thus it will be

As I have mentioned above, first of all, I’ll get rid of the imaginary part of the denominator. Therefore I can write it as

Next, I’ll simplify it. So it will be

Since , so it will be

Hence I can simplify it a bit to get

Therefore I can conclude that is the answer 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!!

## Leave a Reply