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

### 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.

In one of my earlier posts, I have already talked about the addition and subtraction of complex numbers.

##### 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 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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