Complex conjugate numbers. Hello friends, today it’s all about the complex conjugate numbers. Have a look!!

### Complex conjugate numbers

If you are 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**

**Modulus and argument of a complex number**

**Functions of complex variables**

Suppose is a complex number. Then the conjugate of the complex number will be .

Now one of the most obvious properties of the complex conjugate numbers is .

Next, I’ll give some solved examples of complex conjugate numbers.

#### Solved examples of complex conjugate 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 the two complex numbers and as and .

So the conjugate of is . Similarly, the conjugate of is .

Thus the value of will be

Now I’ll expand it to get

If interested, you can read more about the multiplication of complex numbers.

So this gives

Since , I can say that

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

Now I’ll give another example.

##### Example 2

According to Kreyszig (2005)* “Let . Find: . ”

##### Solution

Now here the complex number is . So the conjugate of is .

And here I have to find out the value of .

First of all, I’ll get the value of .

So I’ll start with .

###### Step 1

Thus the value of will be

Now I’ll simplify it. So that means

Since , so this gives

Next, I’ll get the value of .

###### Step 2

Therefore it will be

At first, I’ll remove the imaginary part from the denominator.

Thus it will be

Now I’ll simplify it to get

Since , so this gives

Now I’ll expand the denominator to get

So this means

Also, I know that .

Hence this gives the value of as

Therefore I can say that the real part of is

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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