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Contributor: Lynn Ellis. Lesson ID: 13730
Complex numbers — they aren't even real! But they do have applications in the real world. First, you need to learn arithmetic with them. In this lesson, you will learn how to divide complex numbers.
In this lesson, we will learn about using complex conjugates to divide complex numbers.
You may be wondering what a complex conjugate is, so we will start there.
Complex Conjugates
Let's break down the terms.
Complex numbers have the form (a + bi) where a and b are both real numbers and i^{2} = 1.
Complex numbers allow us to work with square roots of negative numbers since i = √1.
We call i an imaginary number because we can't take the square root of a negative number in the real number system.
We call a the real component of the complex number because it does not include an imaginary number in it.
We call bi the imaginary component because it does contain an imaginary number.
Together they make a complex number. Take a look at this diagram to see how the number systems fit together:
Conjugates are two binomials in the form (m + n) and (m  n).
For example, (x + 3) and (x  3) are conjugates. Notice that the terms are the same, but the operation changes from addition to subtraction.
Give it a try! Play the game below to match up the pairs of conjugates.
Try multiplying these conjugates together and see if you can identify the pattern.
That will happen every time you multiply conjugates. That's important to note, and we will use that fact throughout the rest of this lesson.
OK, now that we know about complex numbers and conjugates, let's put the two words together to define complex conjugates.
Complex conjugates are two complex numbers in the form (a + bi) and (a  bi).
Let's multiply (a + bi)(a  bi)
It looks like this:
(a + bi)(a  bi)
a^{2} + abi  abi  b^{2}i^{2}
a^{2}  b^{2}i^{2}
a^{2} + b^{2}
There is no imaginary part left!
So, you might wonder what we can do with this information and why it is useful to know. There are several ways that we can use complex conjugates in mathematics.
The first one to talk about is dividing complex numbers. Other lessons will explore further applications of complex conjugates, such as finding the modulus (absolute value) of a complex number.
Since this lesson is focused on division, let's look at two different examples of dividing complex numbers.
Example 1
1  9i 
We want to get a single complex number for the answer. 

1  i  
Notice that we have i in both the numerator and denominator. 

We don't want an i in the denominator when we are done. 
1  9i  •  1 + i 
We are multiplying by 1 but in the convenient form of the conjugate of the denominator over itself. 

1  i  1 + i  


Check out the next step to see. 
1  i  9i + 9 
We got the difference of squares in the denominator. 

1 + 1  
That made the i drop out! 

Now we combine like terms. 
8  10i 
Then we put it into the correct form (a + bi) to get the final answer. 

2  
Make sure you simplify fully. 
4  5i 
Here's another example. It doesn't look as nice, because we get fractions in the end, but the process is exactly the same.
Example 2
3  4i  
7 + 2i 
3  4i  •  7  2i  
7 + 2i  7  2i 
21  6i  28i + 8  
49 + 4 
29  34i  
53 
29    34i  
53  53 
Now you try a couple.
[NOTE: The / in the questions below represents the split between the numerator and denominator.]
If you feel like you understand, move to the Got It? section for some more practice. Then try the quiz.