Division: It's Not Just for Real Numbers Anymore

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.


Algebra II, Complex Numbers and Quantities

learning style
personality style
Grade Level
High School (9-12)
Lesson Type
Quick Query

Lesson Plan - Get It!

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  • How does arithmetic change when the numbers aren't real?
  • Can you divide complex numbers like you can divide real 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

  • What are they anyway?

Let's break down the terms.

Complex numbers have the form (a + bi) where a and b are both real numbers and i2 = -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:

number system

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.

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  • Do you know what's really cool about conjugates?

Try multiplying these conjugates together and see if you can identify the pattern.

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  • Did you see that when you multiply conjugates together, you get the difference of squares?

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)

a2 + abi - abi - b2i2

a2 - b2i2

a2 + b2

  • Do you notice that a2 + b2 is a real number?

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  
  • Why?

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.


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


  29 - 34i
  53 53


Now you try a couple.

[NOTE: The / in the questions below represents the split between the numerator and denominator.]

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  • How did you do with those?

If you feel like you understand, move to the Got It? section for some more practice. Then try the quiz.

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