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Divide Complex Numbers

Author: Sophia

what's covered
In this lesson, you will learn how to divide two complex numbers. Specifically, this lesson will cover:

Table of Contents

1. Complex Numbers

A complex number is a number in the form a plus b i, containing both a real part and an imaginary part. The imaginary part is followed by i, which is the imaginary unit, square root of short dash 1 end root.

The process of dividing two complex numbers has many similarities with the process of rationalizing denominators. If you have studied rationalizing denominators, you may be familiar with using a conjugate in order to clear any irrational expressions from the denominator of a fraction. With complex number division, we use complex number conjugates to clear imaginary numbers from the denominator. Before we get to examples, let's review conjugates.


2. Conjugate Review

Recall that a conjugate of a binomial is a binomial with the opposite sign between its terms. Finding the conjugate of a complex number is straightforward. We simply reverse the sign in between the real part and the imaginary part. When we see plus signs, we write minus signs, and vice versa. Here is a table with some complex numbers and their complex conjugates:

Complex Number Complex Conjugate
8 plus 4 i 8 minus 4 i
short dash 6 plus 3 i short dash 6 minus 3 i
7 minus 5 i 7 plus 5 i
short dash 2 minus 9 i short dash 2 plus 9 i

hint
Only the sign in front of the imaginary number changes. Notice that the sign in front of the real number stays the same when writing a complex number and its complex conjugate.

Next, we will see how complex conjugates help us solve division problems involving complex numbers.

term to know
Conjugate
The conjugate of a binomial is a binomial with the opposite sign between its terms.


3. Dividing Complex Numbers

To set up a division problem with complex numbers, we want to write it as a fraction. For instance, to divide 4 plus 7 i by 3 plus 2 i, we write:

fraction numerator 4 plus 7 i over denominator 3 plus 2 i end fraction

Here is where the complex conjugate comes into play. In order to clear imaginary numbers from the denominator, we use the complex conjugate of the denominator to create a second fraction. In this fraction, the complex conjugate will make up the numerator and denominator. Since both the numerator and denominator are identical quantities, the fraction has a value of 1, and can be multiplied by our complex division problem and maintain its value.

big idea
Use the complex conjugate of the denominator to create a second fraction with a value of 1. In order to have a value of 1, the complex conjugate must make up both the numerator and denominator of this fraction. Multiplying the two fractions will not change the value, but it will allow us to simplify our solution.

By creating this fraction, we have really set up a fraction multiplication problem. We'll need to multiply across numerators, and then multiply across denominators. Let's first multiply across numerators and simplify the numerator of our solution as much as we can:

EXAMPLE

Divide 4 plus 7 i by 3 plus 2 i.

fraction numerator 4 plus 7 i over denominator 3 plus 2 i end fraction Multiply by a second fraction with the conjugate 3 minus 2 i in the numerator and denominator
fraction numerator 4 plus 7 i over denominator 3 plus 2 i end fraction times fraction numerator 3 minus 2 i over denominator 3 minus 2 i end fraction Multiply the two fractions
fraction numerator open parentheses 4 plus 7 i close parentheses open parentheses 3 minus 2 i close parentheses over denominator open parentheses 3 plus 2 i close parentheses open parentheses 3 minus 2 i close parentheses end fraction Use FOIL to evaluate numerator
fraction numerator 12 minus 8 i plus 21 i minus 14 i squared over denominator open parentheses 3 plus 2 i close parentheses open parentheses 3 minus 2 i close parentheses end fraction Combine like terms in numerator
fraction numerator 12 plus 13 i minus 14 i squared over denominator open parentheses 3 plus 2 i close parentheses open parentheses 3 minus 2 i close parentheses end fraction Rewrite negative 14 i squared as plus 14
fraction numerator 12 plus 13 i plus 14 over denominator open parentheses 3 plus 2 i close parentheses open parentheses 3 minus 2 i close parentheses end fraction Combine like terms in numerator
fraction numerator 26 plus 13 i over denominator open parentheses 3 plus 2 i close parentheses open parentheses 3 minus 2 i close parentheses end fraction Evaluate denominator

It is in the multiplication of denominators that we'll see why using the complex conjugate is so helpful. While working through this next set of multiplication, pay particular attention when we combine the two imaginary numbers, and when we simplify the squared imaginary unit.

fraction numerator 26 plus 13 i over denominator open parentheses 3 plus 2 i close parentheses open parentheses 3 minus 2 i close parentheses end fraction Use FOIL to evaluate denominator
fraction numerator 26 plus 13 i over denominator 9 minus 6 i plus 6 i minus 4 i squared end fraction Combine like terms in denominator
fraction numerator 26 plus 13 i over denominator 9 minus 4 i squared end fraction Rewrite negative 4 i squared as plus 4
fraction numerator 26 plus 13 i over denominator 9 plus 4 end fraction Simplify denominator
fraction numerator 26 plus 13 i over denominator 13 end fraction Separate into two fractions

By using the denominator's complex conjugate, the i-terms in the denominator will always cancel each other out. Additionally, the i squared term simplifies to a real number. The result is that the denominator is a purely real number, with no imaginary components.

Now we can simplify our fraction to get our final solution:

fraction numerator 26 plus 13 i over denominator 13 end fraction Separate into two fractions
26 over 13 plus fraction numerator 13 i over denominator 13 end fraction Simplify fractions
2 plus i Our solution

EXAMPLE

Divide 2 plus 4 i by 3 minus 5 i.

fraction numerator 2 plus 4 i over denominator 3 minus 5 i end fraction Multiply by a second fraction with the conjugate 3 plus 5 i in the numerator and denominator
fraction numerator 2 plus 4 i over denominator 3 minus 5 i end fraction times fraction numerator 3 plus 5 i over denominator 3 plus 5 i end fraction Multiply the two fractions
fraction numerator open parentheses 2 plus 4 i close parentheses open parentheses 3 plus 5 i close parentheses over denominator open parentheses 3 minus 5 i close parentheses open parentheses 3 plus 5 i close parentheses end fraction Use FOIL to evaluate numerator and denominator
fraction numerator 6 plus 10 i plus 12 i plus 20 i squared over denominator 9 plus 15 i minus 15 i minus 25 i squared end fraction Combine like terms in numerator and denominator
fraction numerator 6 plus 22 i plus 20 i squared over denominator 9 minus 25 i squared end fraction Rewrite plus 20 i squared as negative 20 and negative 25 i squared as plus 25
fraction numerator 6 plus 22 i minus 20 over denominator 9 plus 25 end fraction Combine like terms in numerator and denominator
fraction numerator short dash 14 plus 22 i over denominator 34 end fraction Separate into two fractions
fraction numerator short dash 14 over denominator 34 end fraction plus fraction numerator 22 i over denominator 34 end fraction Simplify fractions
short dash 7 over 17 plus 11 over 17 i Our solution

summary
Complex numbers consist of a real part and an imaginary part. The square root of negative 1 is imaginary, because no real number squared results in a negative number. Conjugates are used when dividing complex numbers to eliminate a complex number from the denominator. Simplify expressions with an i squared term by substituting negative 1 for i squared, multiplying by the coefficient, and writing as a real number with the opposite sign.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

Terms to Know
Conjugate

The conjugate of a binomial is a binomial with the opposite sign between its terms.