Table of Contents |
A complex number is a number in the form , containing both a real part and an imaginary part. The imaginary part is followed by i, which is the imaginary unit, .
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.
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 |
---|---|
Next, we will see how complex conjugates help us solve division problems involving complex numbers.
To set up a division problem with complex numbers, we want to write it as a fraction. For instance, to divide by , we write:
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 byMultiply by a second fraction with the conjugate in the numerator and denominator | |
Multiply the two fractions | |
Use FOIL to evaluate numerator | |
Combine like terms in numerator | |
Rewrite as | |
Combine like terms in numerator | |
Evaluate denominator |
Use FOIL to evaluate denominator | |
Combine like terms in denominator | |
Rewrite as | |
Simplify denominator | |
Separate into two fractions |
Separate into two fractions | |
Simplify fractions | |
Our solution |
EXAMPLE
Divide byMultiply by a second fraction with the conjugate in the numerator and denominator | |
Multiply the two fractions | |
Use FOIL to evaluate numerator and denominator | |
Combine like terms in numerator and denominator | |
Rewrite as and as | |
Combine like terms in numerator and denominator | |
Separate into two fractions | |
Simplify fractions | |
Our solution |
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