Complex Numbers
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 for dividing two complex numbers has many similarities with the process for 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.
Conjugate Review
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:
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.
Dividing Complex Numbers
To set up a division problem with complex numbers, we want to write it as a fraction. To divide by , we write:
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.
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 makeup 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:
It is in the multiplication of denominators where we can see why using the complex conjugate was so helpful. While working through this multiplication, pay particular attention when we combine the two imaginary numbers, and when we simplify the squared imaginary unit.
By using the denominator's complex conjugate, the i-terms in the denominator will always cancel each other out. Additionally, the i^{2} 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:
the conjugate of a binomial is a binomial with the opposite sign between its terms