Now that we know how to add, subtract, and multiply complex numbers, let's learn how to divide them. Whenever I divide by a complex number, I end up with a fraction that has an i in the bottom. So, I'll have something in the numerator divided by a complex number that has a term with i in it. Looking at this fraction, I’m not really sure how to simplify it and come to a solution that makes any sense. This i in the denominator is problematic, I don't know how to deal with it, and I want to get rid of it in any way that I can. To transform this complex denominator into something real, we learned to use the complex conjugate. I'll walk you through how to use the complex conjugate to divide complex numbers and arrive at a solution. Let’s jump right into an example.
I want to find the quotient of these numbers: 31+2i. The first step is to eliminate the i in the denominator. My first step is to multiply both the numerator and the denominator by the complex conjugate of the denominator. Since I want to eliminate the i in the denominator, I will multiply by the complex conjugate, which for 1+2i is 1-2i. Multiplying the numerator and denominator by this conjugate doesn't change the value of the fraction. Expanding this, the numerator distributed into the complex conjugate gives 3-6i. With the denominator, using the foil method, the terms produce 1 from 1*1, -2i from 1*-2i, 2i from 2i*1, and -4i2 from 2i*-2i. Simplifying the i2 term to -1, we get positive 4. Collapsing terms, I am left with 5 in the denominator. I now have 3-6i5.
Next, I expand the fraction into real and imaginary parts, yielding 35 minus 65i. This fraction is already in its lowest terms, so this is the solution. That’s all there is to dividing complex numbers. Let me know if you have any questions.