Hi, and welcome. My name is Anthony Varela, and today we're going to divide complex numbers. So we'll talk about our complex numbers, and we'll talk about their complex conjugate. So that's going to be very helpful and important when dividing complex numbers. And then we'll go ahead and go through an example of dividing complex numbers.
So let's talk about complex numbers and complex conjugates. Well, a complex number generally is a plus bi. It contains a real part, an imaginary part, and then i, this imaginary unit, the square root of negative 1. This is imaginary because any real number squared results in a positive number. So taking the square root of a negative number is not real. It's imaginary.
So here are some examples of complex numbers, 7 plus 3i and negative 2 minus 5i. Well now, let's talk about complex conjugates, and conjugate might be a word that you've heard before. The conjugate of a binomial is a binomial with the opposite sign between its terms. So when we're talking about complex numbers and their conjugates, we can refer to them as complex conjugates.
So all we're going to do is flip the sign in between a and bi. So if our complex number is a plus bi, it's complex conjugate is a minus bi. The complex conjugate to 7 plus 3i is 7 minus 3i, and the complex conjugate to negative 2 minus 5i is negative 2 plus 5i. So notice that no sign is changed here, just the sign in between a and bi.
So when we divide complex numbers, we're goings to be using the denominator's conjugate, and we're going to be using it in a very similar way that we use conjugates to rationalize denominators, if you've ever gone through that process. And so what you do is you multiply this fraction by another fraction, and this fraction has the denominator's conjugate in both its numerator and denominator.
So the conjugate of 3 plus 2i is 3 minus 2i. And since these are equal quantities in both of the numerator and denominator, we're essentially multiplying by 1. And as you'll see when we go through this, the reason why we do this is that our denominator will have no imaginary component, just like rationalizing the denominator then had no irrational component.
So what we really need to do then is multiply across numerators and then multiply across denominators, and then we'll simplify. So multiplying across our numerators-- we'll use FOIL to do this. So multiplying the first two terms, we get negative 12, the outside terms multiply to positive 8i, the inside terms multiply to negative 21i, and the last terms multiply to positive 14i squared.
Well, we can combine the positive 8i and the negative 21i, so we have a minus 13i in there, but how can we think about plus 14i squared? Well, i is the square root of negative 1, so i squared equals negative 1. That's a real number. So what I like to do when rewriting i squared terms is I just like to erase the i squared, but change the sign out in front. So plus 14i squared becomes negative 14.
Well, because that's a real number, we can combine it with negative 12. So I have then negative 26 minus 13i. That is our simplified numerator. We have to also multiply across the denominators. And this is where we're multiplying a complex number by its complex conjugate. So let's see what happens here.
Multiplying our first two terms gives us 9. Multiplying the two outside terms, we have negative 6i. Multiplying the inside terms, we have positive 6i, so you might see where we're going with this, and multiplying our last two terms, we have a minus 4i squared. Well, when we combine negative 6i and positive 6i, that combines to 0. So we just have 9 minus 4i squared.
Well, we know that i squared is a real number. It's negative 1. So we can write minus 4i squared as plus 4. So our denominator simplifies to 13. That is a real number. It has no imaginary part to it. So
Now we can simplify negative 26 minus 13i over 13. We can just divide both this part and this part by 13. This isn't always the case that it works out nicely, but it does here. So negative 26 divided by 13 gives us a negative 2, and negative 13i over 13 gives us a negative i. So that is our simplified quotient to this complex division.
So let's review dividing complex numbers. Well, we used the conjugate, so with our general complex number, a plus bi, the complex conjugate is a minus bi. When we're dividing complex numbers, you use that denominator's conjugate, and we multiply by 1, which means then that we multiply across numerators and multiply across denominators, and you're going to encounter i squared. Remember, that is the real number negative 1.
So thanks for watching this tutorial in dividing complex numbers. Hope to see you next time.