Now that we know how to add subtract and multiply complex numbers, let's learn how to divide them. Well, whenever I divide by a complex number, I'm always going to end up with a fraction that has an I in the bottom. So I'll have something, doesn't really matter what in the numerator divided by a complex number that has a term with I in it. Now looking at this fraction, I'm not really sure how to simplify it and come to a solution that actually makes any sense. So this, I in the denominator is actually bad. I don't know how to deal with it and I want to get rid of it in any way that I can. So how am I going to take this complex denominator and turn it into something real? Well, we just learned that if we have a complex number and multiply it by its conjugate that we're simply left with a real number. So that's actually going to completely solve our problem and give us something that we know how to work with. So I'm going to go ahead and walk you through how to use the complex conjugate to divide complex numbers and come to a solution. Let's go ahead and jump right into an example. So I want to find the quotient of these numbers 3/1 plus two. I now, the first thing I want to do is get rid of that I in the denominator. So my very first step is actually gonna be to multiply both the top and the bottom by my complex conjugate of the bottom. Now, since the bottom is the number that I want to get rid of, I want to get rid of the I and that I'm gonna go ahead and multiply by the complex conjugate. Now, the complex conjugate of one plus two, I, I'm just gonna take that imaginary part and flip the sign. This is gonna be one minus two. I now if I'm multiplying the bottom of my fraction by that, I need to also multiply the top of my fraction as well so that I'm not really changing the value of anything. So let's go ahead and expand this out. So with my numerator, I can take this three and distribute it into that complex conjugate, which will give me three minus six. I. Now with the denominator, since I have two complex numbers, I can go ahead and use foil here. So my first term is that one times one, which will give me one, my outside term, one times negative two, I will give me negative two. I then my inside term two I times one, gives me positive two I and my last term two, I times negative two, I will give me negative four I squared. Now, we know that whenever we're left with an I squared term, we can further simplify that. So this negative four I squared is really just negative four times negative one, which we know is just positive four. So let's go ahead and rewrite our fraction and simplify it as far as we can. So the numerator is still just three minus six. I and then my denominator here, let's look at what terms we have left. Well, I have this one and then I have minus two, I plus two. I, but I know that those middle terms are just going to cancel out. So I don't have to worry about them and then I have this plus four. So I'm simply just left with one plus four and one plus four is just five. So my denominator here is five. OK. So we have completed step number one, we have multiplied the top and bottom by our complex conjugate and simplified as much as we can. Now let's move on to step number two, which is actually to expand our fraction further into the real and imaginary parts. So I want to take the real part and I want to split it from my imaginary part. So looking at my fraction over here, I can just take the numerator and split it keeping that denominator on both terms. So I will really just end up with three fits minus 6/5. I, so I have expanded my fraction into my real and imaginary part. Step two is done. Finally, step three, which is to simplify our fraction to our lowest terms. So I have 3/5 minus six fifths I, and this is actually already in its lowest terms. So that means that this is just my solution. And I've completed step three, I'm completely done and I have my solution. That's all there is to dividing complex numbers. Let me know if you have any questions.