Hey, everyone, after learning how to multiply complex numbers, you may think that dividing complex numbers is up next. And you're right. But before we learn how to divide complex numbers, we actually need to learn about something called the complex conjugate. Now, the complex conjugate might seem a little abstract or maybe even a little useless at first. But I'm going to show you exactly what the complex conjugate is and how we're going to use it in a way that will help us to divide complex numbers. So let's go ahead and get started. Now, in order to find the complex conjugate of some complex number, we simply need to reverse the sign of the imaginary part of our complex number. So if I have some complex number A plus B I, I'm going to look at that imaginary part. So in this case, positive B and I'm simply going to reverse that sign. So the complex conjugate of A plus B I is A minus B I and this is going to work in the reverse direction as well. So if I wanted to find the conjugate of A minus BI, I would again look at that imaginary part and simply reverse the sign. So the complex conjugate of A minus B I is simply A plus B I. So let's take a look at a couple of examples and identify some complex conjugates. So looking at my first example here I have one plus two, I now I wanna look at that imaginary part of my number. So in this case, positive two and simply reverse that sign. So the real part of my number is going to stay the same that doesn't get changed. So I'll have one and then that positive two is going to reverse sign into a negative two. So the complex conjugate of one plus two, I is one minus two, I let's look at another example. So if I have one minus two, I, which is just the answer that we just got in the last one and I wanna find that complex conjugate, I again, am just going to look at that imaginary part. So here I have a negative two and I'm gonna reverse that sign. So again, my real part stays the same. I still have one and then my negative two is going to reverse sign into a positive two. And of course, my I on the end there. So the complex conjugate of one minus two, I is one plus two. I, and you might notice that that's just the reverse of what we did in part. A let's take a look at one more example here, so here I have negative one plus two, I now I just again, want to look at that imaginary part in this case, positive two and reverse that sign. Now the real part of our number is going to stay the same even if it's negative. So this negative one is going to stay a negative one. And then my positive two, since that's my imaginary part is gonna reverse sign to negative two. So the complex conjugate of negative one plus two, I is negative one minus two. I now what do you think will happen if I take a complex number and it's conjugate and I multiply them by each other? Well, let's take a look. So if I have a complex number, so here I have two plus three, I and I'm multiply it by its conjugate to minus three I, I'm gonna need to foil. So let's go ahead and do that. So I need to take my first term two times two and that will give me four. And then my outside terms two times negative three. I that's gonna give me a negative six I and then my inside terms three I times two is gonna give me a positive six I and then of course my last terms positive three I times negative three, I is gonna give me negative nine I squared. Now, whenever we multiply complex numbers remember we need to look for that I squared term. So here I have negative nine I squared and this just becomes negative nine times negative one which we know negative nine times negative one is just positive nine and then I can bring down all of my other terms. So my four comes back down and then I have negative six I positive six. I. So looking at this, you might notice that I have a negative six I and a positive six I in the middle there and you might notice that these are going to cancel out. So if I have minus six, I plus six, I, those are gone, those are going to get canceled. So I'm just left with four plus nine. So the like terms I need to combine four and nine are going to combine to give me 13. So two plus three I my complex number times its conjugate to minus three. I gave me a real number and this is going to happen any time I multiply complex conjugate. So multiplying complex conjugates by foiling is always going to give me a real number. Now, this is going to be really useful for us when we're dividing complex numbers, which we'll see in the next video. Now, something else that you might have noticed here is that this four is really just who is really just the a term of my complex number squared. So this four is really just a squared and then my nine that I have on the end here end here is just my B term three squared as well. So whenever I take complex conjugates and multiply them by each other A plus B I times A minus B I, I'm really just going to get A squared plus B squared. Now, this is not something that you have to memorize, but it can be a helpful shortcut. If you do remember it, that's all for complex conjugates. I'll see you in the next video.