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+bi, 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+bi is a-bi. And this is going to work in the reverse direction as well. So if I wanted to find the conjugate of a-bi, I would again look at that imaginary part and simply reverse the sign. So the complex conjugate of a-bi is simply a+bi. 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 1+2i. Now, I want to look at that imaginary part of my number, so in this case, positive 2, 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 a 1. And then that positive 2 is going to reverse sign into a negative 2. So the complex conjugate of 1+2i is 1-2i. Let’s look at another example. So if I have 1-2i, which is just the answer that we just got a last one, and I want to find that complex conjugate, I, again, am just going to look at that imaginary part. So here I have a negative 2, and I'm going to reverse that sign. So, again, my real part stays the same. I still have 1. And then my negative 2 is going to reverse sign into a positive 2. And of course, my I on the end there. So the complex conjugate of 1-2i is 1+2i. 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 -1+2i. Now I just again want to look at that imaginary part, in this case, positive 2, 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 2, since that's my imaginary part, is going to reverse sign to negative 2. So the complex conjugate of -1+2i is -1-2i. Now what do you think will happen if I take a complex number and its 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 2+3i, and I multiply it by its conjugate, 2-3i, I'm going to need to foil. So let's go ahead and do that. So, I need to take my first term, 2 times 2, and that will give me 4. And then my outside terms, 2 times negative 3 I, that’s going to give me negative 6 I. And then my inside terms, 3 I times 2 is going to give me a positive 6 I. And then, of course, my last terms, positive 3i times negative 3i is going to give me negative 9i2. Now whenever we multiply complex numbers, remember, we need to look for that I squared term. So here I have negative 9i2, and this just becomes negative 9 times negative 1, which we know negative 9 times negative 1 is just positive 9. And then I can bring down all of my other terms. So my 4 comes back down, and then I have negative 6 I, positive 6 I. So looking at this, you might notice that I have a negative 6 I and a positive 6 I in the middle there, and you might notice that these are going to cancel out. So if I have minus 6i, plus 6i, those are gone. Those are going to get canceled. So I'm just left with 4 + 9. So the like terms I need to combine, 45 are going to combine to give me 13. So 2+3i, my complex number times its conjugate, 2-3i, gave me a real number. And this is going to happen any time I multiply complex conjugates. 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 4 is really just, wow, is really just the a term of my complex number squared. So this 4 is really just a2. And then my 9 that I have on the end here is just my b term, 3 squared, as well. So whenever I take complex conjugates and multiply them by each other, a+bi times a-bi, I'm really just going to get a2+b2. 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.