 This video is going to talk about complex numbers. So we have imaginary numbers, which is the square root of negative k. Normally, we don't have a negative under the square root, but when we do, it's an imaginary number, and the imaginary number is at i. So i is equal to that square root of negative 1, and then this is supposed to be i squared, not subscript. i squared is equal to the negative 1. So complex numbers can be written a plus bi, or it happens to be a negative b would be a minus bi. A is a real part, bi is the imaginary part. B is a real number with the imaginary part multiplied with it. So when we look at something like this, we need to just take out that square root of negative 1. So we have the square root of negative 1, and then times the square root of 32, but then we also want to simplify this as far as we can. So we need to find the perfect square, and 32 is the square root of 16, that's a perfect square, times the square root of 2. Yes, 32 is 8 times 4, but 8 has a perfect square in it. So we want the largest perfect square that will go into our number, and then we also have it times this negative 1. So that will give us square root of negative 1 is i, the square root of 16 is 4, and then we have the square root of 2. So let's try again, what if we have a fraction? We can take it the square root of the top or the square root of the bottom, and well, this is going to be the square root of negative 1 times the square root of 4, and then 75 is going to be the square root of the perfect square would be 25 times 3, or the square root of 25 times the square root of 3. This is i for the square root of negative 1, square root of 4 is 2, and then on the bottom we have 5 as the square root of 25, and then times the square root of 3 since it's not a perfect square. So let's move on now to operations. The first operations we want to talk about are addition and subtraction, and you basically do it like you would do any polynomials. You just combine your like terms. The way that I'm going to teach you is to write it in the form of a plus bi, and we'll write b in parentheses so that if it's a negative one, though it should be a negative if we want, like this. Negative 2 plus 3, that's my real parts, plus the 5 minus the 1 is plus a negative 1 if you really want to, times that i, and that will give us negative 2 plus 3 is 1 plus, 5 plus a negative 1 will be 4 times that i. If we do subtraction, think of it like this, we have 7 minus 2, and then plus, so negative 1 is the coefficient here, minus, it says here, then negative 3, but we know if it's minus and negative 3, it's gonna be plus 3, so 7 minus 2 will be 5 plus negative 1 plus 3 would be 2i, or you could just write it 5 plus 2i. Okay, so we, again, what we did different there was, we still left it as a plus bi, but in the parentheses, we were subtracting for a subtraction, and we were adding for an addition. So when we multiply, it's basically just doing foil, but we need to remember that when we have i squared, it's equal to negative 1. We're not gonna have an i in that term when we get done. So let's just do the foil here, 2 times 5 is 10, 2 times i is plus 2i, negative 3i times 5 will be minus 15i, and then negative 3i plus i will be minus 3i squared. This is 10. If we combine these two, we get minus 13i, and then we have minus 3, but remember, i squared is negative 1, so negative 3 times negative 1 really gives us plus 3, and we have our 10 minus our 13i, and simplifying then, that 10 plus 3i will be 13, and then minus our 13i. What happens if we have square roots? You still multiply them, but remember, you multiply what's underneath the radical, like this, 5 times 5 is 25. 5, the negative i square root 3 would be minus 5i square root 3. Then we have positive i square root 3 times 5, so it'd be plus 5i square root 3, and then we have i square root 3 times a negative i square root 3, so it's a negative, positive times positive, i squared, that's the i times the i, and then it's the square root of 3 times 3, or the square root of 9. So we'll do this in a couple of steps, it's 25. Negative 5i square root 3 plus 5i square root 3 cancels each other out, because they're opposites, and then we have minus, and then we have i squared is negative 1, and the square root of 9 here is gonna be 3. What do we have? We have 25 plus, or 28. Negative times a negative 1 gives me the plus 1 times 3, that's how I get positive 3. What do we do here? The most sure way to make sure you get it right is just to rewrite it. 2 minus 3i times 2 minus 3i, and then do the foil. 2 times 2 is 4, 2 times negative 3i would be minus 6i, and we do that again for the inside terms, and then the last terms are gonna be a negative times a negative is a positive, 3 times 3 is 9, i times i is i squared. We have 4 minus 12i plus 9 times a negative 1, and you'll notice that every time it really just changes the sign on that number. This was a plus 9, now it's gonna end up being 4 minus 12i minus 9. 4 minus 9 is gonna be negative 5 minus 12i. When we talk about division, we have to talk about conjugate pairs. Conjugate pairs, if you have a plus bi, its conjugate pair is gonna be a minus bi. So when we multiply those two things, that's like the example we had up here in this second multiplication problem. This was a plus bi, and a minus bi, same bi, but one was positive, one was negative, and what happens to our middle term? They cancel each other out. So we were really left with a squared, 5 times 5, and b squared, which was 3 squared to 3, 2 squared to 3, and then the i squared is with it. So let's do that here. What's the conjugate pair to have one plus 3i? Well, that's gonna be one minus 3i, and if we have an i in there, we really have a square root. So this is what we would say that we have to rationalize. This is how you would rationalize with a complex number. So you multiply the top and bottom by the conjugate, and you distribute the six, so six minus 18i, that's not too bad. Now on the bottom here, one times one would be one, one times negative 3i is minus 3i, one times 3i is plus 3i, and then negative 3i times positive 3i is minus 9i squared, which is really plus 9, because this is a negative one, times that negative nine, and we have one plus nine or 10. Or we could have just said one squared is one, three squared is nine, it's gonna be plus nine, because you always have a negative when you those last two terms. So let's try it again. So if we multiply here, what's the conjugate of two minus 4i? That's gonna be two plus 4i. So two plus 4i, whoops, two plus 4i. If we multiply, and remember this is just like foil, so negative four times two is gonna be negative eight, and then outside terms would be negative four times four, i would be minus 16i, and then plus 16i, and then the plus 32i squared, that's all on the top. And then on the bottom, we're gonna have two times two is four, two times four i is gonna give you plus eight i, and then on the inside that'll give you the minus eight i, and then the last terms are gonna give you minus 16i squared. Well on the top here, we again have something that looks like a difference of squares almost, the 16i and the negative 16i cancel each other out. So we have negative eight, and then remember, i squared changes the sign, so this looks like 32 times negative one is gonna be minus 32, and then on the bottom we have four, but the middle terms cancel each other out because they're opposites, and then this again is times negative one, so negative times a negative will give me plus 16, and that's gonna give us, negative eight, negative 32 is gonna give us negative 40, 16 plus four is gonna give us positive 20, and if we look at that, we're gonna have an answer of negative two.