 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 A minus BI. But typically, we think of it as plus BI. If it happens to be a negative B, we'd pay A minus BI. A is the real part, BI is the imaginary part. B's 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 it again. What if we have a fraction? Well, this is going to be the square root of negative 1. We can take it the square root of the top or the square root of the bottom. So let's take it as 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 term. 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. Negative 2 plus 3, that's my real parts, plus the 5 minus the 1 is plus a negative 1 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, we just need to distribute that negative or think of it like this. We have 7 minus 2 and then plus the negative 1 because it's going to be the I out here. So negative 1 is the coefficient here minus, that's here, the negative 3. Well, we know if it's minus a negative 3, it's going to be plus 3. So 7 minus 2 will be 5 plus negative 1 plus 3 would be 2 I. Or you could just write it 5 plus 2 I. 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 going to 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 2 I. Negative 3 I times 5 will be minus 15 I. And then negative 3 I plus I will be minus 3 I squared. Well, this is 10. If we combine these two, we get minus 13 I. 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 13 I and simplifying then that 10 plus 3 will be 13 and then minus our 13 I. 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 plus a negative I square root 3 would be minus 5 I square root 3. And then we have positive I square root 3 times 5. So it would be plus 5 I 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 a negative 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. This is 25. Negative 5 I square root 3 plus 5 I 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 going to be 3. So what do we have? We have 25 plus 1 times 3 would be 3 or 28. All right. So what do we do here? Well, we could do it a couple of ways. The most sure way to make sure you get it right is just to rewrite it. 2 minus 3 I times 2 minus 3 I. And then do the foil. 2 times 2 is 4. 2 times negative 3 I would be minus 6 I. And we do that again for the inside terms. And then the last terms are going to be a negative times a negative is a positive. 3 times 3 is 9. I times I is I squared. So we have 4 minus 12 I. And then 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 sign. Now it's going to end up being 4 minus 12 I minus 9. 4 minus 9 is going to be negative 5 minus 12 I. And finally we talk about division. And when we talk about division, we have to talk about conjugate pairs. Conjugate pairs, or if you have a plus bi, its conjugate pair is going to 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 squared to 3, and then the I squared is with it. So let's do that here. What's the conjugate pair to 1 plus 3 I? Well, that's going to be 1 minus 3 I. 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 6. So 6 minus 18 I, that's not too bad. Now on the bottom here, 1 times 1 would be 1. 1 times negative 3 I is minus 3 I. 1 times 3 I is plus 3 I. And then negative 3 I plus positive 3 I is minus 9 I squared, which is really plus 9 because this is a negative 1 times that negative 9. And we have 1 plus 9 or 10. Or we could have just said 1 squared is 1, 3 squared is 9. It's going to be plus 9 because you always have a negative when you do those last two terms. So let's try it again. The conjugate to 2 minus 4 I is going to be 2 plus 4 I. So it's a binomial. And on the top here, we also have binomial on the bottom as a binomial. This is like what we were doing with multiplication. Negative 4 times 2 is negative 8. Negative 4 times 4 I would be minus 16 I. 8 times 2 is going to be plus 16 I. That was 8 I plus times 2. And then 8 I times 4 I is going to be plus 32 I squared. And let's multiply out at the bottom. 2 times 2 is 4. 2 times 4 I is plus 8 I. 2 times negative 4 I is minus 8 I. Negative 4 I plus I is going to be minus 16 I squared. Negative 8. On the top we had, we ended up actually having negative 4 times 4 was negative 16. But the 8 times 2 is also 16. So those two canceled each other out. And we ended up with minus 32 I squared, which would be plus 32. Because this is negative 32 times negative 1, because positive 32. And that gives us negative 8 plus 32 is going to be positive 24. And then if we take the 4, the 8 I and the negative I cancel each other out. And then this is minus 16 I. So we end up with plus 16. Again, that's negative 16 times negative 1. So 4 plus 16 is going to end up being 20. And we can reduce that because they're both divisible by 4. And 4 times 6 is 24. And 4 times 5 is 20. And I'm happy with 6 over 5.