 Welcome back to our lecture series Math 1050, college algebra for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. This lecture 16 is the first lecture in chapter 3 for which we are going to be focusing our discussion on quadratic functions. Now, before we get into quadratic functions, there's a little bit we need to talk about with the idea of complex numbers. It really comes down to the following idea. If you look at the very simple quadratic equation x squared plus 1 equals 0, if 1 tries to solve it, you end up transforming this, we'll just subtracting 1 from both sides. You get x squared equals negative 1 and you get stuck right here because there is no real number whose square is equal to negative 1. The basic idea is if your real number was 0, well, 0 squared is 0, that's okay. If you have a positive number and you square, you're going to get positive times positive which is a positive. But if your number was a negative, you're going to negative times a negative which is a double negative, actually a positive again. So real numbers don't square to give us negatives. While the solution to this equation ought to be x equals the square root of negative 1, it turns out that no real number can actually do this. No real number could be the square root of negative 1. Now, that doesn't mean that the square root of negative 1 doesn't exist, and it certainly doesn't tell us that it shouldn't exist. There's no heresy talking about this number, the square root of negative 1. It just means that it shouldn't be called a real number. And it's really unfortunate the terminology, right? There's lots of different number systems out there. We have the natural numbers, the integers, the rational numbers, the real number system, the complex number system we're going to introduce now. We have scalar numbers and vector numbers and matrix numbers. There's lots of different types of numbers, quaternions, octonians. I can tell you lots of different number systems here. Now, unfortunately, it's actually a historical problem while the real numbers get their name. Real numbers sort of in opposition to the imaginary numbers, which many people in the time frame when they would first explore imaginary numbers, they thought that they didn't exist. So they kind of labeled them imaginary because in their snooty way, it's like we're better than those silly people who believe in imaginary numbers, even though they seem to do math better than us. Turns out that imaginary numbers are a very, very useful tool to solve many, many real-life problems. And so the name is sort of like a misnomer. And so I don't want you to focus too much on that, really. I mean, the imaginary numbers are just different types of numbers than real numbers. It's just the name that we've been giving them. And so without getting into a long, philosophical, historical discussion, what really is a number, let's just be satisfied in saying that we have a number, which we call I, which is the square root of negative 1. It's not a real number in the sense of the real number system. It transcends the real numbers. And as such, we call it an imaginary number. Then the complex number system is actually going to be a combination of both a real number and imaginary numbers. So we have this number A, which we'll call this a real number, the real part of our complex number. And then we have this purely imaginary part, this B-I, where B itself is a real number, but a real number times I, we call that an imaginary number. And complex numbers are then the combination of a real part with an imaginary part. And this is often denoted with the C, with this extra line through it. Kind of like the real numbers, but C for complex numbers here. Now, one thing I want to mention is that when it comes to complex numbers, it could be that the imaginary part is 0. Z equals A is considered a complex number. This would just be a complex number with zero imaginary part. That is to say every real number is actually a complex number. But it's also true that if your real part is 0, you just have a purely imaginary number, that's also considered a complex number. So I, 2I, 3I, these are all imaginary numbers, but they're also complex numbers. But you also could have proper complex numbers, that is numbers which are neither real nor complex, like you could take 1 plus 2I. It has a real part of 1, a complex part of 2, sorry, an imaginary part of 2. So I want you to realize that complex number isn't like an either or situation. And this is like, when you look at the real numbers, there is this dichotomy. There's the rational numbers and the irrational numbers. Where the irrational numbers are just those real numbers which are not rational, they can't be written as a fraction. That's not the same situation with complex numbers. With inside the complex number family, you do have real numbers, you do have imaginary numbers, but you also have things which are neither real or imaginary because they're part real, part imaginary, how these hybrids between the two. And that's why they're called complex numbers is because the complex numbers are essentially a two-dimensional type of number. But please rest assured that the complexity of the complex number system is much more imaginary than it actually is real here. And so let me illustrate this to you. If we wanted to start doing arithmetic with complex numbers, we might add or subtract complex numbers. Well, if you come across this and you have no idea what the number I actually is, if you mistook it for a variable, how might you actually try to add together imaginary numbers? You had two imaginary numbers like start two complex numbers, I should say. If you had three plus five I and minus two plus three I, if you didn't know that I was a number, you thought it was a variable, you would just combine like terms. You would take the constant terms together, which is really just the real part of these complex numbers. You get three minus two. And then you would also add together the imaginary parts, which you might think are variables, five I plus three I. And so you just add together like you would any others, combine like terms here, three minus two is a one, five plus three is an eight. So the sum of these two complex numbers is just one plus eight I. What about subtraction? If you're taking six plus four I and you minus from it three plus six I, if you subtract these things, well, since there's two terms in the complex number, I would distribute that negative sign. We get six plus four I, and then we add to it negative three minus six I. So subtraction is just adding the negative, right? And then again, combining like terms, we're gonna get six minus three. We add together the real parts in this case, we subtract them. And then you're going to add together, or in this case, subtract the imaginary parts, four minus six I. And so you end up with three minus two, three minus two I. And that would be then the difference of the complex numbers. So adding and subtracting complex numbers really just comes down to combining like terms. You don't need even to know that I is a square root of negative one. If you knew, if you just thought it was a variable, you would be right, kind of for the wrong reason, but that's okay. The principle of combining like terms applies, whether it's a number or whether it's a variable, and that's how we can add and subtract complex numbers.