 Welcome to our lecture series Math 3120, Transition to Advanced Mathematics for students at Southern Utah University. Throughout this lecture series, I'll be your professor. My name is Dr. Andrew Missildine. Like I mentioned a moment ago, at Southern Utah University, this class is called Math 3120. It has the name of Transition to Advanced Mathematics. As this is the very first video in our lecture series, I think it deserves us to explain, what does it mean to transition to advanced mathematics? I'm assuming if you're watching this video or participating in this lecture series, you probably don't know a lot about advanced notions in mathematics, like logic and set theory and what have you. Your past mathematical experiences have probably been very computationally focused, that is to say, things like calculus, pre-calculus, algebra, trigonometry, things like that. In that setting where computation is king, a math problem looks like the following. We have a problem. We follow some pre-established process, which we call an algorithm, to solve the problem. This process involves some level of mathematical computations, maybe we add numbers, we take the logarithm of something, we take the derivative or integral of a function, what have you. And then we present our solution like, yay, the area under the curve is 17 pi, what have you. This process I just described here is a pretty good template for solving most mathematical problems that you've probably seen before. But I should mention that that perspective of what this is what mathematics is, is actually quite limited. It turns out it's really just the tip of the iceberg when it comes to mathematical reasoning. This is because mathematics is actually more than just numbers and computations, which people erroneously corral mathematics into being, oh, mathematics is about numbers, it's about computations. There's much more to it. Mathematics is more than just a list of strategies for solving difficult problems. Instead, I like to think of mathematics as a lens by which we view the natural human and digital worlds. Mathematics is a philosophy of how we humans interact with the world around us. Mathematics teaches us to look at the world and with two perspectives in mind. We look at the world in a quantitative way and in a logical way. And those are the two tools that mathematics brings that other scientific or philosophical disciplines don't offer. We solve problems using quantitative and logical methods. Now, I'm assuming that the viewer of these lecture videos is what I would call a proto-mathematician. That is, you have some mathematical training and you want to become a professional mathematician someday. I mean, that could be like, that could be a math professor or a statistician, an actuary, a math educator or many other data scientists, many other disciplines as well. But many things would fall under the mathematician umbrella. But at this moment, you're watching this video because we're a proto-mathematician who desires to transition into more advanced mathematics. And as such, it's necessary for you to learn how one communicates inside of the mathematical sciences. After all, by saying mathematics is a lens through which we view the world, we must infer also that mathematics has its own culture, its own history and its own traditions, which it does, it has all of those things. Now, the mathematical culture, I should mention, it of course is going to intersect with the ambient cultures of the human mathematicians who practice it. Humans have culture and it can be sometimes very diverse when you compare different people together. And as such, that informs how people practice mathematicians. There are noticeable subcultures among the mathematical practitioners, some to do with their own personal heritage and some to do with just best practices where they come from. There are many mathematical standards which one could discuss in this lecture series. And I don't want to get into all of like the social science type of things, mostly just focusing from a professional point of view. This is how the typical mathematician approach is something, but be aware of that. And when I say like this is the typical mathematician, there is of course some diversity in there as well. And these are all things that the uninitiated need to know as they transition to advanced mathematics. Perhaps I'm taking a long way to describe this, but to be clear, this transition that I'm keep on talking about means that one has to learn how to communicate inside of the mathematical sciences. Communication involves reading and writing mathematics. It also means hearing and speaking mathematics. That's how we communicate and how do you do those four things with mathematics? There are many perspectives, many aspects that we could explore as we try to understand how communication is unique in the mathematical fields. And that's the main topic of this lecture series, learning how to communicate. So while this class is called Math 3120, we might as well call it Communication 3120. Now, I actually don't know if that's a course number at SUU, if it is, this is not. I'm not really trying to change the name here, but the point is Math 3120 is how do mathematicians communicate with each other? That's what the point of this class is, okay? And so what you're gonna see in each lecture is that each lecture will be broken up basically into three videos. The first video is gonna introduce some type of mathematical or what we could say quantitative content, some new content that's important for the mathematical disciplines. This very first one is gonna be about definitions. We'll talk about that in just a second. The second video in every lecture will then focus on some type of logical content. And then the third video will then focus on communication itself, focusing on communication skills or nuances about communicated mathematics. In particular, they'll try to be related to the topics we learned in the previous two videos of some kind. You'll see that of course as you watch through lecture one. So the main, the first thing to discuss in this transition to advanced mathematics is the ever important meaning of a mathematical definition. Okay? And mathematics, it is of the utmost importance that our language be precise and unambiguous. If we are to craft sound and logical, valid proofs, then we can only do this if we're certain of what the heck we're talking about, right? We can only infer truthfulness from a theorem. If we know what the theorem says, if we can read the proof and be like, oh yeah, that's correct. If we don't understand what we're saying, then no one can infer any truth whatsoever. So it has to be that our language, our communication is clear. Now ambiguity exists when a term can be reasonably interpreted in more than one way. For example, take the word bank for a moment, bank, B-A-N-K in the English language, of course. Now a bank could describe a financial institution, you know, a place where you deposit or withdraw money, but a bank could also refer to the side of a river, all right? So if I just say what does bank mean by itself, you don't know, because it could mean one of two things possibly more, but at least means two things. Consider the following word, W-I-N-D. What is this word? Now, some of you might have said, this is wind, you know, the thing that blows through the sky, but some of you might have also said it was wind, right? Is it a noun or is it a verb? You know, how do you know how to even pronounce the word? You have to know what its meaning is just to even know how to say it. Now, in natural language like English, which we're speaking in right now, context generally is the tool to avoid these ambiguities. Like, you don't just say the word wind all by itself, you typically are going to put in a sentence, like the wind is blowing hard today or I have to wind up my key. I don't know, whatever people wind up, something. But context is then king. Context gives it meaning. Now the word wind is an example of a homograph. It has the same spelling, two different words with the same spelling, two different pronunciation. So if I gave you the pronunciation, you probably could figure out what it was. But then you look at a homonym like bank where the spelling and the pronunciation are identical. And so the only way to tell the difference between these words is by the context. So in mathematics, it's important we realize that context plays a role just like it does in natural language to avoid ambiguities, all right? Context, you know, in mathematical language, sometimes is needed to avoid vagueness. And it's really, really important that we understand the context we use when it comes to mathematical writings. And to give us context, it often is important, often imperative that we define terms in a mathematical sense. Now a definition is a statement that enumerates properties that the defined term possesses. Just to be clear with the definition, because I'm trying to define the word definition right now. I know it's a little bit weird. We can only infer, you know, from this definition, we can only infer from a definition the enumerated properties given by the definition, no more, no less. And so I put on the screen right now two examples of very common early definitions that students would commonly see in a transition to advanced mathematics class. The definition of what does it mean for an integer to be even? And what does it mean for an integer to be odd? Honestly, many of you probably have seen these definitions before. You know, back in primary school, we talked about even and odd numbers, but let's give it a formal definition to avoid any chance of enumeration, excuse me, avoid any chance of enumerations on a big deal. No chance of ambiguity, that's the word I was looking for here. We say that an integer in is an even number if there exists some other number A such that in equals two times A, all right? So an even number, if we read it again, is an integer such that that number N equals two A for some other integer A. That's what an even number is. If I say, oh, this number is even, then you gain this property and nothing else. So if I say this is an arbitrary even number, the only thing we know is that it is divisible by two. That's the only thing we know about it. I guess we should also mention that for an even number, it does have to be an integer. We don't talk about even rational numbers. We don't talk about even real numbers because when you think of the divisibility for rational numbers and real numbers, everything's divisible by two in a manner, speaking, because if you take the number five, you could factor us two times five halves if you look at the rational number system. So built into the definition, we are referring to as an integer, AKA a whole number, which could be positive, negative, or zero. And when we factor the integer, the two factors themselves have to be integers. One of them is two and one is some other integer. We don't care what that is along the integer. That gives us the definition of an even integer. Now, with this definition in hand, we always know what an even integer means when we talk to each other. An even integer means this, nothing else, okay? And even if I call it an even number, the definition has built into it that it has to be an integer. We can't talk about an even rational number like we said before, that would be nonsense. Nonsense, just so you know, has a mathematical meaning, nonsense is when someone uses a mathematical definition incorrectly to say something like, oh, two plus two I is an even complex number, is an example of nonsense, because describing whether an integer is even or not, it has to be an integer. A number can only be even if it's an integer. Other number systems don't have it. It's nonsense to use a definition incorrectly. We have to make sure we avoid ambiguity, but we also have to make sure we avoid nonsense. Now, conversely, an integer we say is odd if the integer n can be written in the form n equals two a plus one for some integer a, okay? So that's what it means to be odd. In particular, an odd number is just one more than an even number. And you've seen examples of this before, right? The numbers four, six, 12, and 20 are all examples of even numbers, and I don't think you need much convincing, but I will just check the definition to be sure. Four is even because it's two times two. I wanna point out here that when we talk about this sum integer a, this integer could be two itself. It doesn't have to be different. We don't know anything about a other than it's an integer, but there is an integer so that four is two times that integer, so four is even. Six is likewise even because it's two times three. 12 is even because it's two times six. 20 is even because it is two times 10. And that's all it means to be even. You're two times a number. Conversely, the numbers one, five, 31, and negative three are all examples of odd numbers. Why is that? Because I can write each of those numbers as an even number plus one. One equals two times zero plus one. Zero is an even number because it's, sorry, two times zero is an even number there. You have five. Five is equal to two times two plus one, right? 31 is equal to two times 15. That gives you 30 plus one. Now you have to be a little bit careful with the negative number here. Look at the definition. Negative three is an even number, excuse me, it's an odd number because it's, because it is positive two times negative two plus one, like so. Negative two, you take two times negative two. That's an even number. It's negative four plus one. It's negative three. That makes it an odd number. And that's all it is to determine whether we have an even number or an odd number. When it comes to a definition, we cannot infer any property from the definition except for those which are explicitly stated in the definition or things that we can logically conclude from the definition, but that gets into the logic side. We'll talk about that later. For example, well, let's say something about this. We'll say one thing. For example, suppose we assume that in is an even number, then we know there exists some integer such that n equals two eight. That's the definition. We can assume that. This is an acceptable inference because it follows by definition. This is a statement you see a lot when written in mathematics by definition. So we make an inference where we'd say something like, oh, n is even, n is even. So n equals two A for some A. You'd say something like that. That follows by definition. That it's not a really deep inference there because it follows directly from the definition. The fact that we could say more from that using logical implications is one of the benefits of logic. Can you say more than that? Well, with the definition, we can't say anything more. We need some logic to get beyond that. Now, on the other hand, with the same even number n here, so let's say that n is even. Therefore, n equals two A. We cannot infer, we can't infer basically anything else. We can't infer that, oh, n equals three B for some B. We don't know that. I mean, it could be true, but it might not be true. I mean, for example, take the number six. Six is equal to two times three, so it's an even number. Six is also equal to three times two. So that is true. That applies for six here, but if we take, for example, the number 10, 10 is two times five, so it's an even number, but on the other hand, 10 does not equal three times B for any integer B. There's no way to do, there's no divisor of three for the number 10, something like that. So this is what I mean by we can only infer what the definition gives us. The definition of an even number gives you that you're divisible by two. It doesn't give you anything else. Now it could be true, but you need more information than what's provided to actually go with it. So you have to be very careful that we only can say what is given by the definition. This holds by definition. Let's consider another definition for a moment. We say that two integers have the same parity if those two integers are both even or they're both odd. And in this situation, we use the word or here, we're saying that at least one of these two things have to be true, they have to both be even or they both be odd. Now you might be like, at least, can they both, how can both happen, right? Can they, a number can't be even and odd, can it? Well, at the current moment, we actually don't know that because for a number to be even, there just has to be some other integer so that the number N equals two A. But to be odd, we just have to require that there's some number two B plus one that happens. Could there be different integers maybe even the same integer, right? Is there some integer A and B so that these can happen simultaneously? Maybe, maybe not. I mean, we probably believe that an integer is either even or odd, but at the current moment, we have improvement such a statement, we allow the possibility that both could happen simultaneously. So when we talk about the word or here, we're saying at least one of these two things is true. And so you're the same parity if you're both even or you're both odd, all right? So the numbers eight and two have the same parity because they're both even numbers. Now, if two numbers are not, if they don't have the same parity, then we say they have opposite parity. That would mean that they're not both even or they're not both odd, which would have to mean that one of them is even, one of them is odd. So take for example, eight and three. Eight and three, they have opposite parity. Eight is an even number, it is not odd and three is an odd number, it is not even. So they have opposite parity. Let's look at one more definition before we end this video. So one thing I should mention about definitions is that when we look at a definition, typically they'll be marked off with some type of header of some kind of title and typically be like definition. So you know that the next paragraph is gonna define something or multiple terms. It typically have a numbering system of some kind. This is quite common mathematics. Now the numbering system we have is actually got three numbers here. The first two numbers represent the chapter and section that we are drawing from from the textbook. Now our lecture series is loosely based upon the book of proof which is an open source textbook about proof writing. I say loosely because, well, for many reasons we won't, we'll sometimes use examples from the book, sometimes we'll use examples from other places. We definitely won't follow the order of the book because after all, this is the first video and we're in section 4.2. This numbering is just to help you know where in the book these things are located and then there's a number that tell you in this lecture we're at item six. So there was like 4.2.1, 4.2.2 and this will include definitions, examples, theorems, et cetera. So look for these numbers from the future. You can reference a theorem or a definition like, oh, by definition 4.2.6, such and such. Another convention we often use with definitions is that the word being defined often will be written in bold. Some people like to use italics to do it. That's perfectly good. You should put something when you're putting a definition there. So it's very clear that this is what we're defining so that people can find it later on if they need to check the definition. This is gonna be another definition about integers. Why do we keep on defining stuff about integers? Well, I'm assuming that by watching this video, we don't necessarily have a background in advanced mathematics. So we have to use a mathematical topic that I can assume the audience knows so that we can talk about things like definitions. So we're using the notion of integers. We know what integers are. Otherwise, we probably wouldn't be watching this video. So it's pretty safe to make some definitions about integers. It's a good starting place. So let A and B be integers. And be aware that when I say this, let A and B be integers. The only thing I can assume about A and B is that they are integers. I can't assume anything else about them. For example, I can't assume that A and B are different integers. They could be the same one. All we can assume is that they're integers because that's what was stated. And so we can only get what the definition of integers implies. So we say that the integer A divides the integer B if there exists some other integer N such that A, N equals B. And that means that A divides B. And this is often denoted using the following notation to try to abbreviate it. We say that A line here B means A divides B. And that's how you read it, A divides B. And so let's see some examples of this. Note that three divides six. And this is true because six is equal to three times two. Two is an integer. And so there is some number, some integer two in this case so that three times two gives us six. This gives us that three divides six. Some other definitions we throw out there. If A divides B, then we say that A is a divisor of B. Therefore, three is a divisor of six. Also, if A divides B, we say that B is a multiple of A. So since three divides six, that means six is a multiple of three. It's also true that two divides six. So two is a divisor of six. And hence six is a multiple of two. And so putting these terms together, we can talk about divisibility of integers. It's also true that four divides 20. That is four is the divisor of 20. And that's because 20 is equal to four times five. There exists an integer so that four times that integer gives you 20. That makes 20 a multiple of four. Finally though, we gotta give you a counter example here. Here's an example where it's not divisible. Seven does not divide 12. There is no, there is no integer such that 12 is equal to seven times that integer. Now, proving that such an integer doesn't exist can be problematic. And at this moment, I'm gonna kinda take it for granted. I think many of you probably will believe that seven does not divide 12 in the integer sense, but how does one actually prove that? I mean, what if you just have to choose a really big number that you've never thought of before and it'll work? We will get to that. Be patient, young grasshopper. We will get to that in the future. All right? Now, one other thing I wanna mention before we close this video on definitions is that we talked about even number just a moment ago, right? We said that N is an even number. This happens if and only if you often abbreviate that statement if and only if you see it a lot. If and only if you'll abbreviate this as IFF. It's not a typo. N is even if and only if N is equal to two A for some integer A. That's the definition. That's why we get that. So being even is exactly the meaning that you can factor in as two times some other integer. But when you look at this, by definition, this is the same thing as saying that two divides N. So it turns out even means you have two as a divisor. That's the same thing as this. So in retrospect, we defined even numbers first and then we defined divisibility. But we could have done the other way around. We could have defined divisibility and then defined even to be using this divisibility notion. This gives us an example of what we call an equivalent definition. That is you could have defined the same topic, the same concept in two different ways. And even though the definitions look different, it turns out that everything that's true about one is true about the other as well. These two definitions of a number being even, that is this statement and this statement are equivalents to each other. And so when one has an equivalent definition, it doesn't matter which definition you use. And so which one do you choose as your definition of even? Like we use this one, which is a little bit more elementary because we hadn't yet introduced the notion of divisibility. That was a choice that we had to make. It was a convention that was made by the mathematical communicator. And that's something that we need to be aware of. It could be that as you're trying to write it's like a mathematical textbook or a mathematical paper, you choose one definition over another, even though we're equivalent because one is more convenient for the other. Maybe it's logically closer to the topic you're searching for. There's a lot of reasons why one can use equivalent definitions and we'll explore many of these in the future.