 As we end lecture one about definitions and statements, before we talk about the main idea of this video, I just wanna give us a quick rundown of how we expect these lectures to work for this lecture series. Like I mentioned previous, I think I mentioned this in video one here, but I'll say it again. Every lecture would be broken up into three videos. The very first video always about some new mathematical topic. And in this lecture, we learned about mathematically, what does it mean to have a definition? The second video will always be about some type of logical topic where in this case, we learned about statements. And then the third one, we'll try to bring things together, harmonize in particular focus on some type of communication topic when possible. So we're gonna have some type of discussion, not necessarily prove anything, but we might see examples and new definitions about things. And for the most part, I will try to connect these mathematical communication topics to the previous things we learned about in that lecture when possible. That won't always be possible, but in this video it is exactly what we're gonna do. Because what I wanna talk about in this video is the notion of a non-equivalence definition. When we first learned about definitions, we ended the video talking about how a number was even if you have N is equal to two A, but we also found out you could say N equals two, excuse me, N is even if two divides N. Both of those mean even. And while this was our definition of even, you could have used this one instead and it wouldn't have changed anything. Logically, they give you the exact same theory. But there do come times in mathematics when another possibility happens. Unfortunately, there are instances when mathematicians use non-equivalent definitions. That is the same word is used to describe two different logical topics, which often have much in common but have some significant distinctions. And I think examples are the best way to explain what's going on here. And the one I see the most often is this fight between what is a natural number. So I'm gonna present to you two definitions of what a natural number is. So definition four to nine, as we call it in our lecture series, this, we say that a natural number is a positive integer or zero, okay? Now, an alternative definition for natural number is that this will be definition four to 10. We say that a natural number is a positive integer. Now, when you look at those definitions, they do seem to agree for the most part on what a natural number is. So for example, if you take the numbers, one, two, three, four, five, six, seven, you keep on going right here. These are all natural numbers. Both definitions agree upon that. No disagreement there, because again, they both say positive integers are natural numbers. And if you take something like X equals negative three, that's not a natural number. It's a negative integer, it's not a natural number. Pi, that's an irrational number, not a natural number, two-fifths. That's a ration number, it's not a natural number. You got the square root of two again, as in this is another irrational number like pi. The main differences is an algebraic number while pi is transcendental, things we won't define right now. You take something like two plus i, a complex number, it's not a natural number. These are all examples of not natural numbers. These two definitions agree upon them exactly. The place they disagree comes down to zero. Is zero natural? Is it a natural number? Well, with the first definition, the answer is yes, because definition, the definition says a natural number is either a positive integer or zero. Zero is included inside of the natural numbers. But if you adopt the second definition of natural number, then it turns out that zero is not a natural number. And so whether zero is a natural number or not comes down to which definition of natural numbers are you using? Now why are there two different definitions that we use to describe the exact same thing? Well, it's because there are compelling reasons on both sides of the debate. And we mathematicians have to accept that others might use a slightly different definition for a commonly studied object, such as natural numbers. For reasons like this, is zero a natural number or not? It's a common practice for mathematical writing to begin with definitions, even on common terms like natural numbers. So that the reader knows exactly what the author is meaning when they say natural number. So if I have a statement that's like, oh, for all natural numbers, such and such property holds. Well, you might ask yourself, does it hold for zero or not? It depends on how you define natural number. And there is a disagreement. Many people disagree on this. Now for the sake of this lecture series, this will be the official definition of natural numbers we use. So in our lecture series, the natural numbers do include the number zero. But if you look at other mathematical documents, they might not include zero as a natural number. You have to be very careful. You have to always check, always be asking those questions because there is no common convention. This has to do with the culture of mathematics. And culturally, the definition of the natural numbers is not universal. Pretty much, I don't know of anyone who's ever excluded a positive integer as being a natural number, but zero can get a bad rap sometimes. But in our lecture series going forward after this video, we will always think of zero as a natural number, even if others do not. That's the convention that we have adopted. All right, let's look at another example of this. Two definitions of the same topic that aren't actually the same. And because they're different, they actually give you a different idea. This time we're gonna define what a prime number is. So two definitions here. We say that a natural number P is a prime number if the only positive divisors of P are one and P itself. That sounds like the usual definition of a prime number. Now, one thing I should mention is that this definition depends on what a natural number is. Now, admittedly, we could define prime numbers for integers. If we were in a number theory course, we would do exactly that. We aren't gonna worry about that right now. We're just using divisibility as an example of how we define things. All right, but notice this definition depends on the word natural number. So if there are two different schools of thoughts on what natural numbers are, and there's two different schools of thought on what a prime number is, you can see that things can start to branch out and get really complicated really quickly, even though we might have simple statements like five is a prime, what does prime mean? What does natural number mean? What does five mean? We have to be very careful in our terms. So it's good practice to begin mathematical documents with definitions. Now, for the sake of a student who's writing homework or something, you probably don't have to write the definition of the terms because they're in the textbook, they're in the lecture notes. They've already been established. But if I'm writing like a mathematical textbook for students I haven't met before, then I should define the terms so they know what I mean when I say something. So one definition of a prime number is that it's a natural number so that the only divisors are one and P, okay? The second definition, we say that a natural number other than one is a prime number if the only divisors of that number P are one and P itself, all right? And so when you compare those definitions much like the definition of natural numbers we saw a moment ago, they agree on nearly everything except for one number. So where do they agree upon? Well, these will both agree that two was a prime number because the only divisors of two are one and two, positive divisors of course, they would agree that three is a prime number. The only way to factor three is one times three. They would agree five, seven, 11, 13, et cetera. Both definitions will agree that these numbers I've just listed on the screen are prime numbers. The disagreement then lies on the one number that I got excluded with the second definition is the number one, a prime. Now there are compelling reasons for both directions. Like if you think of the first definition by the first definition, one actually would be a prime number because when you look at the divisors of one there is only one number, it's one. And so when you look at that definition the only positive divisors of one are one, yeah, and one. It's just the same number said twice. Okay, so that would be a prime by that definition but we exclude one in the second definition of prime numbers. Why? Well because when you look at every other prime number other than one, the ones that we all agree upon they always have two divisors like two gets factors one and two, three gets factors one and three, five gets factors one and five, et cetera. Right, 13 can only one factorization one and 13. For all the other prime numbers which we all agree upon their factorization always involves one and something else that is this P is not equal to one. In the case over here where P could equals one then you get a repeat like oh the only divisors are one and one. The factorization of one is one times one. And so one is somewhat exceptional from all the other primes because it doesn't have a second divisor, it only has one divisor. And so maybe because one is exceptional we should exclude it from being a prime number. So this gives you some of the arguments here. Now for the sake of this lecture series we do adopt the second definition as the official definition of a prime number. So in other words one is not a prime number for the sake of our lecture series. Now like in primary school, elementary school typically prime numbers are defined like this because the one exception might confuse students. This definitely is an easier definition because it doesn't have an exception. Now that exception has good reasons like in number theory when one really studies primes there's good good reasons why we should exclude one as a prime number. But if you're like in the second grade and you're learning about factoring in prime numbers which I might actually be impressive for a second grader but whatever you know if you're third grade, whenever students typically learn about prime numbers it might be difficult to explain why one shouldn't be a prime number. So what we do is we fudge. That is we kind of lie a little bit. We let one be in there temporarily knowing that in the future an exclusion will happen. Why do we have two definitions? Because there was a time where one is appropriate but then as we transition into advanced mathematics maybe we adopt the more mature definition and that's actually if I were to say something about the zero natural number. The reason why we say zero is a natural number there's many reasons for it but I'm gonna stick with this advanced mathematics business we just talked about. Now yes there are some people defined the natural numbers as only the positive integers and not zero. We will include zero and the reason for it is tradition, right? Most people who say zero should not be a positive number generally come down to that's the tradition. They don't usually say it that way but that's the tradition zero is unnatural took people a long time to discover zero, right? As a mathematical concept and that makes it unnatural because it took people a while to discover it and that's a valid argument but that's why sort of like primitive mathematics didn't include zero but we're not looking at primitive mathematics we're not looking at elementary mathematics we're looking at advanced mathematics and from an advanced mathematical point of view there are strong reasons to include zero as a natural number even if historically people did not include it and so we as advanced students of advanced mathematics we exclude one as a prime number we include zero as a natural number and many of these reasons we'll probably explore into the future. So that brings us to the end of lecture one thanks for watching everyone. If you have any questions whatsoever about anything you see in these videos feel free to post your questions in the comments below and I'll be glad to answer them as soon as I can. If you learned anything from this lecture or these videos please like these videos subscribe to the channel to see more videos like this in the future and hopefully I'll see you next time.