 Well, it's because there's an infinite infinity between two integers, or even real numbers. Within zero and one, there's an infinite number of real numbers. So for example, what Ronnie's talking about, if we take the number here, here's a number line. Let's just go like this. Here's zero. Here's one. Here's two. How many numbers, and there's different categories of numbers here. Let's do it better. Let's talk about the real number set. We want to talk about the real number set here. Zero, one, two, three, and negative one, negative two, negative three. We've talked about this real number set. It's one of the first videos, literally video number three out of hundreds of videos I've put out in mathematics. It's the first video, one of the first lessons I put out for mathematics back in 2007 when I started creating math videos. So it's crucial. This is a foundation of, as far as I'm concerned, when we were trying to learn math, high school of math anyway, the beginning of mathematics, is you have to understand what the real number set is, number set. And it's basically human evolution. It's the real number set, the history of mathematics, or the way the rules of mathematics have developed is basically human evolution, a history of human evolution. So the first type of numbers that us human beings really began to grasp were integers, or not integers, sorry, natural numbers, counting numbers, one, two, three, four, five. And we call these natural numbers. Natural numbers, and they're one, two, three, so on and so forth. And then some dude in India defined the number zero, and it was a huge leap in human evolution. It was like the discovery of fire, and apologies if I'm not reading the chat, but it was like the discovery of fire for mathematics. He defined the number zero, and we started looking at how zero works in mathematics. It has a major problem where we can't divide by zero, the universe explodes. But it has a major power, because if we set it equal to zero, set an equation equal to zero, we can factor equations, have things multiplied together and give you zero. That means we can split them up into equaling zero, so we can solve for equations that way. So we made solving for equations way, way easier, right? And because of that, we called it a new category, subset, or a superset, right? And we called the whole number set, because it includes the natural numbers, but as well as zero, right? Human evolution, we learned something new. And then later on, I forget who came up or where they defined integers, right? We started working with integers, which is basically positive and negative whole numbers, right? Two, negative one, zero, one, two, dot, dot, dot. And again, this was a leap in evolution, right? This is like discovering the wheel, right? When people say, what are some of the greatest discoveries, innovations in human history? People go, oh, the computer, sure, the computer was, they go this, this, but people don't realize that two of the major discoveries in human evolution that gave birth to our present civilization or humanity really was fire and the wheel, all right? Those are two of the greatest discoveries in human history, okay, or prehistory, right? So integers are positive and negative whole numbers. And again, these had benefits and problems, right? The problem was, you couldn't, at the time, you couldn't take the even root of a negative number. And they call those imaginary numbers, hundreds of years later, they find out, wow, we should call them complex numbers because the mathematics revealed something about the world to us that we didn't know existed and we didn't understand at the time, we called it imaginary because we thought it was a byproduct of mathematics, the algebra, and then later on, hundreds of years later, you find out, oh my God, this is in life, this is something that exists in the world. We just didn't know how to interact with it, right? And that is happening right now at the moment, right? In mathematics, right now, there are people working on things and mathematics is revealing something to us and the mathematicians, scientists, they don't know what that is, right? What is this thing that's being revealed to us? Is this part of life? And from history, yeah, we know it's part of life, it exists, we just don't know how to read it, interact with it, but the mathematics is revealing it to us, right? That's what integers did, okay? And then above this, you have what's called rational numbers, right? Rational numbers and usually you define this with Q in my part of the world, right? And rational numbers are any numbers that you can write as fractions of integers, right? So the definition of this is fractions of integers, fractions of integers, integers, okay? They usually define, tell people in my part of the world, counting the United States, but these are numbers that either end or repeat and I hate that definition. The reason I don't like that definition because it's not the definition, that's what these numbers are, but the definition should be numbers that you can write as fractions of integers. That allows students to get a better grasp of fractions, right? Not just decimals, right? So once you can manage fractions, you basically rule your own world, right? You power it up, right? And with mathematics, for everything you do, you can undo. With mathematics for things that are, operations that are, there are operations that negate that operation, right? And in this case, if we have a set of numbers that are defined as numbers that you can write as fractions of integers, well, guess what? There are a set of numbers that you cannot write as fractions of integers, right? And these are called irrational numbers, irrational numbers, and these are, it's defined as Q with a line up top. In mathematics, you put a line on top of a letter, usually means not that, right? So Q means rational, this means not rational, right? And these are numbers that you cannot, cannot define, not define, not cannot write as fraction of integers, right, right, as fraction of integers. Pi is one, square root of two is another, square root of any prime number. Prime numbers rules this, right? But for what Joe was saying, right, Zeno's paradox, there's an infinite number of numbers between zero and one, right? But let's look at this from the base up, go through it from human evolution, right? If I was to ask the question, how many natural numbers are there between negative three and three, right? So how many natural numbers between, between, oh my God, I'm just going to go like this, natural numbers between negative three to three. Well the answer is, natural numbers are counting numbers, one, two, three. It doesn't include zero or negative numbers, right? Or fractions. So there are actually only one, two, three natural numbers between negative three and three. So the answer to this is three. And then you can ask the same question, how many whole numbers, how many whole numbers, whole numbers, are there between negative three and three? Well whole numbers is a new category that define the number zero. So zeros include in this. So we've got four whole numbers, right? Four whole numbers between negative three and three. You can continue this, right? How many integers between negative three and three? Well integers are positive and negative whole numbers. So one, two, three, four, five, six, seven. Seven integers between negative three and three. And then you go on to rational numbers, how many rational numbers are there between negative three and three? How many rational numbers between negative three and three? And the answer to that is infinite, infinite, infinite, right? Why is it infinite? For example, let's say because rational numbers are numbers that you can write as fractions of integers, right? So there's one over two here, yeah, that's a rational number, cool. There's one over four here, one over four, and you can continue this. Just in this zone, right? There's infinite number of rational numbers. Between any two rational numbers, there is an infinite number of rational numbers. So if we take this area, let's zoom it out, right? Let's say this line here is one over 32, okay? And this line here would be one over 16, right? One-sixteenth. Well, there's infinite number of numbers here, infinite number, infinite number. Between all these two numbers, there's an infinite number of numbers, right? So huge leap in human evolution from integers to rational numbers. Maybe human technology, right, comparison, I don't know, right? And then you go into irrational numbers and you go, okay, how many irrational numbers are there between negative three and three? How many irrational, rational numbers between negative three and three? That's also infinite. There's an infinite number of irrational numbers between these things. Why is there infinite number of irrational numbers? Because for every one of these things, if you think about it, there is an extra number that is not part of the rational number set because it could be the square root of that number, right? So for example, over here, let's say we had one-third, right? One over three. Well, that's a rational number that exists there, right? And there's an infinite number between zero and one-third. But then there's an additional set of numbers that are not part of the rational numbers such as one over the square root of three. Now that's an irrational number because that's the square root of a prime number, right? But you can't put it into the rational category, you have to put an irrational category. Huge leap forward again, right? So it just grows from there, right? Mathematics is based on some of these most simple concepts you will ever encounter in your life. And as you analyze what they tell you, as you start working with them, just like any tool that you learn how to use, right? If you're new to using a hammer, the only thing you're going to do is, at best, put nails in wood, right? And probably bang your thumb a few times. But as you become a carpenter, you use your tool more and more. You start doing a lot more with hammers than just putting nails on the wall, right? And that goes with every tool, including mathematics, okay? Sort of went off on that one, but I thought it was worth recapping the base of what math is and where this begins and where it can take us, right?