 This is Chichu. Now we're in series four right now and at the beginning of this series I mentioned that I was going to talk a little bit more about zero and infinity basically continuing on with one of the first videos we put out in 2007, the language of mathematics, I think was video number five where I put out a little teaser video on how zero and infinity are related and what the concept of zero infinity really mean. What we're going to do right now is take a look at infinity specifically and take a look at what infinity means and for me infinity has two different definitions, two different meanings or two different ways you're going to look at it, okay? So this is a symbol that we use to represent infinity, the figure eight, right? And the way we look at infinity or the way I look at infinity is basically having two different meanings. One is the countable infinity, something that you can do forever, that goes on forever until the end of time and this infinity where we think about it as something that's countable, something that we can do even though we can't do it because we have a certain amount of lifespan, right? We can't go on forever, but for some reason it's something that you can perceive, right? And that infinity is manageable. The other infinity, the other definition of infinity that we have is where the universe explodes, where the language of mathematics collapses, okay? Where the question really becomes irrelevant, it's something that we cannot comprehend, okay? So those are the two different infinities we're going to talk about right now. Let's take a look at the infinity that goes on forever, something that you can basically count maybe, right? Let's look at the natural number set from the real numbers. Again, one of the first videos that we ever produced, right? So let's take a look at the natural numbers and as we talked about, natural numbers are just basically whole positive numbers that go on forever and that's something you can sort of imagine, right? One, two, three, four, five, six, seven, eight and just continue counting if you feel like it, right? That's something you can imagine, they're just sitting there and just counting higher and higher numbers. So this is one example of infinity, right? And one thing you should keep in mind in this version definition of perspective of infinity is, is that there are different size infinities. There are infinities that could be bigger than other infinities. Just as an example, let's take a look at the whole number set that includes the natural numbers, right? So let's take a look at the whole number set. Again, something we talked about from the real number set, right? Whole number set includes the natural numbers plus zero, right? So the whole numbers, again, it goes on forever, it's infinite, zero, one, two, three, four, five, six, that just continue on, right? The whole number set goes on forever, the natural number set goes on forever, but the whole number set is bigger than the natural number set because it includes one, two, three, it includes all the natural numbers plus one, which is a weird, weird, weird concept to think about where if something's infinite, another infinite could be bigger because it goes infinite plus one, right? And again, that's just one simple example. That's just one more. This whole number set is just one more bigger than the natural number set, right? The integers is double the natural, double the natural numbers plus one, right? So the integers are positive and negative whole numbers including, well, positive and negative natural numbers including number zero, right? So it's zero, and then all the positive numbers, or if you're looking at the video, all the positive numbers and all the negative numbers, right? And that infinity is the integers is twice as big as the natural numbers plus the number zero, right? So this infinity, the integer infinity is bigger than the whole number set, right? The integer set is bigger than the whole number set. And from this, we can go to the rational numbers which are infinitely larger than all of those. So what are the rational numbers? And again, we talked about this in the third and fourth video, I believe, which are any numbers that you can write as fractions of integers, right? So rational numbers aren't twice, three times, four times, ten times, a hundred times bigger than the integers, they're infinitely larger than the integers. Because if you could think about it, take the number zero and take the number one, right? Number zero and number one occur in the whole numbers and the integers and the same length, the same distance occurs in the natural numbers between one and two, right? So take the number zero and one. Between those two numbers, you can put an infinite amount of other numbers, right? And those are your rational numbers because as long as you can represent those numbers as fractions of integers, there's an infinite number that can go there, right? If you want to think about one number that goes between zero and one, is a half, right? Take the average between zero and one, you get a half. Take the average between zero and a half, you get a quarter. Take the average between zero and a quarter, you get one eighth. Average between zero and one eighth is one sixteenth, it's one thirty second, one sixty four. And just continue from there, right? So any two numbers you have, right? Any two numbers that you can take, you can always place a rational number between them, okay? Because you can just take the average of them, okay? So that's the rational numbers. Rational numbers are infinitely larger than integers, right? And integers are larger than whole numbers, bigger infinity than whole numbers, and whole numbers are bigger infinity than natural numbers. And from here we can go to the irrational numbers. Hi, how are you doing? So this is the irrational numbers. And irrational numbers are any numbers that we can't write as fractions of integers, right? Which is the opposite of rational numbers. And for irrational numbers, there's an infinitely larger amount of irrational numbers than there are rational numbers, right? And again, so forth, we just keep on going down in the size of the sets that we have. They're all infinity, but this one is bigger than this one, which is bigger than this one, which is bigger than this one, which is bigger than this one, right? Which is pretty interesting. It's just different size infinities. And there's an amazing documentary out regarding infinity. It's called Forbidden Knowledge, I believe. And if you get a chance, take a look at that documentary, and you'll appreciate this infinity a little bit more. And the other infinity that we're about to talk about. Now, the other infinity, the other way you can think about infinity is where the universe explodes, the laws of mathematics collapse, right? Where there is no question, there is no, there is no, there isn't any way we can relate to what happens with this other infinity. And this other infinity occurs when we divide by zero. And the other way you can think about infinity is if you end up dividing by zero, you get the empty number set. And this symbol is the zero with the line across, just basically means undefined. And instead of this, you could use the infinity symbol here, right? So we can't divide by zero. If we divide by zero, what happens is the laws of mathematics collapse. Basically, the universe explodes as we know it, because mathematics being the language that we're able to quantify the world, quantify the universe, what we're able to interact with, that language collapses. So we have no idea what happens when we divide by zero. It gets an empty number set, right? The question becomes irrelevant. It doesn't make sense. It's something that we cannot comprehend. And if we want to take a look at this, let's go take a look at this concept. This as well, infinity is how it can just continuously grow and what it means when there is nothing, there is no question, there is no interaction when we divide by zero. And what we're going to do is basically go and graph this function f of x is equal to one over x, right? And what we're going to find out is when f of zero, if we set x equal to zero, then f of zero is the empty number set. And this is something that we've got our hands a little bit dirty on in series three as well, right? Series three A and B or maybe it was just series 3B, right? But let's go take a look at a function, what it looks like visually in a Cartesian coordinate system and how we can sort of relate to these two things. Hi. Hi. How are you? Hi. Good. What is your job? I make math videos. I teach math. So I'm doing mathematics. What is your education level? Mathematics. University? Yes, yes. Are you single? No, I'm with someone. Yeah, I have a partner. Sorry.