 In the previous two videos, what we did was take a look at the two different interpretations, two different ways you can look at infinity, right? One being the countable infinity where, you know, you sit down and you count, you know, one, two, three, four, five, six, and you continue this forever and ever and ever, right, until the end of time. The other way you can look at infinity is basically where you got, when for a certain input, when it comes to the unknown, right, where the universe explodes, for a certain X value, we have no Y. We don't, we don't know what f of X is, right? We can't comprehend what that answer is. What we're going to do right now is take a look at what infinity, you know, try to visualize infinity in a different way, right? And this is something that I've used with a lot of my students and for me as well, it was a, it was a wow moment for me when I looked at this thing, when I can't remember who initially introduced this, this to me, someone in some teacher that I had many, many moons ago, right? Or something that I read in the book, I have no idea. But it was, it was one of those things that made me go, wow, wait a second, what? You know, it blew me away. And that's what infinity is, right? It should, it should really blow you away. So for this, just to visualize infinity, just imagine us traveling from point A to B but only going halfway every time. So we're going to travel from point A to point B, but we're only going to go halfway every time. Every step we take is only going to take us half the distance, okay? So from here, half the distance comes out to about here, right? So for our first step, we end up being here. Our second step is going to take us halfway between here and here, right? Which is around here. So we're halfway between this guy and that guy. And again, we're only going to go halfway between here and there. And we're going to continue this forever. We're going to go halfway every time, right? So we're just going to go halfway, halfway, halfway, halfway, halfway, halfway, halfway. Now the kicker with this, with this problem is, with this concept is that we'll never reach B. Theoretically, we'll never reach B, right? You can get very, very close if you only go halfway between the two points every time, but you'll never, ever hit B, right? We always say, you know, okay, we ended up at B, right? Because our feet are only a certain, certain, you know, our feet have a, have a width to them, right? So as soon as we get like an inch closer, the next one we're basically touching B. But in reality, if we only go halfway every time, we'll never hit B, right? So you're going to get close enough that you're maybe a molecule away from B and you're going to cut that in half, right? Now you've got half a molecule and you're going to continuously cut that. You're going to hit the atom length, right? As soon as you had, you hit, you're an atom away from B. Now you can only travel half the distance. So you're going to cut an atom in half. And then after a few more cuts, I have no idea how many cuts you got to do because most of the atom is empty space, right? And you're going to continuously do that and you're going to go to subatomic a particle size, but you're never ever going to get to B, right? And when I thought about this, initially when I was introducing, I was like, well, it doesn't make sense. You're going to get to B anyway, but you're only going halfway every time. Now later on, a few years later, the one, the next level of this that really blew me away. And again, this was a, you know, it wasn't an epiphany. It was just like a wow moment. Like, what does that mean if we never reach B? That held until I got to a concept called Planck's length, Planck's distance. And what Planck's distance tells us is, and this concept in physics, Planck basically introduced the quantum mechanics to us, right? He introduced the concept of quanta, which is basically telling us that information comes to us in packets and the smallest packet of information that could come to us is a ridiculously small number, right? And basically what that meant, what his theory says is that you can never go smaller than a Planck length. So just imagine doing this thing for ever and ever and ever and ever, right? You're going to get to a place where you're a Planck length away from point B. And according to Planck, according to this theory is we can't go any smaller than a Planck length. Now what happens, right? And that sort of introduces the other concept of infinity, which is we don't know. It's an unknown to us. We have no idea what that means, what it means for us to go past that limit of mathematics that we have, right? Anyway, this is one way of looking at infinity that really blew me away. It really, it just, it took me to another level in mathematics where I was trying to understand what all this infinity and zero and nothing and everything and end of time and all this stuff meant. This one gave me a really good appreciation for it, right? And it encompasses both infinities, right? To a certain degree. The infinity that goes on forever, cutting in half, half the distance every time. And the infinity when you hit the Planck length, the Planck, a Planck distance, a quanta away from B, you can't cut that in half. What does that mean? We don't know. Our ability to understand that is limited. We haven't gone that far yet. I hope you like this. We'll talk a lot more about infinity and zero in future videos. For now, this should be a nice second level of teaser for what we did initially in, I guess, video number five in 2007. I'll see you guys in the next video. Bye for now.