The Original Biggest Numbers - Numberphile

So, on Numberphile, Brady, you've got a lot of videos about very, very big numbers. Graham's number, Goodstein sequence, tree three, subcubic graphs, Rayo's number. As well as being very big numbers, just about all of them, maybe with the exception of Graham's number, come out of mathematical logic. And the other thing is that they're all pretty recent discoveries, right? They're all dating from sort of the middle of the 20th century at the absolute earliest. So, a question I was thinking about was that, you know, if there was a Numberphile equivalent 100 years ago or 500 years ago, who were trying to catalog one of the really big numbers people have been thinking about, what would they come up with? Actually, there's a very clear answer where you find those biggest numbers. And the biggest numbers of the ancient world were in India. I think that is absolutely clear. And of all the very big numbers that got contemplated in India, the biggest come out of the tradition of the religion Jainism. So, Jainism is a Indian religion still practiced by millions of people today. But it's also a very ancient religion. It dates back till 2 and 1/2 thousand years BCE. And as part of the sort of mysticism of Jain tradition, they came up with some really, really big numbers. And I thought it might be fun to look at a few. We'll start with ones which represented long periods of time. So, they they put together processes which took a long time to finish and then called some number the length of time it takes to finish the process. So, I'll do an example. We're going to start with a thing called a palya upama, which stands for a pit year. So, it's a length of time measured according to a pit. So, what is the pit? The pit is a cubic pit and it's one yojana wide. You probably don't know what a yojana is. You might have forgotten. It's slightly more than 10 km. So, I'm going to just take it as 10 km. We'll round down a little bit, okay? So, we've got a cubic pit, 10 km by 10 km. Actually, a bit bigger, but we're rounding down a little bit. And then you fill it, the whole thing, with lamb's wool. And then, once every century, you remove a strand of lamb's wool. And the pit year is the length of time it takes you to empty the whole pit. Okay. — So, that's what the pit year is. So, I did a bit of sort of playing around. We can do a bit of a calculation just to give it some sort of idea how many strands of lamb's wool are going to be in there. So, Are we going to press them down and step on them? — Well, so I'm going to just I'm going to go for an underestimate, okay? You're right, at the bottom of a 10 km deep pit of wool, the pressure's going to be pretty high. And those strands down there are going to get really squished together. I'm actually going to assume, just for this calculation, that each strand of lamb's wool occupies a cubic millimeter. And that's got to be a big overstate overstate. It's probably an overstate anyway, but once you factor in the enormous pressure, it's a big overstate. But it's still it's enough to give it let us do a calculation, okay? So, how many strands of lamb's wool, if we assume each one is a cubic millimeter, just need to know how many cubic millimeters there are in a cube 10 km wide? The number of strands of lamb's wool is That's now my 10 km in meters, 10,000 m. Put another thousand on that, and that's now my 10 km in millimeters. We cube it because it's a cubic pit. So, that's the number of strands of lamb's wool. And then we'll multiply by 100 because we're removing one once every century. So, you do the calculation and this is 10 to the 23, 10 to the power 23 years. The pit year, the palya upama, is an absolute minion minimum 10 to the 23 years, okay? But that's just the start. Okay. And they but they weren't using this for any mathematical reason. It was just kind of like, "Oh, it's such a big job, it's going to take me a palya upama to do it. " Like it would be just like vernacular, like it's like a zillion years. Or they using it in some kind of mathematical way? They were they did do mathematics with some very big numbers, which we'll come on to. Um these periods of time um they might have used it in the vernacular, I'm not sure. But what they definitely did is they built the religious mythology out of this. So, um these periods of time were considered to be real periods of time. Um and if you wanted to date, you know, date the universe since the date of creation, this is the kind of unit you would need. In fact, you need much bigger units. So, we'll move on to the next one. The next one is the sagaro upama, which is the ocean year. And this has a nice simple definition. It's 100 million palya upamas. So, it's 100 million of the previous things, okay? So, that's going to be I mean, it's at least 10 to

the 31 years. In the mythology of Jainism, the universe runs on a cycle. And it's the cycle started round about a quadrillion of these ocean years before today, okay? So, the start of the cycle, and all of this is of course an underestimate, was around 10 to the 15, that's my quadrillion, of those ocean years before today. So, that's round about 10 to the 46, of course. That's their sort of big bang, for lack of a better Yeah, I think so. I mean, I think it was a sort of endlessly repeating cycle. So, I don't think they have actually a Well, some people think the big bang does that, too. Well, yeah. Well, indeed, indeed. Um All right. So, that's the start of the cycle. They did also think about periods of time beyond this. Um and in particular, they thought about periods of time which encompass more than an entire cycle sometimes. They had a unit of time called a purvanga, which is defined to be 84 * 100,000, and then everything there is measured in a unit, a sort of fundamental unit, called purvis, which is a number of days, 756 * 10 to the 11 days. This 100,000 is uh the word for that is a lakh. So, it's 84 lakh purvis. This is in days. But that's just the sort of first level. So, the next level is obtained by squaring. So, the next one is one purva, which is 84 Well, I'll just write that as 8,400,000. This time we square it. And then we're counting in purvis again. You can see how it's going to go. We're going to keep increasing that exponent. The next one's called a truti tanga, which is the same thing. And then this goes as far the top one of these, which is their biggest unit of time, as far as I'm aware anyway, is called one shirsha prahelika, which means the top riddle, which I think is a great name for a massive number, which is up to 28. So, we go 8,400,000 to the power 28 * 756 * 10 to the 11. What's that in years? What what sort of exponents we up to here, do you know? Round about 10 to the 206 years. To years, round about. I mean, bearing in mind the you know, the universe as we understand it to be at the moment is 13 billion years ago. So, this is this is, you know, — [snorts] — way, way beyond that. If we sort of extrapolate the current cosmological models, probably we're this period of time would take us past the point where all the supermassive black holes in all the galaxies have evaporated. So, that will be, you know, a very dark and empty universe by that point, if it still exists. Like it seems very arbitrary. So, this original number, the one purvanga, so the 84 lakh purvis, was said to be the lifespan of the original founder of Jainism. And that's like a really long time, obviously. — It's over a quintillion years. Right. — Yes. So, a lot of the mythology of Jainism happens over these sort of time scales, which really no one, very few other people, think about, yeah. So, anything else? — Yes, there is. We haven't got to the biggest numbers yet. Oh, we're going bigger? We're going bigger. All right. You got more paper? I think more paper, yeah. — So, as well as contemplating very, very long time scales, the ancient Jains also developed a theory of very big numbers just for their own sake, not really representing anything particular, just really as a an investigation into immense numbers. And they they classified them in different ways. And in particular, at the upper end, they had the concept of an unenumerable number. And the idea of an unenumerable number is that it's a finite number, but it is so big that for practical purposes, it's basically infinite. I think that's the general idea. And maybe it's worth saying that in modern mathematics, we don't really have that idea. So, there's a description of the first unenumerable number, which is comes out of a book written around 1,000 CE by someone called Nemichandra, and the book is called Trilokasara, which I understand translates to the essence of three worlds. And in this book, he gives this fantastic description of a really big number, the first unenumerable number. Okay. And it takes as its starting point the geo the sort of mystic geography of the plane on which we all live, okay? And so, in the middle of this plane is an island called This is Jambu Island.

That's where we live. It's very big. Its width in the traditional measure of Yojanas is 100,000. Translating into miles, that's over half a million miles wide. So we got this big island around half a million miles wide. And then outside of the island we've got an ocean going all the way around, right? That's called the salt ocean. And then outside that ocean we've got another sort of continental island, sort of annular island. That's fire flame bush island. And then outside that, you can see where it's going, outside that we've got another ocean. And then island and so on. And this carries on I mean, there's different accounts, but for this thought experiment we this carries on indefinitely, okay? But it's not just that um we've got these islands and oceans and islands and oceans. Their size is very important. The first island is round about half a million miles wide. And then the first ocean is double that. So I've not drawn this to scale. The first ocean is double that. So it's round about a million miles wide. And then the next island is double that. So it's round about 2 million miles wide. And then the next one's double that and so on. So each one is double the width. So exponential growth just baked into the geography of the place we're working in. Right. So that's the background. That's the setting for this thought experiment. So then the first thing we do, we dig a cylindrical pit under the first island, Jambu Island. And what we do is fill that pit with mustard seeds. So this whole thing is going to be a quantity of mustard seeds, okay? So the depth of the pit is 1,000 Yojanas, which is 5,000 miles or something. So a 5,000 mile deep pit under the entire island. The rule is that the height of the mountain needs to be 1/11 of the circumference of the circle. Of course. Obviously, right? I brought some mustard seeds. Would you like to just see how big they are? Yeah, go on then. Just in case you've never seen a mustard seed. — Yeah. So that's Those are mustard seeds. Yeah, they're pretty small. I mean, I don't know if it's exactly the same kind of mustard seeds they were having in my bed. They're pretty small things. I mean, it's roughly speaking the same size of a grain of sand. I've got to get rid of these mustard seeds. Okay. — Okay. Uh oh, they've gone all over the table. That was bound to happen. Oh, yeah. Well, that happened. Okay. Uh we're surrounded by mustard seeds, but not as many as um as they were about to be appearing in this pit. At the moment, this is about 5,000 miles deep. And then this thing is 1/11 of the circumference of the pit. So that's It's really tall. Yeah. Like it's I mean, it's thousands of miles tall. — So yeah, very tall. Very yeah, okay. So it's very very tall. Already that mountain of mustard seeds is big enough that you can fit planet Earth in it like loads of times. I mean, it's already a massive number, okay? But we're just getting started. So And now this is the clever bit. Because what you do now is you take this collection of mustard seeds, right? And you put the first one on the island. And the second one in the first ocean. And the third one on the next island. And the next ocean. island. And so on. And you keep doing that until you've completely exhausted the whole mountain. Right? And that's taken you to some Eventually you've got to some other island or ocean. Depending on if it was an even or odd number of — Exactly. That's yeah. Exactly. And then you do the same thing all over again. So you build you dig a pit the same depth about 5,000 miles deep under that whichever disc you've reached which by now will be very wide. Actually I did a sort of back of the envelope calculation and it's something like Okay, how wide is it? Something like 10 to the 40 light years wide. Right. Okay. Right. It's that sort of thing, okay? Which um you know, bearing in mind the the observable universe is um 10 to the 11 light years wide or something. — Right. It's enormously bigger. So you make a circular ditch under the under the continent you've reached. And you build a mountain of mustard seeds again. Uh so that gives us a new a new mountain. Yeah. A one which is 1/11 again as our uh 1/11 of the circumference, yeah. — Yeah, yeah. Of the circumference, which is 10 to the 40 light years. — Or the diameter years, yeah. Oh, yeah, circumference even, yeah. — Yeah, but I mean, when you're multiplying a number like that by uh pi, it doesn't make it doesn't make much difference. So um yeah, I mean, at this sort of scale, these sorts of numbers, it sort of stops mattering whether you're measuring in millimeters or light years just because the the numbers are bi- a bit so big. This is the second mountain of mustard seeds, which is you know so enormous that you know, the the

observable universe is just an invisible speck compared to it, okay? And then what do you think we do? We distribute those seeds out ring by ring by ring. We do. And then that takes you to another place where you build another mountain. Uh you keep doing it. How many times do you repeat the process? Uh the answer is the cube of the number of mustard seeds in the original mountain. And that and a back of the envelope calculation suggests that's around 10 to the 45 seeds in the first mountain. So we repeat that process the cube of 10 to the 45 times. — [snorts] — And if you back of the envelope tell big this final number is. — So the final number, yes. So uh maybe I should say I should credit the mathematician and historian Radha Charan Gupta who in 1992 did a sort of modern mathematical analysis of this situation, which is what I'm following here. And he So he did the uh the calculation. And I mean, this the number that we get after completing this, that is the first unenumerable number. Let me try and say it in Sanskrit. Uh so Jaghanya Parita Asankhyata, the first unenumerable number. I mean, it is so big that if you were to try to, you know, write it as a power of a tower of 10s or something you can't because the tower is just too tall. We can use Knuth arrow notation. And I could I'll just explain this in a moment. So let's say it's something like this number, 10 to the 10 to the 45 two Knuth arrows 10 to the 135. So this is an approximate value for the first unenumerable number. What that means is if we built a tower an exponential tower of X's, and I want my height of this tower to be 10 to the 135. So far taller than we could ever draw. And then each value of X is this number 10 to the 10 to the 45. That is about the scale of the first unenumerable number in Jain traditional mathematics. We've talked about some big numbers even in this room before, you know, I mean, we could you listed some of them at the start of the video. — Yeah. Is When we talk about tree threes, Graham's number, something like that. Is this in that ballpark? Is it still not really coming close? Is it — I think it's fair to say that all the ones I've listed at the start are much bigger. Um but it is big enough that it defeats our attempts to write it down using just traditional mathematical notation. So we need Knuth arrows or uh something. I mean, to consider numbers on this scale you have to develop the kind of machinery which can then take you to the places like Graham's number. So they were on you know, on the road towards that sort of territory. It's a testament to the size of those big Graham's numbers that you did this crazy thing that we would sort of almost laughing at how big it is. And then you at the end you said, "Oh, no, it's still not close to those ones. " Yeah, that's right. I mean, the these numbers are on another scale. But I think it's fair to say that this came thousands of years earlier, right? Thousands of years earlier. So it's taken us It actually took you know, there was a big hiatus in terms of the biggest numbers people had thought of. Um it was the ancient Jains with this number and others. They had others which are sort of around this. They went a little bit bigger than this, actually. This is the most fun one to talk about. It's almost like you needed things like Knuth notation and some of the new notation that mathematicians use now before you could start playing and inventing those bigger numbers. And they Yeah, they just didn't have access to that. Uh I mean, they started to So they did start talking about like repeating processes large numbers of times. And the sort of process is an abstract arithmetical operation. And that's sort of how you do it, right? I mean, that's what the Knuth arrow is. You talk about you've got some arithmetical uh operation and then you say, "Okay, I'm going to iterate that a large number of times. " I've seen an account of um ancient Jain writers who might have gone as far as three Knuth arrows. 10 three Knuth arrows 38. So that's sort of two levels beyond exponentiation, right? The two Knuth arrows is a level beyond exponentiation. That's two levels beyond. No one else thought about numbers anywhere near as big as this. So there was there's a real sort of hiatus from the ancient Jains who were doing this thousands of years ago. And the rest of the world only really caught up in the sort of second half of the 20th century. So they were the record holders for most of history. If you enjoy seeing Richard here on Numberphile and big numbers are, well, your thing, then you really need to check out Richard's new book called huge

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