The Rest Is Science - (Finite) Numbers So Large They'd Destroy You

Episode Date: February 10, 2026

It starts as a friendly challenge: who can name the biggest number? The only rule? Infinity doesn’t count. What follows is a journey through the biggest finite numbers ever imagined. From Archimedes’ grains of sand to Graham’s Number, a sequence so vast it stretches the limits of human comprehension, Professor Hannah Fry and Michael Stevens tumble through this strange landscape of scale, tracing how mathematicians have pushed counting to its absolute edge. But beyond vast calculations, perhaps this is less about numbers and more about us. Why do humans push at the limits of finitude at all? How do we represent the biggest numbers in existence? Why can’t our brains feel the difference between a million, a billion, and a trillion? And when do big numbers affect our ability to empathise with others? ------------------- For more information about Cancer Research UK, their research, breakthroughs and how you can support them, visit ⁠⁠https://cancerresearchuk.org/restisscience⁠⁠ Cancer Research UK is a registered charity in England and Wales (1089464), Scotland (SC041666), the Isle of Man (1103) and Jersey (247). A company limited by guarantee. Registered company in England and Wales (4325234) and the Isle of Man (5713F). Registered address: 2 Redman Place, London, E20 1JQ. ------------------- Find The Rest Is Science all over the internet by ⁠⁠clicking here.⁠⁠ ------------------- Video Producer: Adam Thornton + Oli Oakley Video & Social: Bex Tyrrell Assistant Producer: Imee Marriott Senior Producer: Lauren Armstrong-Carter Head Of Digital: Samuel Oakley Exec Producer: Neil Fearn Learn more about your ad choices. Visit podcastchoices.com/adchoices

Hello, welcome to The Rest of Science. I'm Hannah Frye. And I am Michael Stevens. Today, we're going to play a game, Hannah and I. Who can name the biggest number? Infinity. Infinity not allowed. Okay. Finite numbers only. Finite numbers only. Only a number that if you had enough time you could count to and be done and then move on to something else. All right. Well, I mean, that seems like quite simple rules.

No infinity. No, infinity is too easy. There are different sizes of infinity, and we will cover them soon. But what is almost more terrifying to me, to be honest, are just large finite numbers, numbers that you could count to if you lived forever. You'd reach the end. But yet, their magnitude is beyond incomprehensible. Hey, but first, let's address the jellyfish in the room. Okay, go on.

Is it a jellyfish on your plaster? Is that your daughter's plaster? Yeah, it's the only band-aids I have. Like, I'm a grown man. I actually, I shouldn't say that. I don't know anything about grown men. I know about myself. And I don't use Band-Aids very often.

But yesterday, I walked into a tree branch. And that sounds fake, but the truth is that I was just like walking across a parking lot. I thought, oh, I'll stay on the crosswalk. And I turned and went right into this low tree branch. And luckily, there's no welter bump, but it scraped the skin. So I actually just went home. I'm like, I can't walk into the store with a bloody head wound. Like, that would be lower.

If it was a pharmacy, you're probably all right. But yeah. You better work not a pharmacy. It was a hardware store. It would look like, sir, I think before you buy anything else here, you need to maybe get a hard hat or something. We need a little chat with you about help and safety. So I wore a hat yesterday because I was embarrassed. but I just, especially on the show,

I want to be a good role model for the baldies out there. You know, I'm not ashamed, you know, flaunt what your mama gave you. That's what I do. And my mom gave me this. Baldness comes from the maternal inheritance. How old were you when it went? I'd say about 17.

Oh, that is young. That's when people were like, wow, your hair line is really high. And I'm like, yeah. And then by college it was here. and now it doesn't exist. It just, it never, it's just empty. But I've got the side. I've got the sides.

Yeah. They keep me a little bit warm. Just where you need it. Okay, you ready for our game? Yeah, enough about my head. Okay, let's talk about numbers. Can I start? Go on then.

This episode is brought to you by Cancer Research UK. So when most people think of naked mole rats, their unusual relationship to cancer probably isn't the first thing that comes to mind. But maybe it should be because it is incredibly rare for them to develop cancer, which could be partly down to their unique immune system, or it might be the way that their cells respond to damage. So scientists are studying their biology for its cancer-fighting secrets. It's a reminder that discoveries can sometimes come from places you don't expect. Cancer Research UK is the world's largest charitable funder of cancer research. Thousands of scientists of doctors and nurses

work across more than 20 countries to help turn discoveries in the lab into new tests, new treatments and new innovations. And the impact is clear. Over the past 50 years, the charity's pioneering work has helped double cancer survival in the UK, meaning more people living longer, better lives free from the fear of cancer. For more information about Cancer Research UK, their research, their breakthroughs, and how you can support them, visit Cancer Researchuk. forward slash rest is science.

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Remember, the game is to name the largest finite number. I would say that finite is a number that you can count to given enough time. Yeah, one that eventually stops. It eventually stops. And that excludes any kind of infinity. Because infinity isn't some number that you reach. It is the act of never stopping. I mean, there's a bit of a debate about whether infinity is a number at all, rather just a concept or a collection of concepts.

But this is not an episode on infinity. We're going to do that later in glorious, weird detail. Right. Let's start with the biggest number. The biggest number you can think of. The biggest number, I think we should start with eight. It's pretty big in some ways. Sure.

And let me justify why I'm starting at eight. It's famously been found that the most chunks of information a person can store in their short-term memory is seven. Really? Yeah, it's literally from like the most cited psychology paper ever. And it was a study of like how many words or things or like meaningful chunks can a person keep in their head right away short-term. You'd give someone a list of like a grocery list, banana, eggs, butter, blah, blah, blah, blah. seven they can do but eight it's like just across cultures

across not quite across ages we're mainly talking here about like younger adults seven was the max but wait was seven the max or seven was the average because I'm sorry I don't like to big myself up here but I think I could I think I could beat seven

but mostly because I'd be using like memory techniques thank you I take that back seven was the average okay okay all right so we're going an eight. But it's an average that doesn't have a whole lot of skew. Like, it's not like there's long tails in either direction. It's kind of like everyone's pretty close. There aren't like a lot of people who can do 25. Yeah. And there aren't a lot of people who could only do two. Eight more than you can hold in your head, supposedly. I sort of want to test you. You want to test me? Yeah, I do. I do want to test you. Because I think that you, I've seen you memorize scripts before. And I think, I think,

I think you might be better at memorizing eight. All right. Oranges. Coffee. Squirrels. Mushrooms. Cards. Teaspoons.

Pins. Strawberries. How many of was that? I don't know. I'll just say them back. We've got coffee, oranges and teaspoons. squirrels and mushrooms and pens and cards.

That wasn't pens. Was it pins? Pins, you're right. Sorry, you're right, you're right. Just that's an accent thing. I did a bit of a bit of a like painting a picture. You know, I put the teaspoon in the coffee cup. I had the squirrel live in a mushroom house.

Nice. So I was cheating using some ancient techniques. Okay, here we go. Come. ketchup? Justice, green, tomorrow, tetrahedron, bicycle, haircut, irony, third. Okay, all right. Justice, ketchup, green, irony, tomorrow, third, tetrahedronedron, missing one.

You're missing two. Oh, damn. How many did I get? You got seven. You got exactly seven. There's the list. You missed bicycle and haircut, which we've also demonstrated two other really famous psychological phenomena, which are that people tend to remember the first and last parts of lists, but not the middle. Wow.

There you go. There you go. Okay. Talking of lists of words, here's a number for you. 180,000. That's apparently the number of words in the English language. 180,000 words in our language. That's a lot more than eight.

Look, we're getting there. We're getting bigger and bigger as we go. How many words do you think you know? Not 180,000, that's for sure. Yeah, no, same here. I wonder if there's a test that can be done. I'm sure it wouldn't be like 180 words are shown and you say you define it or not.

I think it'd be like, we'll test you on like a thousand and from there we can extrapolate how much of the language you know. I would love to know that. I want to know how many the average adult knows. I'm going to guess about like 80,000. Wow. Okay. The average native English speaker actually knows between 20 and 35,000 words.

20 and 35,000. And you'll only need about 10,000 to have conversations. So we're all doubly prepared. I mean, that doesn't seem like very many. It doesn't seem like very many. But why doesn't it? Because it is a lot. Like, it's, that's a lot.

20 to 35,000. Um, I guess it feels like compared to like the amounts of money we read about in the news, it doesn't sound like a big number. It doesn't feel like a big number. I would love, we should do this someday. Maybe not like on a podcast, but we should just sit down and list every word we can think of. Could you, could you list 35,000 words in one sitting? Just like, oh, let's see, have I done, have I done yesterday?

Yeah, yeah, shoot. I mean, the cheats way would just be to start off with the word one and then go two and then go three. Oh, no, of course. Does that count, though? No, of course. Compound words is cheating. By constructing number names, you can go way past 35,000. How about 35,000 and one?

Exactly. Okay, bigger numbers, bigger numbers still. Got one, I got one. That's going to way, way beat your 180,000. And this is going off script, so you better be ready. Go on. I was just looking this up last night. One billion.

That's, that is, that, that's a, that's a big number. That's a big number. And here's what's special about one billion. Yeah. That, that kind of puts us up against another limit. One billion is about how many heartbeats anything gets in its life. Oh, that's a beautifully poetic idea. Because of course, if you're a teeny tiny mouse, your heart beats,

faster, but your life is shorter. That's right. And if you're a human, you know, heart rate does correlate with longevity. Really fast heart rate is not great. I mean, a really slow one isn't great either. But in general, we find that, yeah, faster heart rates are found in animals that don't live as long. Animals that live a long time, turtles, slow heart rate. And so when you do the math, it equals out and we all get around a billion.

plus or minus a billion, like chickens get about two billion. But today's chickens are quite engineered for our pleasure. A factor of two. I'm not, I don't care about a factor of two. If it's one billion, half one billion, I'm fine with that. That's still about a billion. It's within an order of magnitude. And it's kind of, yeah, almost too poetic.

Like we each get a billion, whether you're tall or short, a mouse or a whale. Here's your billion. Do what you want with it. Hmm. Have the best life possible. I like the idea that there's some sort of quota. I sort of think that about words sometimes. There is actually a set number of words that I will speak in my entire lifetime. And all I've got to do is work out the order of them. Yeah, that's right. You've got them all in a bag. You can build whatever you want with them. Yeah. Yeah. The one number that comes up a lot actually when you're talking about big numbers is, I don't know, like the number of stars in the galaxy. Right. which is actually sort of not that big.

It's about 100 billion, somewhere between 100 billion and 400 billion. Okay, I love that we keep getting bigger and bigger. This is like very fun. Okay, so 100 billion stars, that's 100 times more than I'm going to get to have heartbeats. But not as high as the number of trees on Earth, which is 3 trillion.

That's, I mean, that's a whole order of magnitude bigger. Isn't that cool? I've talked about that before in videos because it's just, it's so surprising. And it's also poetic because it's like, you know what? Outer space man, like grow up. We've got more trees here than our entire galaxy has. And like that makes me really proud to be an earthling.

Yeah. Three to four trillion trees, Hannah. The other one that comes up quite a lot is the number of grains of sand. That's something that people like to use as a big number. Oh, yeah. You know what? And I actually calculated some things about grains of sand. Like grains of sand comes up all the time when you're reading about big numbers or the history of mathematics because of Archimedes' little paper.

Did they call the papers back then? A treatise? What do you call a thing that's written 2,000 years ago that's eight pages long? A treatise. I think a treatise. Yeah. Yeah. Okay.

We're speaking, of course, about the sand reckoner. And I'm sure we're both pretty familiar with it. But for the audience out there, it's a cool story. Basically, it feels like back in Archimedes' time, which was like the third century BC, okay, the 300 to 200 BC area, there was this probably like an idiom that like you could not even name the number of grains of sand on Earth. Because in their numbering system, a myriad was the biggest, which is 10,000. There weren't names for numbers above 10,000.

So the number of grains of sand on the entire plant, come on. A mathematician could never even come up with a name or a symbol for that number that made sense and followed a system. And what Archimedes did in the Sand Reckoner was he said, I bet I can. In fact, I did. I can name you numbers and give you ways to reach them that surpass the number of grains of sand on Earth and in fact surpass the number of grains of sand that would fit in the universe. Because this is a thing, it's like there's the sort of separation of the number of things, number of actual objects, right? Because that obviously exists. It was more that like the way of naming them, the ability

of maths stopped. There was like not a finite number in the sense of objects, but there was a finite limit to what maths could do. Yes. That is such an important pivot point in mathematical history. The like, we can count things. But using math and language, we can go beyond what can be counted or what we can even imagine there being. Because the universe is not full of grains of sand. And yet, if it were, Archimedes calculated that it would contain about 10 to the 63 grains of sand. That's a one followed by 63 zeros. He did something quite clever, actually, to get there, because you had myriad, 10,000, did you say. And they would have myriads of myriads,

so like 10,000, 10,000s, as it were. Sure. But the way that he got there was he was sort of saying, okay, well, imagine you've got a myriad of myriads, and then you sort of put that in a box. And now you get a myriad of myriads of those boxes. So he was sort of kind of raising numbers to powers before that stuff had been existed.

Remember, zero wasn't even a thing at this point, right? I know. I mean, the Romans who came after were still using their silly numerals, right? The way that they counted stuff, they did not have this easy decimal, you know, positional system that we have at the moment. I know. And so I recommend that you go and read it. It's only, like I said, eight pages long. And it's fun because it does feel like an early viral YouTube educational video. you know, because he's like, okay, guys, like, I'm going to try to do this and, you know, you could read some other little things that have been written about it, but like, I'm going to guess

that the distant stars are as far away from the sun as the, I don't remember all the ratios, but he had to make a lot of assumptions about how big a grain of sand was and how many Greek stadiums could fit inside the universe. And he always tried to overestimate so he could be like, this is an upper bound. Like, the real number will be smaller, but that's fine because I'm trying to show you that I can think of some big numbers. Yeah. He also, I mean, the actual universe itself, this is before they even decided that the sun was the center of the solar system, let alone the universe, right? I know.

And that's what's also, I think, so important about being familiar with the sand reckoner. It's that Archimedes went ahead and assumed that the sun was the center of the solar system. So when you have this whole like, oh, we all thought that the Earth was the middle until recently, it's like, no, in 300 BC, in the 200 BCs, like some guy was like, well, obviously the sun's in the middle, we go around it. Anyway, 10 to the 63 is a really big number. That's how many grains of sand Archimedes calculated could fill the universe as he knew it. We know the same universe. We see the same distant stars. I mean, we can see actually further because of telescopes.

But the number of grains of sand I calculated, this will help us go even higher, that could actually fill the observable universe, is more like three or four times 10 to the 85. Oh, okay. Because the number of particles in the observable universe is 10 to the 80, which on the surface sounds like quite similar numbers. 10 to the 80, 10 to the 85 sound quite similar. But when you get to the number of particles, you've still got, what is it, 10 to the 5 to go, 100,000 to go. You need to do that 100,000 times over.

Yeah, yeah. And so I guess the number of particles is smaller because particles do not pack the universe. But the sand in our example does. And then, of course, because this is a very early version of a YouTube educational video, at the end of the sand reconner,

our committee says, if you enjoyed that content, please hit that like, and then subscribe, right? Yeah. Well, actually, he does, but then he finishes with box for box because this was old YouTube, you know.

This was a long time ago. And he was like, oh, and click here on this annotation to watch Leave Brittany Alone. No, but it's really fun. And it was not the largest number we found in ancient texts. There are Indian and Chinese texts that come up with names for even larger, numbers. There's a few different stories, but I think one of my favorites is the future Buddha. This is

Prince Siddhartha and he wanted to marry this really beautiful princess Gopha, but her father was like, I'm not sure about this guy. I'm not sure about this kid. It's still this pampered prince. He's never done a day's work in his life. Is he actually capable of doing anything? And so to win her hand, the challenge was set that he had to compete against other suitors in like all of the manly stuff. So archery, wrestling and arithmetic. That was the main role. And it came down to the showdown between him and this mathematician

who was called Arjuna. And Arjuna tries to stump the prince and he's like, okay, do you know any numbers beyond the Cotee? And a Cotee was 10 million, right? Saddamath doesn't just say yes. He basically on the spot, supposedly, this is how the story goes,

starts to construct this numerical system that is so incredibly complex that it makes everybody's head spin. He comes up, he starts counting essentially in multiples of 10. So he has the Koti, which is 10 million. Then he has the Ayuta, which is a billion. Then the Nyuta, which is 100 billion.

And it keeps going, keeps going, keeps going, until he comes up with the Talakshana, which is 10 to the 53. And he doesn't stop there. He then enters this second numbering system, goes through more tiers and more tiers. It's not that you're multiplying by numbers, you're adding additional zeros on in the top, right?

So you're kind of using an exponent, is what the mathematicians would say. And then eventually he gets to a number that is one followed by 421 zeros. This is known as Buddha's number, and it's so big that if you turned every single particle in the universe into another universe and counted all of the particles in those universes,

you would still be nowhere near this number. And I mean, in conclusion, he won the math battle, he got the girl. Deservidly. Deservant. That was winged me, I've got to be honest. A one followed by more than 400 zeros. We've gone past to Google. We had.

We as a species went past to Google long before the Greeks. Yeah, yeah. Long before Google.com, the search engine. Google, by the way, is a one followed by 100 zeros. Sort of like a nice, neat, cute little number. Quite small, actually, in comparison to what we're describing here. Speaking of nice round numbers, 10 to the 100, which is a one followed by 100 zeros is a Google, a one followed by 200 zeros is called a Gargookle.

Is it? Yeah. There's a whole field of naming big numbers called Googleology. And it's pretty fun. And if you're ever like trying to go to sleep or you can't sleep, just look up names of big numbers. And everyone's like, we need to agree on these so that they become official. Gascuilian, yeah, has not, has not, is that. It's one that, you know, you sort of say in joke, it's never, it hasn't yet been adopters as an official number, but I'm, I'm holding out hope for it.

Thing is, at this point, though, all of these stories are essentially people trying to come up with names for big numbers. And it's like, let's just make a name for it. But these numbers don't actually really relate to very much, apart from maybe these theoretical ideas of the number of grains of sand in the universe. There are very real objects and very real situations in which you do reach these unfathomably large numbers, right? Anytime that you're dealing with a combination of something.

I'm teeing you up here, Michael. You're teeing me up, yeah. What a perfect tee up for me to share one of my favorite little factoids. I talked about this in a video many years ago, and it's the scale of 52 factorial, written as 52 with an exclamation point after it, and that simply means mathematically

every number from 1 to 52, every integer, from 1 to 52 multiplied together. So 1 times 2 times 3 times 4 times 5, all the way up to 52, which is the number of cards and a deck of cards. And in probability theory, 52 factorial is also the number of ways you can arrange 52 cards uniquely,

where the arrangement means something like the top card is the ace of spades, the next one is the two of spades and so on, right? You could do that. You could also put the King of Hearts at the top and change nothing else, and that's a whole new order. How many of these unique orders are there? There are 52 factorial.

And 52 factorial, I think, is a great place for us to start talking about how inconceivable the sizes of these numbers are. Because you mentioned that the number of particles in the universe is a one followed by about 80 zeros. Well, 52 factorial is an 8 followed by 67 zeros. These visualizations of 52 factorial came from Scott Cheapel, and they scare me to think about.

All right, so set a timer for 52 factorial seconds, and do this at the equator, standing on the equator of Earth. Just stand there, start the timer, and do nothing. Let it go and wait a billion years. After a billion years have passed, take one step forward. Let's say you're traveling east. Fine. And also you can walk on water.

Anyway, wait another billion years. The clock is running this entire time. You wait another billion years and you take another step. Hold on. We're going here for one second represents one. one unique order that a deck of cards can be in. That's right. That's right.

Okay. We've already got to a billion years. Yeah. We've already passed. A billion years have to pass before you even do anything. You take one step around the equator every billion years. By the time you have walked all the way around the earth, take one drop of water out of the Pacific Ocean and set it aside.

And again, you wait a billion years to take one more step. Once you've gone all the way around the world again, you take one more drop, a single drop out of the Pacific Ocean, and you keep this up until the Pacific Ocean is empty. And at that point, you place a sheet of paper on the ground, and you refill the Pacific Ocean, and you keep waiting a billion years for each step. after you've gone all the way around again,

you know, you take one dropout, this whole process continues until the Pacific Ocean is empty again, and you put a second piece of paper on the ground. By the time the stack of paper reaches the sun, there will still be eight times 10 to the 67 seconds left. What? If you put all the paper away, you start the whole process again,

and you do this whole process of walking around the earth, one step every billion years, taking one drop out after each trip around the earth, refilling the ocean, putting a sheet of paper on the ground, repeat, repeat, repeat. Do that a thousand times, you will be one third of the way done. Two-thirds of the time on your timer will still be there. So hold on, hold on, no, no, no. You have to do a complete loop of the earth before you take one drop. That's right.

Every drop, it also has a complete loop of the earth. And then once you feel emptied all the oceans, then you get one sheet of paper. That's right. And you start all over. Wow. Two sheets of paper, three sheets, four sheets. Once it reaches the sun, you are still a thousand. You have to do that a thousand more times before you're even a third of the way through 52 factorial seconds.

So, I mean, the conclusion of that then is that if you shuffle a deck of cards, you can pretty much guarantee that no other human who has ever existed or ever will exist has effectively landed on that same one second as you, right? Has got that exact same configuration as you have because there are so many. Isn't that weird? Like, a deck of cards that's been properly shuffled has never been in the same order as any other shuffled deck of cards. If you want to feel unique, go shuffle a deck of cards.

You've just created something that has never existed, in order that has never been seen and will never be seen again. I like that so much. Probably. I like that so much. Those analogies, those ways to understand how big these numbers are. I mean, you sort of have to turn it into time, really, don't you, to be able to get a grasp of it.

But I think Buddha himself or Sadatah, who was coming over all these big numbers I was talking about a moment ago, he had an example of this about how you have a bird with a silk scarf. and once every number of years, every hundred years, you would, the bird would fly past a mountain with a silk scarf. And eventually, eventually, eventually, the whole mountain will be worn away by the... Oh, worn away by just the scarf touching it.

Once every hundred years. I don't think it was an exact precise calculation of how big these numbers were, but it was a sort of, as you say, a visualization, a way to start imagining the, like, vast, of these numbers. The thing is, I mean, all of these numbers that we've described so far,

that 52 factorial is like, is phenomenal. It's not as big as some of the other ones numbers we've mentioned, though. I mean, not by long stretch. No, 52 factorial is just, what, 8 times 10 to the 67th. But thousands of years ago,

Indian mathematicians were talking about 10 to the 400. It's also not, we don't stop there. There are numbers that are even bigger than that, so big that you, I mean, they're quite literally inconceivable. Quite literally, you are not capable of even talking about them in terms of the number of zeros because they are just way too big. I think the most famous example of that is Graham's number. Now, okay, Graham's number is a little bit difficult to explain where it comes from, but I'm going to give it a go, okay? It's a number that arises from a mathematical theory called Ramsey theory. And essentially, if you imagine that you

You've got, it's all about cubes. It's all about joining together the corners of cubes. That's what it all comes from. Like connecting them with lines. Connecting them with lines, exactly. So, okay, let's imagine that you've got a square, just a flat square. I mean, a cube in two dimensions. You've got the lines around the outside, but you can also have diagonal lines

which are connecting up the diagonal corners. Okay, you could color all those in, right? You could make some of them red. You could make some of them blue. you can colour them whatever way you like. That's all very nice and simple. Now, if you include an additional dimension, if you go up to three dimensions,

so you have a normal cube, you can imagine now that square with the diagonal cross on it appears on every face of your three-dimensional cube, but you also have additional crossings on the inside where you're connecting up the opposite and diagonal corners from within the cube. Okay, you could colour in all of the three. those, blue and red, how are you wanted? Now, if you're a mathematician, why stop there? Why stop at

three dimensions? You can describe what four dimensional cubes look like. It's just, you know, the coordinate system, you just add an extra zero on the end. You could do five dimensions, you can do six dimensions, you can do as many dimensions as you like. You can start talking hypothetically about, I mean, enormous numbers of dimensions, but ultimately the idea is the same. It's a cube, and you're talking about joining up all of the corners. Now, there was this question in Ramsey theory, which was going back to that square, that original square where you have a cross in the middle of the square and all the corners are connected. This question of Ramsey theory, I'm really simplifying here slightly, a lot, actually. I'm really glad you are, by the way, because I've read the Wikipedia page for Graham's number and it does not simplify.

It just jumps right into, hey, here's a bunch of words and a cube and a square. and like you get it. You get it. And on you go. I mean, we're getting to the point now. This is the field of combinatorics, by the way. And there are going to be mathematicians listening to this who know way more commentatorics than me

and who are, I'm sure, going to write in angrily about the way that I'm absolutely butchering this description of Graham's number. But I'm doing my best. Okay, so go with me. Okay, so here is the question. If you colour in all of those links, blue and red and do whatever,

is there a point at which you cannot find one of those original squares, two-dimensional squares, where all of those links are the same colour. In 3D, there's a way to colour it in that you can avoid it. The question is, what dimension do you have to go to until it becomes absolutely inevitable that you will find these slices through your cubes, your hypercubes, where all of the nodes are connected and they're all the same colour? That's essentially the question. It sounds completely theoretical and it absolutely is. is pure mathematicians really enjoy coming up with these challenges for themselves and then spending their entire lifetimes trying to solve them, right?

Yeah, I was going to say, no one was actually like, please help. I have a higher dimensional cube I need to decorate with blue and red. What are those things? Garlands? Garlands, exactly. Yeah, no one was saying that. I think this is a thing, actually. I think that there is a misconception that what mathematicians do all day is just count really big numbers. and I mean that is what we're doing in this episode but actually what I've had to do all day

is come up with crazy questions for themselves, puzzles about mini-dimensional cubes and the colouring of edges on it. Anyway, okay, so here was the challenge, right? Is like, what's the number of dimensions at which point you cannot avoid this? You cannot avoid finding a slice where all of the links are the same colour.

And Graham came up with an upper bound. He said, okay, well, I know it's more than six, and I know that it's less than this number, which I'm going to call Graham's number. Now, Graham's number is called Ronald Graham is so gigantic that it is, you cannot explain it in terms of zeros anymore. It is, you have a whole new notation, a new way to describe how numbers relate to one another in order to even be able to describe what it is. here's the way that this extra notation works. So if you have 3 plus 3 plus 3, that's the same as 3 times 3, right? If you had 3 times 3 times 3, that's the same as 3 to the power of 3,

which you could also write as 3 up 3. Because it's sort of like you write the 3 up. Oh, okay, okay. But when you have this up arrow, you can go a bit further because you could say 3 up up 3, which is 3. to the power of three to the power of three. Oh, yes. Okay.

Up arrow notation. It's like another operation after exponentiation. Exactly. Now, the thing is, is that these get very big, very, very quickly. So three up three is 27, three to the power three. But three up up three is $7.6 trillion. Whoa. Just one arrow brings us into the trillion.

An additional arrow, exactly. I mean, it's crazy. So when you get to three up, up, up three, you have got three to the power of 7.6 trillion, which is already a ridiculous, massive, crazy number, okay? That's three times itself seven trillion times. Yeah, yeah, exactly. Seven point six trillion times, exactly. That is already a giant number, right?

Three to the power of 7.6 trillion is, I mean, it destroys 52 factors. It makes your crazy number look like a speck in the ocean, right? I mean, not even that. Dwarfs it way more than that. So it's sort of like there's no comparison. Yeah, if you compare them, 52 factorial is pretty close to zero compared to where we already are with just what, three up, up. Three up, up up, yeah, exactly. Now, the way that you make Graham's number is you say, okay, we're just going to call the new number.

We're just going to call it G1. just this new number, and that is three, up, up, up, up, up, up, up, up, up, up, up, up, up, up, up, G1 ups, three. Okay, so three and, and four ups, and then three. Okay, it's just, it's, it's, it's, it's so ridiculous. That was G2. G1 ups. G1 ups, right? That sounds like an amazing nickname, by the way. G1 ups. Oh, that's G1 Ups. How you doing, man? He's a big dude. But we're still out at Graham's number. Oh, no, we're nowhere near.

We haven't even started. So G2 is three to the G1 ups three. G3 is three G2 ups, three. And you carry on going over and over again until you get to G64. G64, which is three, 664. G63 ups. Yeah. Which itself was G62 ups, which itself was C61 ups, which itself was da-da-da-da-da-da-da-da-da-da.

And remember, three-ups absolutely dwarfs your 52 factorial number. I heard something like the number, Graham's number is so large if you actually could imagine it. Just imagine it. Your brain would become a black hole. That's not, I mean, that's not theoretical. That's, I mean, people have literally done the calculations to this in the sense that there's a limit to the amount of information which you can measure that your brain can hold. And if you do that, the density of information is so big that you exceed the sports child radius of your own head.

So yeah, if you could, if you can imagine this number, your head turns into a black hole. However, two things I will say. We know, we know the solution to this problem is between six and Graham's number. in the year 2000 or so, someone actually worked out that it's between 11 and Graham's number now. So we're getting closer. We're narrowing it down, right?

Sorry, 6, 7, 8, 9 and 10. You're out of the running. But I love that this isn't just a fun game or a story. Graham's number had a purpose, which is that it was an upper bound on a mathematical problem. It's not just, wow, here's this big number. I hope I win the girl. It was, hey, I'm doing math,

and I've found an unhelpfully large boundary for the answer. But here is the answer. I tell you what we do know about Graham's number, though. It ends in a seven. I read that, and I find that really impressive that we can find sequences within it. So we know the last few digits of it. It's not like we know how it starts, but not how it ends.

We can tell you it ends in a seven. Yeah. I mean, this is the thing it's a proper, like it's a proper, number that exists, it's just completely beyond our comprehension. And yet it is still finite. If you had enough time, you could count to it. And then you would be done and you'd have to find something else to do. But what we're going to do after the break is we're going to move on.

Because for a while, Graham's number was thought to be the largest number ever imagined. The largest finite number you could count to, you know, could count to. But after the break, we're going to look at two mathematicians locked in a battle to find even bigger numbers. This episode is brought to you by Cancer Research UK. We often think of beating cancer as treatment, but imagine stopping it before it begins. After years of work, Cancer Research UK scientists are launching a clinical trial of lung Vax, the first vaccine designed to prevent lung cancer. It builds on TracerX, the world's largest cancer evolution study, which tracked lung cancer cells over many years to uncover the disease's earliest warning signs.

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sci-fi vision calls it the best show streaming right now we're running out of time and we still don't know the rules don't miss what the movie blog calls something you need to watch saving those children is how we all go home from binge all episodes exclusively on paramount plus welcome back hopefully you are suitably refreshed by that ad break after the mind melting number weirdness of the first half thing is there is this idea of like naming

larger and larger numbers. I mean, there's something quite delightful in it, isn't there, Michael? Yeah, there is. I mean, it's a battle. You know, it's a battle of the wits. But it's really trippy to think that we're reaching numbers that have no physical significance. Like, we still haven't left our solar system as a species. And yet mentally, we've left the universe.

We're talking about numbers that are larger than the number of combinations of particles that could fit in the observable universe. There is no reason, ostensibly, to worry about these numbers. And yet we can because our brains are like the most bizarre vessel ever. But don't you think that that's what's so delightful, so delightful about human curiosity is that even though there is no point, even though it just melts your brain completely to even try and conceive of them,

let alone actually successfully do so, all the same, we still. kind of want to. Maybe there's no point, but it's like asking, well, what's the point in an eagle living? You know, we can get into philosophical discussions of purpose and it's like, it's just the nature of the beast. It's the nature of the universe. And for us, that role is stuff like this. What if I loved you? What if I counted beyond the universe? That's just what we do. Yeah. What if this benchmark that you set or has been set by people before by by buddha and archimedes can be beaten that was the great idea uh between two mathematicians who had their own showdown this is at mit they wanted to go so

far beyond what graham's number had had i mean the bar essentially set by graham's number which until that point was the largest number that had appeared in a mathematical paper so this is the idea. This is MIT in 2007. There are these two mathematicians called Adam Elgar and Augustine Rayon. And they're like, okay, let's take each other on. Let's have the ultimate jewel. But rather than being weapons involved, let's just come up with the biggest numbers we possibly write down. Graham's number is a good threshold, but let's see if we can go further. So Adam Elgar, he's sort of the challenger in all of this. He comes up with an idea that It's actually kind of similar in some ways to the sort of the basis behind Graham's number.

He has this idea of creating the mathematicians call them trees, but essentially it's dots and lines that are connected with each other. And he comes up with a way of setting up a number that is the number of combinations of a different way that you can join dots and lines in different colours together. It's really impressive. Everyone finds him extremely. excellent and intelligent as a result of this. What I find impressive isn't just that, you know, a big number was described, but that it could be

shown that this number was larger than Graham's number. Like, how cool. Like, we're not just going there. We're kind of like making a map. Yeah. And yet, Rayo comes in and wins the competition. Rayo comes in and wins the competition. And he does it with this absolute genius move.

He's like, I'm not going to, I'm not going to play a lot. around with dots and lines. I'm not going to mess around with combinatorics. No, no, no, no. What I'm going to do is I'm going to say, all right, Graham's number, your number, Adam Lager, all of that. They are real numbers and they can be described using symbols.

And some of them need more symbols than others. Graham's number, for instance, needs actually quite a lot of symbols to properly describe. But think of all of them ups. Rayo is like, okay, if I say that there's like a category of all of the numbers that can be described by up to a Google of symbols, right? So like the number 453 needs three symbols, 453.

52 factorial also needs three symbols, 52, and an exclamation mark. Graham's number needs a lot more because you've got all of them ups. You've got a lot. Yeah, sure. There's a lot of G1, G2, G whatever. So Rayo says, okay, well, look, if you count the number of symbols that you need to describe this number, right? And let's say you've got like a category,

like all of the numbers that need less than a Google of symbols to describe them. That's all there. I'm going to say my number is the smallest number that cannot be described by a Google of symbols. So all of those numbers in there, I'm going to do that, plus one, basically. That's essentially one. way it did. Which is brilliant because you just, it's, it's just so impervious to any, any, any, but what if? Because look, fine, I can compress the number of symbols required to represent a number. I could say, you know what, Graham's number? Let's just represent it with, um, a really bold G. Now it only takes one symbol. And he's like, yeah, I know, but my number, Rayo's number is defined as the one that's, that's, that's, can't in your system. No matter how much you compress it, I'm always

beyond you. I mean, the thing is, we could come up with our own number. We could come up with a Fry Stevens number, which is the smallest number that's larger than any number that can be named in expression of the language with a Google Plex symbol or less. I mean, you can, you can't out Rayo, Rayo. Yeah, could you say the smallest number that cannot be described in a system using Rayo's number of symbols? You might run into a paradox. I think there might be some secular logic going on in a minute. Yeah. But anyway, I mean, this is all fun in games, right?

This is all fun and games. It is really fun in games. But yet there's something so important in this because we're trying to describe and kind of give some scale to these large numbers. But there are much, much smaller numbers that we as a society and as a species need help understanding.

Even the difference between a million and a billion is something that the more we talk about big numbers in our real lives that really do count things like dollars, like people. We just become numb to them. And it's a struggle, but yet it's so important that we help people picture how large these quantities are. The difference in a million and a billion is one that I always think of because, I mean, they sort of sound so similar. They're just different by one letter in a way.

And again, if you turn it into time, I think suddenly it becomes a bit more natural. The difference between a million seconds, a million seconds is 11 days. A billion seconds is 31 years. Yeah. I mean, it's like they're gigantically different. There was this study back in 2013 where people were investigating exactly this idea. Can people really conceive of the difference of these numbers? This is by David Landy.

And they had a number line. number line had a thousand on it and it had a billion and they asked people to place one million on it and about 40% of people placed one million halfway halfway between a thousand and a million and in reality a million was barely a pixel above 1,000 I know I know you need a thousand millions to get a billion yeah and of course people put it in the middle I would have thought that they would because it's in the middle you go thousand million billion that's it that's how the naming works and yet they're so far apart yeah a million seconds is 11 days a billion seconds is 31 years a trillion seconds is 31,000 years. Is it? Which is just it's just times a thousand

because a trillion isn't like the next number after a billion. It's the next name for a number after a thousand billion. And so I think that politically and journalistically, we should start pushing to get people to use only one kind of number. Like just let's only talk in billions. So don't say the national debt is a trillion and we're cutting two million in funding because those both sound like they are close to each other. They've got alien in the names.

Trillions and millions are the difference between a thousand. thousand billion and 0.002 billion. If you saw those together, you'd go, that doesn't make a difference. I saw a really amazing visualization about how rich Elon Musk is. Yeah. And I think it's, I mean, it goes back to that number line, right? Like to get that answer correct, what you needed to do was to cut up that line into a thousand pieces and just choose one of them. That would be where a million is. This idea that, you know, Musk is worth. I mean, by some estimates close to a trillion, if not over. It's so gigantic.

It's not just like a bit bigger. It's absolutely inconceivable. I mean, quite literally inconceivable, the difference between these numbers. But it also, I think this ends up really mattering when it comes to charities and not-for-profits, trying to get support for people. This is like something that's been really noted.

I think that we inevitably honing on stories about individuals way more than we do about large numbers. You know, the statistic doesn't really draw empathy from us in quite the same way. There was one really interesting

study by, this is by Paul Slovak who wanted to try and understand, like, in what ways do we stop caring? And he presented participants with these various humanitarian cases and he would have a picture of a

and ask about the amount of donations people wanted to do. And he found that if you show that exact same picture, but underneath it, say, there are a million people like this who are also suffering, donations went down, not up. Which is really extraordinary. Like, this is counterintuitive to us, which on the one hand is what makes the fact that these mathematicians are doing this for fun,

all the more impressive, I think, or all the more, I don't know, it makes me love the strangeness and curiousness of humanity more. But at the same time, I think it really demonstrates how we are not wired for this stuff. Like, this is not innate to us.

That's right. Yes, we can be proud that we're capable of describing numbers this large, but yet we aren't really wired to feel it. I once worked with a charity and they said something somewhat similar, they said, the thing that helps donations the most isn't statistics or numbers, and it's also not any kind of extreme case.

It's not as effective to show the story of a guy who overcame some hardship and just climbed Kilimanjaro. It's more effective to say, this guy overcame the hardship because of your donations, and because of that, he was able to take his daughter to the park. that that means so much more to people than, oh, he climbed a mountain. I haven't climbed a mountain. So why do I care that this other guy did? But to not be able to make dinner for your son, like that matters so much more than any number we can come up with.

There's Hans Rosling, who was just this absolutely extraordinary statistician and global health advocate. his daughter, Anna Rosling, who wrote the book Factfulness, she also is, I think, really very aware of this tension. That on the one hand, you need the statistics in order to make the bigger argument, to make the sort of the data-driven logical case, but that ultimately, without the emotional side of it, you know, when people don't connect with big numbers, we just don't.

So what she has is something, that she calls the bird's eye view and the worms eye view. So there's one of her websites, this amazing thing where you see all of the maps, you see the largest statistics, but at any moment you can zoom in and find individual stories of the people who are actually affected. And I think that that's the most impactful way that I've ever seen these two things tied together, knowing that our brains really don't work in the same way as those mathematicians' brains do, not when it comes to having empathy

towards other people. So today we've reached the largest described finite number. But then we kind of like found something even bitter. And that's what I love about this show. Yeah, that's what I love about this show too. Also, the fact that we didn't just decide to describe the largest finite numbers to you

by just reading off all these zeros. Imagine if we had, if we had just been like, okay, eight, how about 100, how about a billion? How about a trillion? We just kept doing that with no, explanation. Hey look, we haven't launched our members only podcast yet. Maybe that could be the first episode. That could be a members only episode. Michael and Hannah try to beat each other with larger and

larger finite numbers until one of them falls asleep. You can read out your square root of four book for us. Ooh, yes, the square root of four to a million decimal points. All right. Well, thank you so much for watching and listening to us on the rest of science. Make sure you're following wherever you get your podcast. Be sure to like and subscribe. on YouTube. Smash that like button, Archimedes. Smash it, hit the bell, and sign up for our newsletter

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See you next time. Next time.