(Finite) Numbers So Large They'd Destroy You

Hello, welcome to The Rest of Science. I'm Hannah Fry. 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. OK, go on. What's that? Is it a jellyfish on your plaster? Is that your daughter's plaster? the 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 welt or 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 if it was a pharmacy you're probably all right but um but yeah but it was 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 health 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 when that's when people were like, wow, your hairline 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. OK, you ready for our game? Yeah, enough about my head. OK, 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 cancerresearchuk.org forward slash rest is science. remember that 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 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 just a concept or a collection of concepts but we're but this is not an episode on infinity we're going to do we're going to do that later in in glorious weird detail right let's uh let's start with your 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 give someone a list of like a grocery list, banana, eggs, butter, 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? I'm sorry. I don't like to big myself up here, but I think I could beat seven. But mostly because I'm using like memory techniques. Thank you. I take that back. Seven was the average. OK, OK. All right. So we go in 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 eights 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 um i've seen you memorize scripts before and i think i think um i think you might be better at memorizing eight all right oranges coffee um squirrels uh mushrooms um cards uh teaspoons pins strawberries how many 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 pens? Pens. You're right. Sorry. You're right. 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. Yeah. I had the squirrel live in a mushroom house. Nice. So I was cheating using some ancient techniques. OK, here we go. ketchup justice green tomorrow tetrahedron bicycle haircut irony third okay all right um justice ketchup green irony tomorrow third tetrahedron i'm missing one you're missing two oh damn how many did i get 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 uh that people tend to remember the first and last parts of lists but not the middle wow there you go there 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. It is. 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. OK. The average native English speaker actually knows between 20 and 35,000 words, 20 and 35,000. thousand and you only need about ten thousand 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 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 perfect 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 uh yesterday uh yeah i did shoot uh 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 because compound words no 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 um that's gonna 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 uh 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 put 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 plus one billion, I'm fine with that. That's still about a billion. It's it's within an order of magnitude. And it's kind of 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. 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. that's right you've got them all in a bag you can build whatever you want with them yeah yeah um 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 we're getting we're i love that we keep getting bigger and bigger this is like very fun OK, 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 three trillion. 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 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 them papers back then? A treatise? What do you what do you call a thing that's written 2000 years ago that's eight pages long? I think it's a treatise. Yeah. Yeah. OK, 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 B.C., OK, the 300 to 200 B.C. area, there was this probably like an idiom that like you 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 planet, 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 the 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 um 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, as you say, and they would have myriads of myriads. So like 10,000, 10,000, as it were. But the way that he got there was he was sort of saying, OK, 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. 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, OK, guys, 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 the 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 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 BC, it's 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 as 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're you've still got what is it? One hundred 10 to the five to go. One hundred thousand to go. You need to do that one hundred thousand 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 Reckoner, our committee says, if you enjoy that content, please hit that like button and subscribe. Right. Yeah. 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 Britney 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 there's a few different stories but i think one of my favorites is um the future buddha this is prince siddhartha and he wanted to marry this really beautiful princess gopa but her father was like I'm not sure about this guy. I'm not sure about this kid. He'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 coty which and a coty was uh was 10 million right siddhartha doesn't just say yes he's he basically on the spot supposedly how this is how the story goes starts to construct this numerical system that is uh so incredibly complex that it makes everybody's head spin he comes up we start 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 Nayuta, 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 like enters this second numbering system, goes through more tiers and more tiers. He's sort of, it's not that you're multiplying by numbers, you're adding additional zeros on 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. Deservedly. Deservedly. a one followed by more than 400 zeros we've gone past a google we have we had gone we as a species went past a 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 Gar Google. Is it? Yeah. There's a whole field of naming big numbers called Googleology. And it's pretty fun. 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. Gasquillian. Yeah. Has not. Is that? is one that you know you sort of say in joke it's never it's it hasn't yet been adopted as an official number but i'm i'm holding out hope for it the 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 in 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 eight followed by 67 zeros. These visualizations of 52 factorial came from Scott Sheepiel, 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 unique order that a deck of cards can be in. That's right. That's right. Okay. We 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 take one drop out, 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 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've filled emptied all the oceans then you get one sheet of paper that's right and 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 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, an order that has never been seen and will never be seen again. I like that so much. 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 Siddhartha, who was coming up with 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, the bird would fly past a mountain with a silk scarf. And eventually, eventually, eventually, the whole mountain will be worn away. 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 vastness of these numbers. Thing is, I mean, all of these numbers that we've described so far, that 52 factorial is like it's phenomenal. It's not as big as some of the other numbers we've mentioned, though. I mean, not by a long stretch. No, 52 factorial is just what, eight 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 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, OK, Graham's number is a little bit difficult to explain where it comes from, but I'm going to give it a go. OK, it's a number that arises from a mathematical theory called Ramsey theory. um and essentially if you imagine that you've got uh it's all about cubes it's all about joining together the corners of cubes that's where it 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 the the lines around the outside but you can also have diagonal lines which are connecting up the the diagonal corners okay you could color all those in right you can make some of them red you can make some of them blue you can you know color them whatever way you like that 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. OK, you could colour in all of those blue and red, however 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 um you have the a cross in the middle of the square and all the corners are connected this question of ramsey theory i'm i'm really simplifying here slightly um 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 they're going to be mathematicians listening to this who know way more combinatorics 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 just go with me okay so here is the question if you color 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 color in 3d there's a way to color it in that you can avoid it the question is what 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. There 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 the thing, actually. I think 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 mathematicians do all day is come up with crazy questions for themselves, puzzles about many dimensional cubes and the colouring of edges on it. Anyway, OK, so here is the 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 uh 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 what is called ronald graham is so gigantic that it is uh 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 three plus three plus three, that's the same as three times three, right? If you had three times three times three, that's the same as three to the power of three, which you could also write as three up three, because it's sort of like you write the three up oh okay okay but when you have this up arrow you um you can go a bit further because you could say three up up three which is three to the power of three to the power of three oh three to the power of 27 up arrow notation it's like a 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 get big what just one arrow brings us into the trillion exactly i mean it's crazy so um 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. OK, 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 seven point six trillion is I mean, it destroys 52 factorial. It makes it makes your crazy number look like looks like a speck in the ocean. Right. I mean, not even that. That dwarfs it way more than that. It's sort of like there's no there's no comparison. If you compare them, 52 factorial is pretty close to zero compared to where we already are with just what, three up, up, up, three up, up. Yeah, exactly. Now, the way that you make Graham's number is you say, OK, 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, three. so three and and four ups and then three okay so it's already absolutely massive then g2 is three up up up up up up up up g1 ups three okay it's just it's it's so ridiculous that was 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 we're still 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 63 up G63 ups. Yeah. which itself was g62 ups which is 61 ups which itself was 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 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 um exceed the sports chart radius of your own head so yeah if you could 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 uh 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 six seven eight nine and ten 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 yeah for the answer but here is the answer i tell you what we do know about graham's number though um it ends in a seven that 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 we can tell you it ends in a seven yeah it's i mean And 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 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. Radiotherapy is over a century old, but it is still changing. Cancer Research UK helped lay the foundations of radiotherapy in the early 20th century and has driven progress ever since. Radiotherapy remains one of the cornerstones of cancer treatment today. 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Just search for The Book Club every Tuesday and hopefully we will see you there. 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 ego 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 just what we do Yeah What if this benchmark that you set or has been set by people before by Buddha and Archimedes can be beaten That was the great idea between two mathematicians who had their own showdown. This is at MIT. They wanted to go so far beyond what Graham's number 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 at MIT in 2007. There are these two mathematicians called Adam Elger and Augustine Rayo. And they're like, okay, let's take each other on. Let's have the ultimate duel. But rather than being weapons involved, let's just come up with the biggest numbers we can possibly write down. Graham's number is a good threshold, but let's see if we can go further. So Adam Elger, he's sort of the challenger in all of this. He comes up with an idea that is 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 colors together. OK, 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 gonna i'm not gonna play around with dots and lines i'm not gonna mess around with combinatorics no no no no no what i'm gonna do is i'm gonna say all right graham's number your number admilgar all of that can't 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 to properly describe it think of all of them ups ray 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 um 52 factorial also needs three symbols five two and an exclamation mark graham's number needs a lot more because you've got all of them ups yes 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 what he 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 reo's number is defined as the one 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 come up with a fry stevens number which is uh the smallest number that's larger than any number that can be named in the expression of the language with a google plex symbol or less i mean you can you can you can't outray yeah 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 and games, right? This is all fun and games. It is really fun and 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 we 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 it's a struggle. But yet it's so important that we help people picture how large these quantities are. The difference between a million and a billion is one that I that I 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. 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. This 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, about halfway between a thousand and a million. And in reality, a million was barely a pixel above one thousand. I know. I know. You need a thousand millions to get a billion. 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? 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 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 billion and point oh, oh, two 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 it. 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 hone in 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 Slovic, 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 person 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 our 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 is 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 worm's 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's the most impactful way that I've ever seen these two things tied together, knowing that that our brains really don't work in the same way as those mathematicians brains do, not when it comes to having empathy for towards other people. So today we've reached the largest described finite number. But then we kind of like found something even bigger. 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 um decide to describe the largest finite numbers to you by just reading off all these zeros because imagine if we had if we had just been like okay eight how about a hundred 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 podcasts. Be sure to like and subscribe on YouTube. Smash that like button, Archimedes. smash it hit the bell and sign up for our newsletter at the rest is dot com slash science if you would like to answer any of your questions especially on our thursday episodes our field notes episodes where we i mean we're even more rambling and meandering than we are on this one um you can send us in anything you like to the rest is science at goalhanger.com see you next time next time The End in ancient history. To get a flavour of the series, here is a clip from our episode with none other than Stephen Fry. It is one of my favourite subjects, the story of the Greeks and the siege of Troy and Odysseus' return home, of course. I say Greeks, Homer called them the Achaeans, the Danaeans, the Argives. The word Greeks is a much later one, but it refers really to the Mycenaeans, a warrior aristocracy, essentially, obsessed with honor and reputation that would give them an eternal glory, a kleos, as they call it. It's the kleos that's in the name of so many Greeks, you know, Cleopatra and all the Heracles, who's Hercules, you know, Hera's glory. He was actually named Heracles because she hated him because he was a love child of Zeus and she never liked Zeus's love child, her husband, her errant husband. And so as an attempt to placate her, Tiresias, because he was born in Thebes, suggested that he change his name, as a baby this was, to Heracles, the glory of Hera. But it didn't help much. It didn't help at all. And then Athena even put her on Hera's breast when Hera was asleep because it would bond them if he suckled her milk. But she woke and saw it and tossed him away and her breast milk spread across the sky to form the Milky Way. I didn't know that story. Because galaxy, of course, is from the Greek for milk. Galactic, as in lactic. Right. So the chocolate makers are right. Anyway, this is completely separate. Lovely. Keep going. Don't stop. Well, we really hope you enjoyed that clip. To hear more on the Bronze Age apocalypse and how it shaped the ancient Greek epics, just subscribe to Empire wherever you get your podcasts.