(Finite) Numbers So Large They'd Destroy You

7:12Hannah Fry:Eight's more than you can hold in your head, supposedly. I sort of want to test you.

7:16Michael Stevens:You want to test me?

7:17Hannah Fry: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 you might be better at memorizing eight. All right. Oranges. Coffee.

7:35Hannah Fry:Squirrels. Mushrooms.

7:44Hannah Fry:cards, teaspoons, pins, strawberries. How many was that?

7:52Michael Stevens:I don't know. I'll just say them back. We've got coffee, oranges and teaspoons, squirrels and mushrooms and pins and cards.

8:01Hannah Fry:That wasn't pins.

8:03Michael Stevens:Was it pins?

8:04Hannah Fry:Pins, you're right. Sorry, you're right. That's an accent thing.

8:08Michael Stevens:I did 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.

8:17Hannah Fry:Nice.

8:18Michael Stevens:So I was cheating using some ancient techniques. Okay, here we go.

8:26Michael Stevens:Ketchup, justice, green, tomorrow, tetrahedron, bicycle, haircut, irony, third. Okay.

8:40Hannah Fry:All right. um justice ketchup green irony tomorrow third tetrahedron i'm missing one you're missing two oh damn

8:58Michael Stevens:how many did i get seven 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.

9:18Hannah Fry: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. It is. Look, we're getting there. We're getting bigger and bigger as we go. How many words do you think you know?

9:41Hannah Fry:Not 180 ,000, that's for sure.

9:44Michael Stevens: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 1 ,000 and from there we can extrapolate how much of the language you know.

10:00Hannah Fry:I would love to know that. I want to know how many the average adult knows.

10:04Michael Stevens:I'm going to guess about like 80 ,000.

10:07Hannah Fry:Wow. Okay. The average native English speaker actually knows between 20 and 35 ,000 words.

10:14Michael Stevens:20 and 35 ,000. And you only need about 10 ,000 to have conversations. So we're all doubly prepared.

10:22Hannah Fry:I mean, that doesn't seem like very many.

10:24Michael Stevens:It doesn't seem like very many. But why doesn't it? Because it is a lot. Like it's that's a lot. Twenty to thirty five thousand. 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.

10:39Hannah Fry:It doesn't feel like a perfect number.

10:41Michael Stevens: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 thirty five thousand words in one sitting? Just like, oh, let's see. Have I done? Have I done yesterday? day. Yeah, shoot.

10:58Hannah Fry: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. No.

11:10Michael Stevens:By constructing number names, you can go way past 35 ,000. How about 35 ,001?

11:16Hannah Fry:Exactly. OK, bigger numbers, bigger numbers still.

11:19Michael Stevens: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

11:27Hannah Fry:looking this up uh last night one billion that's that is that that's a that's a big number that's

11:36Michael Stevens:a big number and here's what's special about one billion yeah that that that kind of put puts us up against another limit one billion is about how many heartbeats anything gets in its life

11:50Hannah Fry: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.

11:58Michael Stevens: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

12:36Hannah Fry:a factor of two if it's one billion plus one billion i'm fine with that that's still about

12:39Michael Stevens: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.

12:56Hannah Fry: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.

13:10Michael Stevens:Yeah, that's right. You've got them all in a bag. Now you can build whatever you want with them.

13:15Hannah Fry:Yeah, yeah. The one number that comes up a lot, actually, when you talk 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.

13:31Michael Stevens:OK, we're getting, 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.

13:40Hannah Fry: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

13:47Michael Stevens: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

14:10Hannah Fry: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.

14:17Michael Stevens: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 call a thing that's written 2 ,000 years ago that's eight pages long? A treatise.

14:41Hannah Fry:I think a treatise.

14:42Michael Stevens:treatise yeah yeah okay we're speaking of course about the sand reckoner um 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 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.

15:31Michael Stevens: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 it's like that there's

15:54Hannah Fry:the sort of separation of the number of things number of actual objects right because that 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.

16:15Michael Stevens: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

16:43Hannah Fry:by 63 zeros he did something quite clever actually to get there because um you had myriad 10 000 as you say and they would have myriads of myriads so like 10 000 10 thousands as it were sure but but 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?

17:26Hannah Fry:The way that they counted stuff, they did not have this easy decimal, you know, positional system that we have at the moment.

17:33Michael Stevens: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.

18:15Michael Stevens:but that's fine because i'm trying to show you that i can think of some big numbers yeah he also

18:20Hannah Fry:i mean the the actual universe itself there was this is before they even decided that the uh that the sun was the center of the solar system let alone the universe right i know and that's what's

18:32Michael Stevens: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 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.

19:20Hannah Fry: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.

19:46Michael Stevens: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.

19:55Hannah Fry:And then, of course, because this is a very early version of a YouTube educational video, at the end of the Sand Reckoner, Archimedes says, if you enjoyed that content, please hit that like button and subscribe, right?

20:07Michael Stevens: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 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.

20:35Hannah Fry:There's a few different stories, but I think one of my favourites is the future Buddha. This is Prince Siddhartha. And he wanted to marry this really beautiful princess, Goppa. 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 rule. And it came down to this showdown between him and this mathematician who was called Arjuna.

21:17Hannah Fry:And Arjuna tries to stump the prince and he's like, OK, do you know any numbers beyond the Koti? And a Koti was 10 million, right? Siddhartha 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 Nayuta, which is 100 billion. And he 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 sort of it's not that you're multiplying by numbers you're adding additional zeros on the top right so you're 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.

22:23Hannah Fry: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.

22:41Michael Stevens:Deservedly.

22:42Hannah Fry:Deservedly. Don't wing me over, I've got to be honest.

22:45Michael Stevens:A one followed by more than 400 zeros. We've gone past a Google.

22:50Hannah Fry:We have.

22:51Michael Stevens:We as a species went past a Google long before the Greeks. Yeah, yeah. Long before Google.com, the search engine.

22:59Hannah Fry: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.

23:09Michael Stevens: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 Gargugle. 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 and 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

23:37Hannah Fry:gasquillian yeah has not has not uh it's that it's 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 The thing is, up until 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?

24:15Hannah Fry:anytime that you're dealing with a combination of something. I'm teeing you up here, Michael.

24:21Michael Stevens: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 one to 52, every integer from one 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?

25:04Michael Stevens: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 Cheepiel 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.

25:54Michael Stevens: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

26:18Hannah Fry: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

26:33Michael Stevens:to pass. 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.

27:20Michael Stevens: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

28:17Hannah Fry: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 you start all over wow two sheets of paper three sheets four sheets once

28:28Michael Stevens: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

28:42Hannah Fry: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.

28:59Michael Stevens: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.

29:16Hannah Fry: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 then eventually, eventually, eventually, the whole mountain will be worn away by the...

29:57Michael Stevens:Oh, worn away by just the scarf touching it. Once every hundred years.

30:01Hannah Fry: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.

30:23Michael Stevens: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.

30:34Hannah Fry: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 them 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 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.

31:34Hannah Fry:Okay, you could colour all those in, right? You could make some of them red, you could make some of them blue, you can, you know, 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 those blue and red however you wanted.

32:12Hannah Fry: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 could do six dimensions, you could 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 the cross in the middle of the square and all the corners are connected.

32:52Hannah Fry:This question of Ramsey theory, I'm really simplifying here slightly, a lot, actually.

32:57Michael Stevens: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.

33:09Hannah Fry: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 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's 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?

33:44Hannah Fry: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. There is, pure mathematicians really enjoy coming up with these challenges for themselves and then spending their entire lifetimes trying to solve them, right?

34:17Michael Stevens: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?

34:26Hannah Fry: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 was 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.

34:59Hannah Fry:And Graham came up with an upper bound. He said, OK, 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 you cannot explain it in terms of zeros anymore. 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

36:24Michael Stevens:is 7.6 trillion whoa get big just one arrow brings us into the trillion arrow exactly i mean it's

36:33Hannah Fry: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.

36:50Michael Stevens:That's three times itself, 7 trillion times.

36:53Hannah Fry:Yeah, yeah, exactly. 7.6 trillion times. Exactly. That is already a giant number, right? Three to the power of 7.6 trillion is, I mean, it destroys 52 factorial. It makes your crazy number look like a speck in the ocean, right? I mean, not even that. Dwarfs it way more than that. It's sort of like there's no comparison.

37:17Michael Stevens:If you compare them, 52 factorial is pretty close to zero compared to where we already are with just

37:24Hannah Fry:what three up 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 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

38:01Michael Stevens:nickname by the way g1 ups oh that's g1 ups how you doing man he's a big dude but we're still

38:08Hannah Fry:not at gram's number we're oh no we're nowhere near we haven't even started so g2 is three to the g1 ups three um g3 is three g2 ups three and you carry on going over and over again until you

38:26Michael Stevens:get to g64 g64 which is 363 up g63 ups yeah which itself was g62 ups which itself is 61

38:41Hannah Fry:ups which itself was da-da-da-da-da-da-da-da-da-da-da-da. And remember, three ups absolutely dwarfs your 52 factorial number.

38:51Michael Stevens: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.

39:01Hannah Fry: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 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

39:45Michael Stevens: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

40:05Hannah Fry:But here is the answer. I'll tell you what we do know about Graham's number, though. It ends in a seven.

40:11Michael Stevens: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.

40:24Hannah Fry:Yeah. I mean, they think it's a proper number that exists. It's just completely beyond our comprehension.

40:35Michael Stevens: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.

41:20Michael Stevens:This episode is brought to you by Cancer Research UK.

41:24Hannah Fry: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.

41:36Michael Stevens:Radiotherapy remains one of the cornerstones of cancer treatment today. Every year, millions of people worldwide benefit from Cancer Research UK's work to make it more precise.

41:46Hannah Fry:Scientists are still refining how radiotherapy is delivered. And one example is an experimental treatment called flash radiotherapy, which delivers radiation in fractions of a second, up to a thousand times faster than standard radiotherapy.

42:01Michael Stevens:And early studies suggest that speed could make a real difference. Flash radiotherapy may cause up to 50 % less damage to healthy cells.

42:10Hannah Fry:But scientists don't yet know why healthy cells seem to be spared, so Cancer Research UK are working to answer that. Understanding it could be key to reducing side effects in the future.

42:23Michael Stevens:For more information about Cancer Research UK, their research and breakthroughs, and how you can support them, visit cancerresearchuk.org forward slash the rest is science.

42:59Michael Stevens:Book now at Hilton.com. Hilton, for this day.

43:28Michael Stevens:at indeed.com slash podcast. Terms and conditions apply.

43:39Hannah Fry: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?

43:57Michael Stevens:Yeah, there is. I mean, it's 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.

44:34Hannah Fry: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.

44:52Michael Stevens: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's just what we do. Yeah.

45:20Hannah Fry: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 Augustin Rayo. And they're like, okay, let's take each other on. Let's have the ultimate duel.

46:02Hannah Fry: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 colours together, okay?

46:43Hannah Fry:It's really impressive. Everyone finds him extremely excellent and intelligent as a result of this.

46:49Michael Stevens: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.

47:04Hannah Fry:Yeah.

47:05Michael Stevens:And yet, Rayo comes in and wins the competition.

47:09Hannah Fry:Rayo comes in and wins the competition. And he does it with this absolute genius move. He's like, I'm not going to play around with dots and lines. I'm not going to mess around with combinatorics. No, no, no, no, no. What I'm going to do is I'm going to say, all right, Graham's number, your number, Adamel Girl, 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 it. Think of all of them ups. raya was 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 52 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 a all of the numbers uh that need less than a google of symbols to describe them that's all there i'm gonna say my number is the smallest number that cannot be described by a google of symbols.

48:36Hannah Fry:So all of those numbers in there, I'm going to do that, plus one, basically. That's essentially what he did.

48:43Michael Stevens:Which is brilliant because you just, 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 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 can't in your system. No matter how much you compress it, I'm always beyond you.

49:13Hannah Fry:I mean, the thing is, we could come up with our own number. We could come up with a Frye-Stevens number, which is 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't out Rayo, Rayo.

49:28Michael Stevens: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.

49:40Hannah Fry: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.

49:48Michael Stevens:It is. 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 a struggle, but yet it's so important that we help people picture how large these quantities are.

50:30Hannah Fry:The difference between 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 uh they uh 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 halfway 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

51:28Michael Stevens: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.

51:52Hannah Fry:Is it?

51:53Michael Stevens: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 billion and point 002 billion. If you saw those together, you'd go, that doesn't make a difference.

52:40Hannah Fry: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.

53:27Hannah Fry: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 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

54:45Michael Stevens: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. Oh. That that means so much more to people than, oh, he climbed a mountain.

55:36Michael Stevens: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.

55:49Hannah Fry: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 the bird's eye view and the worm's eye view so there's one of her websites this amazing thing where you 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.

56:44Hannah Fry:And I think that 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.

57:01Michael Stevens: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.

57:11Hannah Fry: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.

57:20Michael Stevens: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.

57:29Hannah Fry:Hey, look, we haven't launched our members only podcast yet. Maybe that could be the first episode.

57:33Michael Stevens: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.

57:41Hannah Fry:You can read out your square root of four book for us.

57:44Michael Stevens:Ooh, yes. The square root of four to a million decimal points.

57:49Hannah Fry: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.

58:02Michael Stevens:smash it hit the bell and sign up for our newsletter at the rest is.com slash science

58:08Hannah Fry:if you would like to answer any of your questions especially on our thursday episodes uh 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

58:56Michael Stevens:We'll see you next time. new wraps today in app or at order.sweetgreen.com available at participating locations only.