The Original Biggest Numbers - Numberphile

[00:00] So on number file Brady, you've got a

[00:02] lot of videos about very very big

[00:04] numbers. Graham's number, Goodstein

[00:06] sequence, tree three, subcubic graphs,

[00:09] rayo's number, as well as being very big

[00:11] numbers. Just about all of them maybe

[00:12] with the exception of Graham's number

[00:14] come out of mathematical logic. And the

[00:17] other thing is that they're all pretty

[00:19] recent discoveries, right? They're all

[00:22] dating from sort of the the middle of

[00:24] the 20th century at the absolute

[00:26] earliest. So a question I was thinking

[00:29] about was that you know if there was a

[00:31] number file equivalent a 100red years

[00:33] ago or 500 years ago who were trying to

[00:36] catalog what are the really big numbers

[00:38] people have been thinking about what

[00:39] would they come up with actually there's

[00:41] a very clear answer where you find those

[00:43] biggest numbers and the biggest numbers

[00:46] of the ancient world were in India I

[00:49] think that is absolutely clear and of

[00:52] all the very big numbers that got

[00:54] contemplated in India uh the biggest

[00:58] come out of the tradition of the

[01:01] religion Janism. So Janism is a Indian

[01:04] religion still practiced by millions of

[01:06] people today but it's also a very

[01:09] ancient religion. It dates back till um

[01:11] 2 and a half thousand years BCE and as

[01:14] part of the sort of mysticism of Jane

[01:18] tradition they came up with some really

[01:20] really big numbers and I thought it

[01:21] might be fun to look at a few. We'll

[01:23] start with ones which which represented

[01:27] long periods of time. So they they they

[01:29] put together um processes which took a

[01:33] long time to finish and then called some

[01:36] number the length of time it takes to

[01:38] finish the process. So I'll do an

[01:39] example. We're going to start with a

[01:41] thing called a paleopama which stands

[01:44] for a pit year. So it's a length of time

[01:47] measured according to um a pit. So what

[01:51] is the pit? The pit is a cubic pit and

[01:54] it's one yojana wide. You probably don't

[01:58] know what a yojana is. You might you

[01:59] might have forgotten. It's slightly more

[02:01] than 10 km. So I'm going to just take it

[02:02] as 10 km. We'll round down a little bit.

[02:04] Okay. So we've got a cubic pit 10 km by

[02:07] 10 km by 10 km. It's actually a bit

[02:09] bigger, but we're rounding down a little

[02:10] bit. And then you fill it the whole

[02:12] thing with lamb's wool.

[02:17] And then once every century you remove a

[02:21] strand of lambs wool and the pit year is

[02:24] the length of time it takes you to empty

[02:26] the whole pit. Okay, so that's that's

[02:30] what the pit year is. So I did a bit of

[02:32] sort of playing around. We can do a bit

[02:33] of a calculation just to give some sort

[02:36] of idea how many strands of lambs wool

[02:38] are going to be in there. So

[02:39] >> are we going to press them down and step

[02:41] on them?

[02:41] >> Well, so I'm going to just I'm going to

[02:44] go for an underestimate. Okay, you're

[02:46] right. At the bottom of a 10 km deep pit

[02:50] of wool, the pressure is going to be

[02:51] pretty high and those strands down there

[02:53] are going to get really squeegeed

[02:54] together. I'm actually going to assume

[02:56] just for this calculation that each

[02:58] strand of lamb's wool occupies a cubic

[03:01] millimeter. And that's got to be a big

[03:04] overstate overestimate. It's probably an

[03:06] overestimate anyway, but once you factor

[03:08] in the enormous pressure, it's a big

[03:09] overestimate, but it's still it's enough

[03:11] to give it let us do a calculation.

[03:13] Okay. So, how many strands of lamb's

[03:15] wool if we assume each one is a cubic

[03:17] millimeter? Just need to know how many

[03:18] cubic millimeters there are in a cube 10

[03:22] km wide. The number of strands of lambs

[03:24] wool is that's my 10 km in meters.

[03:27] 10,000 m. Put another th00and on them.

[03:30] That's now my 10 km in millimeters. We

[03:33] cube it because it's a cubic pit. So,

[03:36] that's the number of strands of lambs

[03:37] wool. And then we'll multiply by 100

[03:40] because we're removing one once every

[03:42] century. So you do the calculation and

[03:44] this is 10 to the 23 10 the^ 23 years.

[03:48] The pit year the paleo palmer is

[03:52] an absolute minimum minimum 10^ the 23

[03:55] years. Okay. But that's just the start.

[03:58] >> Okay. And they but they weren't using

[04:00] this for any mathematical reason. It was

[04:01] just kind of like oh it's such a big job

[04:03] it's going to take me a poly yo palmer

[04:06] to do it. like it it would be just like

[04:07] vernacular like oh like a zillion years

[04:10] or or were they using it in some kind of

[04:12] mathematical way?

[04:13] >> They were they did do mathematics with

[04:17] some very big numbers which we'll come

[04:18] on to. Um these periods of time

[04:21] um they might have used it in the

[04:23] vernacular I'm not sure but what they

[04:25] definitely did is they uh they built the

[04:26] the religious mythology out of this. So

[04:29] um these periods of time were considered

[04:31] to be real periods of time. Um and if

[04:34] you wanted to date, you know, date the

[04:36] the universe since the uh since the the

[04:39] date of creation, this is the kind of

[04:41] unit you would uh uh need. In fact, you

[04:44] need much bigger units. So, we'll move

[04:46] on to the next one. The next one is the

[04:48] Suro prama, which is the ocean year. And

[04:52] this has a nice uh simple definition.

[04:54] It's 100 million paleoparms. So, it's a

[04:56] 100 million of the previous things.

[04:58] Okay. So that's going to be I mean it's

[05:00] at least 10^ the 31

[05:03] years in in the mythology of Janism. The

[05:07] universe runs on a cycle and

[05:10] it's the cycle started round about a

[05:13] quadrillion

[05:15] of these ocean years before today. Okay.

[05:19] So the start of the cycle and of all of

[05:21] this is of course an underestimate was

[05:23] around 10^ the 15 that's my quadrillion

[05:27] of those ocean years before today. So

[05:29] that's round about 10 to the 46. Of

[05:32] course

[05:32] >> that's their sort of big bang for lack

[05:34] of a better

[05:35] >> Yeah I think so. I mean I think it was a

[05:37] sort of endlessly repeating cycle. So I

[05:39] don't think they have actually

[05:40] >> some people think the big bang does that

[05:42] too but

[05:42] >> well yeah well indeed indeed. Um all

[05:44] right. So that's the that's the start of

[05:46] the cycle. They did also think about

[05:49] periods of time beyond this. Um and in

[05:53] particular they thought about periods of

[05:54] time which

[05:57] encompass more than an entire cycle.

[05:59] Sometimes they had a unit of time called

[06:01] a pervanga which is defined to be 84 *

[06:07] 100,000 and then everything there is

[06:10] measured in a unit a sort of fundamental

[06:13] unit called pervies which is a number of

[06:16] days 756* 10 the 11 days. This 100,000

[06:19] is uh the word for that is a lack. So

[06:22] it's 84 lakh pervies. This isn't days

[06:26] but that's that's just the sort of first

[06:28] level. So the next level is obtained by

[06:33] squaring. So the next one is one perver

[06:37] which is 84. Well I'll just write that

[06:40] as 8,400,000.

[06:42] This time we square it and then we're

[06:44] counting in peries again. You can see

[06:46] how it's going to go. We're going to

[06:47] keep increasing that exponent. The next

[06:50] one's called a truy tanga which is the

[06:53] same thing. And then this goes as far

[06:56] the top one of these which is their

[06:58] biggest unit of time as far as I'm aware

[07:01] anyway is called one shera pelica which

[07:05] means the top riddle which I think is a

[07:08] great name for a massive number which is

[07:10] up to 28. So we go 8,400,000

[07:15] to the^ 28 * 756 * 10 11.

[07:19] >> What's that in years? What what sort of

[07:20] exponents we up to here? Do you know?

[07:23] round about 10^ the 206

[07:26] >> years 10^ the 26 years round about I

[07:29] mean bearing in mind that you know the

[07:30] universe as we understand it to be at

[07:33] the moment is 13 billion years ago so

[07:34] this is this is you know way way way

[07:36] beyond that if we sort of extrapolate

[07:38] the current cosmological uh models

[07:43] probably we're this this period of time

[07:46] would take us past the point where all

[07:49] the super massive black holes in all the

[07:51] galaxies have evaporated So that will be

[07:54] you know a very dark and empty universe

[07:57] by that point if it still exists

[07:59] >> like it seems very arbitrary. So this

[08:00] original number the one pervanga so the

[08:03] 84 lakh pervies

[08:06] was said to be the lifespan of the

[08:10] original founder of Janism

[08:12] >> and that's like a really long time

[08:14] obviously

[08:14] >> it's over a quintilion years

[08:16] >> right

[08:16] >> yes so a lot of the mythology of Janism

[08:20] happens over these sort of time scales

[08:22] which really no one very few other

[08:24] people think about yeah

[08:25] >> is there anything else

[08:26] >> yes there is we haven't got to the

[08:27] biggest numbers yet

[08:28] >> oh we're going bigger

[08:29] >> we're going bigger Yeah. All right.

[08:30] Paper.

[08:31] >> I think more paper. Yeah.

[08:37] >> So, as well as contemplating very very

[08:40] long time scales, the ancient James also

[08:44] developed a theory of very big numbers

[08:47] just for their own sake, not really

[08:49] representing anything particular, just

[08:50] really as a an investigation into

[08:53] immense numbers. And they they they

[08:55] classified them in different ways. And

[08:57] in particular at the upper end they had

[08:59] the concept of an uninnumerable number.

[09:02] And the idea of an unnumerable number is

[09:06] that it's a finite number but it is so

[09:10] big that for practical purposes it's

[09:12] basically infinite. I think that's the

[09:14] that's the general idea and and maybe

[09:16] it's worth saying that in in modern

[09:18] mathematics we don't really have that

[09:19] idea. So there's a there's a description

[09:22] of the first unnumerable number which is

[09:24] comes out of a book written around 1,000

[09:26] CE by someone called Nemichandra and the

[09:31] book is called trilocasara which I

[09:33] understand translates to the essence of

[09:35] three worlds and in this book he gives

[09:38] this fantastic description of a really

[09:40] big number the first unnumerable number

[09:44] okay and it takes as its starting point

[09:47] the jog the sort of mystic geography of

[09:52] the the plane on which we all live.

[09:54] Okay. And so in the middle of this plane

[09:58] is an island called this is Jamboo

[10:02] Island. That's where we live. It's very

[10:05] big. Its width in the traditional me

[10:08] measure of yojan is 100,000. Translating

[10:11] into miles that's over half a million

[10:13] miles wide. So we got this big island

[10:15] around half a million miles wide. And

[10:17] then outside of the island, we've got an

[10:21] ocean going all the way around, right?

[10:23] That's called the salt ocean. And then

[10:25] outside that ocean, we've got another

[10:27] sort of continental island or annular

[10:30] island. That's Fireflame Bush Island.

[10:33] And then outside that, you can see where

[10:35] it's going. Outside that, we've got

[10:36] another ocean. And then outside that,

[10:38] we've got another island. And so on. And

[10:41] this carries on. I mean there's

[10:43] different accounts but for this thought

[10:45] experiment this carries on indefinitely.

[10:47] Okay. But it's not just that um we've

[10:50] got these islands and oceans and islands

[10:52] and oceans and islands and oceans. Their

[10:53] size is very important. The first island

[10:56] is around about half a million miles

[10:58] wide. And then the first ocean is double

[11:01] that. So I've not drawn this to scale.

[11:03] The first ocean is double that. So it's

[11:05] around about a million miles wide. And

[11:07] then the next island is double that. So

[11:10] it's around about 2 million miles wide.

[11:12] And then the next one's double that and

[11:14] so on. So each one is double the width.

[11:17] So exponential growth just baked into

[11:20] the the geography of the place we're

[11:22] working. Right? So that's the

[11:24] background. That's the setting for this

[11:26] thought experiment. So then the first

[11:27] thing we do, we dig a cylindrical pit

[11:30] under the first island, Jamboo Island.

[11:32] And what we do is fill that pit with

[11:35] mustard seeds. So this whole thing is

[11:37] going to be is going to be a quantity of

[11:39] mustard seeds. Okay. So the depth of the

[11:41] pit it's a thousand yanas which is 5,000

[11:44] miles or something. So 5,000mi deep pit

[11:47] under the entire island. The rule is

[11:49] that the height of the mountain needs to

[11:52] be 111th of the circumference of the

[11:57] circle.

[11:58] >> Of course.

[11:58] >> Of course. Obviously. Right. I brought

[12:00] some mustard seeds. Would you like just

[12:01] see how big they are?

[12:02] >> Yeah. Go on then. Just in case you've

[12:03] never seen a mustard seed.

[12:05] >> Yeah. So that's those those are mustard

[12:07] seeds.

[12:08] >> Yeah, they're pretty small.

[12:09] >> They're pretty small. I mean, I don't I

[12:10] don't know if it's exactly the same kind

[12:12] of mustard seeds they were having in my

[12:13] bed. They're pretty small things. I

[12:14] mean, it's roughly speaking the same

[12:16] size of a grain of sand. I've got to get

[12:18] rid of these mustard seeds.

[12:19] >> Okay.

[12:20] >> Okay.

[12:22] >> Oh, they've gone all over the table.

[12:24] >> That was bound to happen, was it?

[12:25] >> Oh dear. Yeah.

[12:26] >> Bound to happen. Okay. We were

[12:28] surrounded by mustard seeds, but not as

[12:29] many as um that were about to be

[12:33] appearing in this pit.

[12:34] >> At the moment, this is about 5,000 mi

[12:37] deep. And then this thing is

[12:39] >> 11th of the circumference of the pit.

[12:43] >> So that's so that's

[12:44] >> it's really tall.

[12:45] >> Yeah. Like it's

[12:46] >> I mean it's thousands of miles tall.

[12:47] Thousands of miles tall.

[12:49] >> Yeah. Very tall. Yeah.

[12:50] >> It's very very very tall. Already that

[12:52] mountain of mustard seeds is big enough

[12:54] that you could fit planet Earth in it

[12:56] like loads of times and it's already a

[12:58] massive number. Okay, but we're just

[13:00] getting started. So, and now this is the

[13:02] clever bit because what you do now is

[13:06] you take this collection of mustard

[13:07] seeds, right? And you put the first one

[13:09] on the island and the second one in the

[13:12] first ocean and the third one on the

[13:15] next island and the next one on the next

[13:17] ocean and the next one on the next

[13:18] island and so on. And you keep doing

[13:20] that until you've completely exhausted

[13:23] the whole mountain. Right. And that's

[13:26] taken you to some eventually you've got

[13:28] to some other island or or ocean

[13:31] >> depending on if it was an even or odd

[13:32] number of states.

[13:33] >> Yeah. Exactly. And then you do the same

[13:36] thing all over again. So you build you

[13:39] dig a pit the same depth 5,000 mi deep

[13:42] under that whichever disc you've reached

[13:46] >> which by now will be very very wide

[13:49] >> actually

[13:50] I I did a sort of back of the envelope

[13:52] calculation and it's something like okay

[13:55] how wide is it something like 10^ the

[13:57] 10^ the 40 light years wide

[14:01] >> right okay

[14:02] >> right it's that it's that sort of thing

[14:04] okay which um you know bearing in mind

[14:07] the the the observable universe is um 10

[14:11] to the 11 light years wide or something

[14:15] is enormously bigger. So you make a

[14:17] circular ditch under the under the under

[14:19] the continent you've reached and you you

[14:21] build your mountain of mustard seeds

[14:23] again. So that gives us a new a new

[14:25] mountain.

[14:26] >> Yeah. One which is

[14:27] >> 111th again is our uh

[14:29] >> one 11th of the circumference. Yeah.

[14:30] >> Yeah. Yeah. The circumference which is

[14:32] 10^ the 10^ the 40 lighty years

[14:34] >> or the diameter which is 10 to the 10 to

[14:35] the 40 lighty years. Oh yeah, the

[14:37] circumference even.

[14:38] >> Yeah, but I mean when you're multiplying

[14:40] a number like that by pi, it doesn't

[14:42] make it doesn't make much difference. So

[14:45] yeah, I mean at this sort of scale,

[14:46] these sorts of numbers, it sort of stops

[14:48] mattering whether you're measuring in

[14:50] millimeters or light years just because

[14:52] the the numbers are big so big. This is

[14:54] the second mountain of must sees which

[14:56] is you know so enormous that you know

[14:59] the the observable universe is just an

[15:02] invisible speck compared to it. Okay.

[15:04] And then what do you think we do? We

[15:06] distribute those seeds out ring by ring

[15:08] by ring by ring.

[15:09] >> We do. We do. And then that takes you to

[15:11] another another place where you build

[15:13] another mountain. Uh you keep doing it.

[15:15] How many times do you repeat the

[15:17] process? Uh the answer is the cube of

[15:21] the number of mustard seeds in the

[15:23] original mountain. And that a back of

[15:26] the envelope calculation suggests that's

[15:28] around 10^ the 45 seeds in the first

[15:32] mountain.

[15:33] So we repeat that process the cube of

[15:37] 10^ the 45 times

[15:40] >> and have you back of the enveloped how

[15:42] big this final number is. So the final

[15:44] number. Yes. So maybe I should say I

[15:47] should credit the mathematician and

[15:49] historian Radha Char Gupta who in 1992

[15:52] did a sort of modern mathematical

[15:54] analysis of this situation which is what

[15:55] I'm following here. And he so he did he

[15:58] did the uh the calculation and I mean

[16:02] this the number that we get after

[16:04] completing this that is the first

[16:06] unnumerable number. Let me try and say

[16:09] it in Sanskrit. uh so jaga parita

[16:13] asamata the first uninable number I mean

[16:16] it is so big that if you were to try to

[16:20] you know write it as a power of a tower

[16:23] of tens or something you can't because

[16:27] the the tower is just too tall we can

[16:29] use canth arrow notation and I I'll just

[16:32] explain this in a moment so let's say

[16:34] it's something like this number 10 to

[16:36] the 10 to the 10 to the 45 two can

[16:41] arrows 10^ the 135. So this is an

[16:45] approximate value for the the first

[16:47] uninumeral number. What that means is if

[16:50] we built a tower an exponential tower of

[16:53] x's and I want my height of this tower

[16:56] to be 10^ the 135. So far taller than we

[16:59] could ever ever draw. And then each

[17:01] value of x is this number 10 the 10^ the

[17:06] 10 45. That is about the scale of the

[17:11] first unnumerable number in Jane

[17:13] traditional mathematics.

[17:14] >> We've talked about some big numbers even

[17:16] in this room before you know we could

[17:19] you listed some of them at the start of

[17:20] the video.

[17:20] >> Yeah.

[17:21] >> Is when we when we talk about tree

[17:23] threes numbers that sort of thing.

[17:26] >> Is this in that ballpark? Is it still

[17:28] not really coming close? Is it

[17:30] >> I think it's fair to say that all the

[17:31] ones I listed at the start are much much

[17:33] bigger. Um but it is big enough that it

[17:37] defeats our attempts to write it down

[17:39] using just traditional mathematical

[17:41] notation. So we need canoe arrows or uh

[17:44] something I mean to consider numbers on

[17:46] this scale

[17:48] you have to develop the kind of

[17:49] machinery which can then take you to

[17:52] places like Graham's number. So they are

[17:54] you know on the road towards that sort

[17:56] of territory. It's a testament to the

[17:58] size of those big big Grahams numbers

[18:01] that you did this crazy thing that we

[18:03] were just almost almost laughing at how

[18:05] big it is and then you at the end you

[18:06] said oh no it's still not close to those

[18:08] ones.

[18:08] >> Yeah, that's right. I mean those numbers

[18:10] are on on another scale. But I think

[18:13] it's fair to say that this came

[18:16] thousands of years earlier, right?

[18:18] Thousands of years earlier. So it's

[18:19] taken us it actually took you know there

[18:22] was a big hiatus in terms of the biggest

[18:24] numbers people had thought of. Um it was

[18:27] the ancient James with this number and

[18:29] others. They had others which are sort

[18:31] of around around this. They went went a

[18:33] little bit bigger than this actually.

[18:34] This is the most fun one to talk about.

[18:36] >> It's almost like you needed things like

[18:38] canth notation and some of the new

[18:40] notation that mathematicians use now

[18:43] before you could start playing and

[18:45] inventing those bigger numbers. Yeah.

[18:47] They just didn't have access to that. uh

[18:49] >> I mean they started to so they did start

[18:52] talking about like repeating processes

[18:56] large numbers of times and the sort of

[18:58] process is an abstract arithmetical

[19:00] operation and that's sort of how you do

[19:02] it right I mean that's what the canth

[19:03] arrow is you talk about you've got some

[19:05] arithmetical operation and then you say

[19:09] okay I'm going to iterate that a large

[19:11] number of times I've seen an account of

[19:16] um ancient Jane writers We might have

[19:19] got as far as three kith arrows 10 three

[19:24] kith arrows 38. So that's sort of two

[19:28] levels beyond exponentiation. Right? The

[19:30] two kith arrows is is a level beyond

[19:32] exponentiation. That's two levels

[19:34] beyond. No one else thought about

[19:36] numbers anywhere near as big as this. So

[19:39] there was there's a real sort of hiatus

[19:42] from the ancient js who were doing this

[19:43] thousands of years ago. and the rest of

[19:45] the world only really caught up in the

[19:47] sort of second half of the 20th century.

[19:49] So they were the record holders for most

[19:50] of history. If you enjoy seeing Richard

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[20:41] you can watch a 20inute video with

[20:44] everyone's name. Will you end up on a

[20:46] sea or an island? There's only one way

[20:48] to find out. There are links down below

[20:51] and everyone can watch it, patron or

[20:53] not. Have fun with it. It's uh it's

[20:55] quite the experience.