[0:00] So on number file Brady, you've got a
[0:02] lot of videos about very very big
[0:04] numbers. Graham's number, Goodstein
[0:06] sequence, tree three, subcubic graphs,
[0:09] rayo's number, as well as being very big
[0:11] numbers. Just about all of them maybe
[0:12] with the exception of Graham's number
[0:14] come out of mathematical logic. And the
[0:17] other thing is that they're all pretty
[0:19] recent discoveries, right? They're all
[0:22] dating from sort of the the middle of
[0:24] the 20th century at the absolute
[0:26] earliest. So a question I was thinking
[0:29] about was that you know if there was a
[0:31] number file equivalent a 100red years
[0:33] ago or 500 years ago who were trying to
[0:36] catalog what are the really big numbers
[0:38] people have been thinking about what
[0:39] would they come up with actually there's
[0:41] a very clear answer where you find those
[0:43] biggest numbers and the biggest numbers
[0:46] of the ancient world were in India I
[0:49] think that is absolutely clear and of
[0:52] all the very big numbers that got
[0:54] contemplated in India uh the biggest
[0:58] come out of the tradition of the
[1:01] religion Janism. So Janism is a Indian
[1:04] religion still practiced by millions of
[1:06] people today but it's also a very
[1:09] ancient religion. It dates back till um
[1:11] 2 and a half thousand years BCE and as
[1:14] part of the sort of mysticism of Jane
[1:18] tradition they came up with some really
[1:20] really big numbers and I thought it
[1:21] might be fun to look at a few. We'll
[1:23] start with ones which which represented
[1:27] long periods of time. So they they they
[1:29] put together um processes which took a
[1:33] long time to finish and then called some
[1:36] number the length of time it takes to
[1:38] finish the process. So I'll do an
[1:39] example. We're going to start with a
[1:41] thing called a paleopama which stands
[1:44] for a pit year. So it's a length of time
[1:47] measured according to um a pit. So what
[1:51] is the pit? The pit is a cubic pit and
[1:54] it's one yojana wide. You probably don't
[1:58] know what a yojana is. You might you
[1:59] might have forgotten. It's slightly more
[2:01] than 10 km. So I'm going to just take it
[2:02] as 10 km. We'll round down a little bit.
[2:04] Okay. So we've got a cubic pit 10 km by
[2:07] 10 km by 10 km. It's actually a bit
[2:09] bigger, but we're rounding down a little
[2:10] bit. And then you fill it the whole
[2:12] thing with lamb's wool.
[2:17] And then once every century you remove a
[2:21] strand of lambs wool and the pit year is
[2:24] the length of time it takes you to empty
[2:26] the whole pit. Okay, so that's that's
[2:30] what the pit year is. So I did a bit of
[2:32] sort of playing around. We can do a bit
[2:33] of a calculation just to give some sort
[2:36] of idea how many strands of lambs wool
[2:38] are going to be in there. So
[2:39] >> are we going to press them down and step
[2:41] on them?
[2:41] >> Well, so I'm going to just I'm going to
[2:44] go for an underestimate. Okay, you're
[2:46] right. At the bottom of a 10 km deep pit
[2:50] of wool, the pressure is going to be
[2:51] pretty high and those strands down there
[2:53] are going to get really squeegeed
[2:54] together. I'm actually going to assume
[2:56] just for this calculation that each
[2:58] strand of lamb's wool occupies a cubic
[3:01] millimeter. And that's got to be a big
[3:04] overstate overestimate. It's probably an
[3:06] overestimate anyway, but once you factor
[3:08] in the enormous pressure, it's a big
[3:09] overestimate, but it's still it's enough
[3:11] to give it let us do a calculation.
[3:13] Okay. So, how many strands of lamb's
[3:15] wool if we assume each one is a cubic
[3:17] millimeter? Just need to know how many
[3:18] cubic millimeters there are in a cube 10
[3:22] km wide. The number of strands of lambs
[3:24] wool is that's my 10 km in meters.
[3:27] 10,000 m. Put another th00and on them.
[3:30] That's now my 10 km in millimeters. We
[3:33] cube it because it's a cubic pit. So,
[3:36] that's the number of strands of lambs
[3:37] wool. And then we'll multiply by 100
[3:40] because we're removing one once every
[3:42] century. So you do the calculation and
[3:44] this is 10 to the 23 10 the^ 23 years.
[3:48] The pit year the paleo palmer is
[3:52] an absolute minimum minimum 10^ the 23
[3:55] years. Okay. But that's just the start.
[3:58] >> Okay. And they but they weren't using
[4:00] this for any mathematical reason. It was
[4:01] just kind of like oh it's such a big job
[4:03] it's going to take me a poly yo palmer
[4:06] to do it. like it it would be just like
[4:07] vernacular like oh like a zillion years
[4:10] or or were they using it in some kind of
[4:12] mathematical way?
[4:13] >> They were they did do mathematics with
[4:17] some very big numbers which we'll come
[4:18] on to. Um these periods of time
[4:21] um they might have used it in the
[4:23] vernacular I'm not sure but what they
[4:25] definitely did is they uh they built the
[4:26] the religious mythology out of this. So
[4:29] um these periods of time were considered
[4:31] to be real periods of time. Um and if
[4:34] you wanted to date, you know, date the
[4:36] the universe since the uh since the the
[4:39] date of creation, this is the kind of
[4:41] unit you would uh uh need. In fact, you
[4:44] need much bigger units. So, we'll move
[4:46] on to the next one. The next one is the
[4:48] Suro prama, which is the ocean year. And
[4:52] this has a nice uh simple definition.
[4:54] It's 100 million paleoparms. So, it's a
[4:56] 100 million of the previous things.
[4:58] Okay. So that's going to be I mean it's
[5:00] at least 10^ the 31
[5:03] years in in the mythology of Janism. The
[5:07] universe runs on a cycle and
[5:10] it's the cycle started round about a
[5:13] quadrillion
[5:15] of these ocean years before today. Okay.
[5:19] So the start of the cycle and of all of
[5:21] this is of course an underestimate was
[5:23] around 10^ the 15 that's my quadrillion
[5:27] of those ocean years before today. So
[5:29] that's round about 10 to the 46. Of
[5:32] course
[5:32] >> that's their sort of big bang for lack
[5:34] of a better
[5:35] >> Yeah I think so. I mean I think it was a
[5:37] sort of endlessly repeating cycle. So I
[5:39] don't think they have actually
[5:40] >> some people think the big bang does that
[5:42] too but
[5:42] >> well yeah well indeed indeed. Um all
[5:44] right. So that's the that's the start of
[5:46] the cycle. They did also think about
[5:49] periods of time beyond this. Um and in
[5:53] particular they thought about periods of
[5:54] time which
[5:57] encompass more than an entire cycle.
[5:59] Sometimes they had a unit of time called
[6:01] a pervanga which is defined to be 84 *
[6:07] 100,000 and then everything there is
[6:10] measured in a unit a sort of fundamental
[6:13] unit called pervies which is a number of
[6:16] days 756* 10 the 11 days. This 100,000
[6:19] is uh the word for that is a lack. So
[6:22] it's 84 lakh pervies. This isn't days
[6:26] but that's that's just the sort of first
[6:28] level. So the next level is obtained by
[6:33] squaring. So the next one is one perver
[6:37] which is 84. Well I'll just write that
[6:40] as 8,400,000.
[6:42] This time we square it and then we're
[6:44] counting in peries again. You can see
[6:46] how it's going to go. We're going to
[6:47] keep increasing that exponent. The next
[6:50] one's called a truy tanga which is the
[6:53] same thing. And then this goes as far
[6:56] the top one of these which is their
[6:58] biggest unit of time as far as I'm aware
[7:01] anyway is called one shera pelica which
[7:05] means the top riddle which I think is a
[7:08] great name for a massive number which is
[7:10] up to 28. So we go 8,400,000
[7:15] to the^ 28 * 756 * 10 11.
[7:19] >> What's that in years? What what sort of
[7:20] exponents we up to here? Do you know?
[7:23] round about 10^ the 206
[7:26] >> years 10^ the 26 years round about I
[7:29] mean bearing in mind that you know the
[7:30] universe as we understand it to be at
[7:33] the moment is 13 billion years ago so
[7:34] this is this is you know way way way
[7:36] beyond that if we sort of extrapolate
[7:38] the current cosmological uh models
[7:43] probably we're this this period of time
[7:46] would take us past the point where all
[7:49] the super massive black holes in all the
[7:51] galaxies have evaporated So that will be
[7:54] you know a very dark and empty universe
[7:57] by that point if it still exists
[7:59] >> like it seems very arbitrary. So this
[8:00] original number the one pervanga so the
[8:03] 84 lakh pervies
[8:06] was said to be the lifespan of the
[8:10] original founder of Janism
[8:12] >> and that's like a really long time
[8:14] obviously
[8:14] >> it's over a quintilion years
[8:16] >> right
[8:16] >> yes so a lot of the mythology of Janism
[8:20] happens over these sort of time scales
[8:22] which really no one very few other
[8:24] people think about yeah
[8:25] >> is there anything else
[8:26] >> yes there is we haven't got to the
[8:27] biggest numbers yet
[8:28] >> oh we're going bigger
[8:29] >> we're going bigger Yeah. All right.
[8:30] Paper.
[8:31] >> I think more paper. Yeah.
[8:37] >> So, as well as contemplating very very
[8:40] long time scales, the ancient James also
[8:44] developed a theory of very big numbers
[8:47] just for their own sake, not really
[8:49] representing anything particular, just
[8:50] really as a an investigation into
[8:53] immense numbers. And they they they
[8:55] classified them in different ways. And
[8:57] in particular at the upper end they had
[8:59] the concept of an uninnumerable number.
[9:02] And the idea of an unnumerable number is
[9:06] that it's a finite number but it is so
[9:10] big that for practical purposes it's
[9:12] basically infinite. I think that's the
[9:14] that's the general idea and and maybe
[9:16] it's worth saying that in in modern
[9:18] mathematics we don't really have that
[9:19] idea. So there's a there's a description
[9:22] of the first unnumerable number which is
[9:24] comes out of a book written around 1,000
[9:26] CE by someone called Nemichandra and the
[9:31] book is called trilocasara which I
[9:33] understand translates to the essence of
[9:35] three worlds and in this book he gives
[9:38] this fantastic description of a really
[9:40] big number the first unnumerable number
[9:44] okay and it takes as its starting point
[9:47] the jog the sort of mystic geography of
[9:52] the the plane on which we all live.
[9:54] Okay. And so in the middle of this plane
[9:58] 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
[19:53] here on Number File and Big Numbers are
[19:56] well your thing, then you really need to
[19:58] check out Richard's new book called Huge
[20:01] Numbers. The cover will probably look
[20:04] like one of these depending on where you
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[20:08] or pre-order right now depending on
[20:11] where you live and when you're watching
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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.