---
title: 'The Original Biggest Numbers - Numberphile'
source: 'https://youtube.com/watch?v=sJqceEImtew'
video_id: 'sJqceEImtew'
date: 2026-06-28
duration_sec: 1258
---

# The Original Biggest Numbers - Numberphile

> Source: [The Original Biggest Numbers - Numberphile](https://youtube.com/watch?v=sJqceEImtew)

## Summary

This video explores the concept of very large numbers in ancient Indian mathematics, specifically within the Jain tradition. It contrasts these historical numbers with modern large numbers like Graham's number, highlighting the Jain tradition's early development of concepts like 'unnumerable numbers' and iterative processes for generating them.

### Key Points

- **Introduction to Ancient Big Numbers** [0:00] — The video discusses that many very large numbers from modern mathematics (Graham's number, etc.) originate from mathematical logic and are recent discoveries. The question is posed: what were the biggest numbers in the ancient world, with the answer being found in India, particularly in the Jain tradition.
- **Jainism and Its Big Numbers** [1:01] — Jainism is an ancient Indian religion, still practiced today, that developed very large numbers as part of its mysticism. These numbers were used to represent long periods of time, such as the 'palya' (pit year) and 'sagaropama' (ocean year).
- **The Palya: A Pit Year** [2:04] — A 'palya' is defined as the time it takes to empty a cubic pit (10 km per side) filled with lamb's wool, removing one strand every century. An estimate gives a minimum of 10^23 years.
- **The Sagaropama: An Ocean Year** [4:48] — A 'sagaropama' is 100 million 'palya's, leading to a time scale of at least 10^31 years. These were used in Jain cosmology to describe the age of the universe and its cycles.
- **The Palyopama: A Unit of Time** [6:01] — The 'palyopama' is a unit of time defined as 84 lakh (8,400,000) 'palya's, which is then squared to create larger units. The largest unit is the 'shirsha prahelika' (top riddle), which is 8,400,000^28 * 756 * 10^11 days, approximately 10^206 years.
- **Unnumerable Numbers** [8:37] — The Jains also developed a theory of very large numbers for their own sake, classifying them as 'unnumerable' (finite but practically infinite). The first such number is described in the book 'Trilokasara' by Nemichandra (c. 1000 CE).
- **The First Unnumerable Number: A Thought Experiment** [9:54] — The thought experiment involves a series of concentric islands and oceans, each doubling in size. A cylindrical pit is dug under the first island (Jambu Island) and filled with mustard seeds. The process is repeated, creating a mountain of mustard seeds, which is then distributed to create a new mountain, and so on.
- **The Scale of the First Unnumerable Number** [15:21] — The process is repeated the cube of the number of mustard seeds in the original mountain. The final number is approximately 10^(10^(10^45)) with two Knuth arrows, or 10^(10^(10^135)). This is an enormous number, far beyond the scale of the observable universe.
- **Comparison to Modern Large Numbers** [17:31] — While this number is huge, it is still much smaller than modern numbers like Graham's number. However, it represents a significant historical achievement, as the Jains were the first to think about numbers on this scale, thousands of years before modern mathematics.
- **Historical Record Holders** [19:49] — The Jains were the record holders for the largest numbers for most of history, until the development of modern mathematical notation in the 20th century. The video concludes by promoting Richard's new book 'Huge Numbers' and thanking Patreon supporters.

## Transcript

So on number file Brady, you've got a
lot of videos about very very big
numbers. Graham's number, Goodstein
sequence, tree three, subcubic graphs,
rayo's number, as well as being very big
numbers. Just about all of them maybe
with the exception of Graham's number
come out of mathematical logic. And the
other thing is that they're all pretty
recent discoveries, right? They're all
dating from sort of the the middle of
the 20th century at the absolute
earliest. So a question I was thinking
about was that you know if there was a
number file equivalent a 100red years
ago or 500 years ago who were trying to
catalog what are the really big numbers
people have been thinking about what
would they come up with actually there's
a very clear answer where you find those
biggest numbers and the biggest numbers
of the ancient world were in India I
think that is absolutely clear and of
all the very big numbers that got
contemplated in India uh the biggest
come out of the tradition of the
religion Janism. So Janism is a Indian
religion still practiced by millions of
people today but it's also a very
ancient religion. It dates back till um
2 and a half thousand years BCE and as
part of the sort of mysticism of Jane
tradition they came up with some really
really big numbers and I thought it
might be fun to look at a few. We'll
start with ones which which represented
long periods of time. So they they they
put together um processes which took a
long time to finish and then called some
number the length of time it takes to
finish the process. So I'll do an
example. We're going to start with a
thing called a paleopama which stands
for a pit year. So it's a length of time
measured according to um a pit. So what
is the pit? The pit is a cubic pit and
it's one yojana wide. You probably don't
know what a yojana is. You might you
might have forgotten. It's slightly more
than 10 km. So I'm going to just take it
as 10 km. We'll round down a little bit.
Okay. So we've got a cubic pit 10 km by
10 km by 10 km. It's actually a bit
bigger, but we're rounding down a little
bit. And then you fill it the whole
thing with lamb's wool.
And then once every century you remove a
strand of lambs wool and the pit year is
the length of time it takes you to empty
the whole pit. Okay, so that's that's
what the pit year is. So I did a bit of
sort of playing around. We can do a bit
of a calculation just to give some sort
of idea how many strands of lambs wool
are going to be in there. So
>> are we going to press them down and step
on them?
>> Well, so I'm going to just I'm going to
go for an underestimate. Okay, you're
right. At the bottom of a 10 km deep pit
of wool, the pressure is going to be
pretty high and those strands down there
are going to get really squeegeed
together. I'm actually going to assume
just for this calculation that each
strand of lamb's wool occupies a cubic
millimeter. And that's got to be a big
overstate overestimate. It's probably an
overestimate anyway, but once you factor
in the enormous pressure, it's a big
overestimate, but it's still it's enough
to give it let us do a calculation.
Okay. So, how many strands of lamb's
wool if we assume each one is a cubic
millimeter? Just need to know how many
cubic millimeters there are in a cube 10
km wide. The number of strands of lambs
wool is that's my 10 km in meters.
10,000 m. Put another th00and on them.
That's now my 10 km in millimeters. We
cube it because it's a cubic pit. So,
that's the number of strands of lambs
wool. And then we'll multiply by 100
because we're removing one once every
century. So you do the calculation and
this is 10 to the 23 10 the^ 23 years.
The pit year the paleo palmer is
an absolute minimum minimum 10^ the 23
years. Okay. But that's just the start.
>> Okay. And they but they weren't using
this for any mathematical reason. It was
just kind of like oh it's such a big job
it's going to take me a poly yo palmer
to do it. like it it would be just like
vernacular like oh like a zillion years
or or were they using it in some kind of
mathematical way?
>> They were they did do mathematics with
some very big numbers which we'll come
on to. Um these periods of time
um they might have used it in the
vernacular I'm not sure but what they
definitely did is they uh they built the
the religious mythology out of this. So
um these periods of time were considered
to be real periods of time. Um and if
you wanted to date, you know, date the
the universe since the uh since the the
date of creation, this is the kind of
unit you would uh uh need. In fact, you
need much bigger units. So, we'll move
on to the next one. The next one is the
Suro prama, which is the ocean year. And
this has a nice uh simple definition.
It's 100 million paleoparms. So, it's a
100 million of the previous things.
Okay. So that's going to be I mean it's
at least 10^ the 31
years in in the mythology of Janism. The
universe runs on a cycle and
it's the cycle started round about a
quadrillion
of these ocean years before today. Okay.
So the start of the cycle and of all of
this is of course an underestimate was
around 10^ the 15 that's my quadrillion
of those ocean years before today. So
that's round about 10 to the 46. Of
course
>> that's their sort of big bang for lack
of a better
>> Yeah I think so. I mean I think it was a
sort of endlessly repeating cycle. So I
don't think they have actually
>> some people think the big bang does that
too but
>> well yeah well indeed indeed. Um all
right. So that's the that's the start of
the cycle. They did also think about
periods of time beyond this. Um and in
particular they thought about periods of
time which
encompass more than an entire cycle.
Sometimes they had a unit of time called
a pervanga which is defined to be 84 *
100,000 and then everything there is
measured in a unit a sort of fundamental
unit called pervies which is a number of
days 756* 10 the 11 days. This 100,000
is uh the word for that is a lack. So
it's 84 lakh pervies. This isn't days
but that's that's just the sort of first
level. So the next level is obtained by
squaring. So the next one is one perver
which is 84. Well I'll just write that
as 8,400,000.
This time we square it and then we're
counting in peries again. You can see
how it's going to go. We're going to
keep increasing that exponent. The next
one's called a truy tanga which is the
same thing. And then this goes as far
the top one of these which is their
biggest unit of time as far as I'm aware
anyway is called one shera pelica which
means the top riddle which I think is a
great name for a massive number which is
up to 28. So we go 8,400,000
to the^ 28 * 756 * 10 11.
>> What's that in years? What what sort of
exponents we up to here? Do you know?
round about 10^ the 206
>> years 10^ the 26 years round about I
mean bearing in mind that you know the
universe as we understand it to be at
the moment is 13 billion years ago so
this is this is you know way way way
beyond that if we sort of extrapolate
the current cosmological uh models
probably we're this this period of time
would take us past the point where all
the super massive black holes in all the
galaxies have evaporated So that will be
you know a very dark and empty universe
by that point if it still exists
>> like it seems very arbitrary. So this
original number the one pervanga so the
84 lakh pervies
was said to be the lifespan of the
original founder of Janism
>> and that's like a really long time
obviously
>> it's over a quintilion years
>> right
>> yes so a lot of the mythology of Janism
happens over these sort of time scales
which really no one very few other
people think about yeah
>> is there anything else
>> yes there is we haven't got to the
biggest numbers yet
>> oh we're going bigger
>> we're going bigger Yeah. All right.
Paper.
>> I think more paper. Yeah.
>> So, as well as contemplating very very
long time scales, the ancient James also
developed a theory of very big numbers
just for their own sake, not really
representing anything particular, just
really as a an investigation into
immense numbers. And they they they
classified them in different ways. And
in particular at the upper end they had
the concept of an uninnumerable number.
And the idea of an unnumerable number is
that it's a finite number but it is so
big that for practical purposes it's
basically infinite. I think that's the
that's the general idea and and maybe
it's worth saying that in in modern
mathematics we don't really have that
idea. So there's a there's a description
of the first unnumerable number which is
comes out of a book written around 1,000
CE by someone called Nemichandra and the
book is called trilocasara which I
understand translates to the essence of
three worlds and in this book he gives
this fantastic description of a really
big number the first unnumerable number
okay and it takes as its starting point
the jog the sort of mystic geography of
the the plane on which we all live.
Okay. And so in the middle of this plane
is an island called this is Jamboo
Island. That's where we live. It's very
big. Its width in the traditional me
measure of yojan is 100,000. Translating
into miles that's over half a million
miles wide. So we got this big island
around half a million miles wide. And
then outside of the island, we've got an
ocean going all the way around, right?
That's called the salt ocean. And then
outside that ocean, we've got another
sort of continental island or annular
island. That's Fireflame Bush Island.
And then outside that, you can see where
it's going. Outside that, we've got
another ocean. And then outside that,
we've got another island. And so on. And
this carries on. I mean there's
different accounts but for this thought
experiment this carries on indefinitely.
Okay. But it's not just that um we've
got these islands and oceans and islands
and oceans and islands and oceans. Their
size is very important. The first island
is around about half a million miles
wide. And then the first ocean is double
that. So I've not drawn this to scale.
The first ocean is double that. So it's
around about a million miles wide. And
then the next island is double that. So
it's around about 2 million miles wide.
And then the next one's double that and
so on. So each one is double the width.
So exponential growth just baked into
the the geography of the place we're
working. Right? So that's the
background. That's the setting for this
thought experiment. So then the first
thing we do, we dig a cylindrical pit
under the first island, Jamboo Island.
And what we do is fill that pit with
mustard seeds. So this whole thing is
going to be is going to be a quantity of
mustard seeds. Okay. So the depth of the
pit it's a thousand yanas which is 5,000
miles or something. So 5,000mi deep pit
under the entire island. The rule is
that the height of the mountain needs to
be 111th of the circumference of the
circle.
>> Of course.
>> Of course. Obviously. Right. I brought
some mustard seeds. Would you like just
see how big they are?
>> Yeah. Go on then. Just in case you've
never seen a mustard seed.
>> Yeah. So that's those those are mustard
seeds.
>> Yeah, they're pretty small.
>> They're pretty small. I mean, I don't I
don't know if it's exactly the same kind
of mustard seeds they were having in my
bed. They're pretty small things. I
mean, it's roughly speaking the same
size of a grain of sand. I've got to get
rid of these mustard seeds.
>> Okay.
>> Okay.
>> Oh, they've gone all over the table.
>> That was bound to happen, was it?
>> Oh dear. Yeah.
>> Bound to happen. Okay. We were
surrounded by mustard seeds, but not as
many as um that were about to be
appearing in this pit.
>> At the moment, this is about 5,000 mi
deep. And then this thing is
>> 11th of the circumference of the pit.
>> So that's so that's
>> it's really tall.
>> Yeah. Like it's
>> I mean it's thousands of miles tall.
Thousands of miles tall.
>> Yeah. Very tall. Yeah.
>> It's very very very tall. Already that
mountain of mustard seeds is big enough
that you could fit planet Earth in it
like loads of times and it's already a
massive number. Okay, but we're just
getting started. So, and now this is the
clever bit because what you do now is
you take this collection of mustard
seeds, right? And you put the first one
on the island and the second one in the
first ocean and the third one on the
next island and the next one on the next
ocean and the next one on the next
island and so on. And you keep doing
that until you've completely exhausted
the whole mountain. Right. And that's
taken you to some eventually you've got
to some other island or or ocean
>> depending on if it was an even or odd
number of states.
>> Yeah. Exactly. And then you do the same
thing all over again. So you build you
dig a pit the same depth 5,000 mi deep
under that whichever disc you've reached
>> which by now will be very very wide
>> actually
I I did a sort of back of the envelope
calculation and it's something like okay
how wide is it something like 10^ the
10^ the 40 light years wide
>> right okay
>> right it's that it's that sort of thing
okay which um you know bearing in mind
the the the observable universe is um 10
to the 11 light years wide or something
is enormously bigger. So you make a
circular ditch under the under the under
the continent you've reached and you you
build your mountain of mustard seeds
again. So that gives us a new a new
mountain.
>> Yeah. One which is
>> 111th again is our uh
>> one 11th of the circumference. Yeah.
>> Yeah. Yeah. The circumference which is
10^ the 10^ the 40 lighty years
>> or the diameter which is 10 to the 10 to
the 40 lighty years. Oh yeah, the
circumference even.
>> Yeah, but I mean when you're multiplying
a number like that by pi, it doesn't
make it doesn't make much difference. So
yeah, I mean at this sort of scale,
these sorts of numbers, it sort of stops
mattering whether you're measuring in
millimeters or light years just because
the the numbers are big so big. This is
the second mountain of must sees which
is you know so enormous that you know
the the observable universe is just an
invisible speck compared to it. Okay.
And then what do you think we do? We
distribute those seeds out ring by ring
by ring by ring.
>> We do. We do. And then that takes you to
another another place where you build
another mountain. Uh you keep doing it.
How many times do you repeat the
process? Uh the answer is the cube of
the number of mustard seeds in the
original mountain. And that a back of
the envelope calculation suggests that's
around 10^ the 45 seeds in the first
mountain.
So we repeat that process the cube of
10^ the 45 times
>> and have you back of the enveloped how
big this final number is. So the final
number. Yes. So maybe I should say I
should credit the mathematician and
historian Radha Char Gupta who in 1992
did a sort of modern mathematical
analysis of this situation which is what
I'm following here. And he so he did he
did the uh the calculation and I mean
this the number that we get after
completing this that is the first
unnumerable number. Let me try and say
it in Sanskrit. uh so jaga parita
asamata the first uninable number I mean
it is so big that if you were to try to
you know write it as a power of a tower
of tens or something you can't because
the the tower is just too tall we can
use canth arrow notation and I I'll just
explain this in a moment so let's say
it's something like this number 10 to
the 10 to the 10 to the 45 two can
arrows 10^ the 135. So this is an
approximate value for the the first
uninumeral number. What that means is if
we built a tower an exponential tower of
x's and I want my height of this tower
to be 10^ the 135. So far taller than we
could ever ever draw. And then each
value of x is this number 10 the 10^ the
10 45. That is about the scale of the
first unnumerable number in Jane
traditional mathematics.
>> We've talked about some big numbers even
in this room before you know we could
you listed some of them at the start of
the video.
>> Yeah.
>> Is when we when we talk about tree
threes numbers that sort of thing.
>> Is this in that ballpark? Is it still
not really coming close? Is it
>> I think it's fair to say that all the
ones I listed at the start are much much
bigger. Um but it is big enough that it
defeats our attempts to write it down
using just traditional mathematical
notation. So we need canoe arrows or uh
something I mean to consider numbers on
this scale
you have to develop the kind of
machinery which can then take you to
places like Graham's number. So they are
you know on the road towards that sort
of territory. It's a testament to the
size of those big big Grahams numbers
that you did this crazy thing that we
were just almost almost laughing at how
big it is and then you at the end you
said oh no it's still not close to those
ones.
>> Yeah, that's right. I mean those numbers
are on on another scale. But I think
it's fair to say that this came
thousands of years earlier, right?
Thousands of years earlier. So it's
taken us it actually took you know there
was a big hiatus in terms of the biggest
numbers people had thought of. Um it was
the ancient James with this number and
others. They had others which are sort
of around around this. They went went a
little bit bigger than this actually.
This is the most fun one to talk about.
>> It's almost like you needed things like
canth notation and some of the new
notation that mathematicians use now
before you could start playing and
inventing those bigger numbers. Yeah.
They just didn't have access to that. uh
>> I mean they started to so they did start
talking about like repeating processes
large numbers of times and the sort of
process is an abstract arithmetical
operation and that's sort of how you do
it right I mean that's what the canth
arrow is you talk about you've got some
arithmetical operation and then you say
okay I'm going to iterate that a large
number of times I've seen an account of
um ancient Jane writers We might have
got as far as three kith arrows 10 three
kith arrows 38. So that's sort of two
levels beyond exponentiation. Right? The
two kith arrows is is a level beyond
exponentiation. That's two levels
beyond. No one else thought about
numbers anywhere near as big as this. So
there was there's a real sort of hiatus
from the ancient js who were doing this
thousands of years ago. and the rest of
the world only really caught up in the
sort of second half of the 20th century.
So they were the record holders for most
of history. If you enjoy seeing Richard
here on Number File and Big Numbers are
well your thing, then you really need to
check out Richard's new book called Huge
Numbers. The cover will probably look
like one of these depending on where you
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