Okay, yeah, I've got another big number for you Brady: 1.76 times 10 to the 67. Okay so let me
ask you a question, can you think of a number- you know, completely random number that no one
in the history of humanity has ever thought of? - (Brady: A number no- I think I could do that quite easily.)
Yeah? Okay, I agree. So what would you have to do? What would- how would you do it?
(Well I would just have to make it big and I would just- I would choose like, you know, nine or ten digits and)
(randomly choose those digits.) - Okay. So I think nine or ten digits is probably a bit on the low side,
I think. So this is what this number is all about. It's about thinking of numbers that no one in the
whole history of humanity has ever thought of. This is an estimate that I've come up with based on
stuff that I've seen on the internet. So basically if you go to a number that is bigger than this,
so pick a random number that's bigger than this value, then there's a 99% chance that that number
has never been thought of by anybody ever in the history of humanity. - (Quick interjection before you)
(start: you told me nine to ten digits was a bit on the low side. If I choose a random ten digit number,)
(random, who on Earth will have thought of that in the history of humanity?) - So there will be
some probability that- so okay, so I need to justify this clearly, right it's where this
comes from and I think it's going to be something that's going to spark a lot of debate, we could
argue about whether this is a good estimate or not. I think one thing's for sure, if you
if you think you can get away with a nine digit number and I'm saying, well try a 68 digit number,
then that's definitely going to be a good ballpark. So we're probably in safe territory going this big-
(I think you're high-balling it, but anyway.) - Okay let's see then, let's see where does it come
from? Where's the maths come from behind this? Snd we can argue all day about the assumptions that have
gone into it. So the first thing you've got to ask is, is how many numbers has humanity thought
of? What's the sample size of numbers that humanity have thought of? And this is probably one bit where
people are going to really argue about- about my assumptions but I'm going to make them and that's
it, you could argue. - (How are we defining 'thought of' by the way? Just-)
Here we go, we've already started.
Well let's just- okay so I'm going to say, my assumption is going to be that a typical person
thinks of a number every two minutes of their waking life. Now that's probably an overestimate,
but I want to overestimate- well we can redo it again different number if you want. But let me
go let me go for for that right? I think I want to overestimate because I want to be guaranteed right
that I'm in a good place. So let me make that my assumption. Okay so typical person, life expectancy
today I think is 73.2 years. How many days is that? Well let's just multiply that by 365 and a quarter.
How many hours? That's 24 hours; but we're already awake for around two thirds of the day
so I need to factor in two thirds. And I'm saying- so this is the number of hours that a typical
person would be awake for in their lifetime, and if we say that they think of a number every two
minutes then I should multiply by 30. Now clearly I know that's probably a bit of an overestimate,
people aren't thinking about numbers as much when they're young, Numberphile viewers will
probably think about way more numbers obviously. So we've got to do something
so this is this is the assumption I'm going to go with. - (Unique numbers? Or are we coming to that?)
No, I'm not suggesting everybody's thinking of different numbers, we're all- most of us are
all going to think about lots of low numbers right? I'm a Liverpool fan and this is all I'm thinking
about the moment is the number four, because I think we're gonna win four trophies this season.
(I thought you were going to think of four because Liverpool just beat Man United four-nil.)
Well, there's also that, there's also that. Lots of fours going off in Liverpool at the moment. Anyway so so
let's just say this is the number of numbers that a typical person will think of in their
lifetime - obviously they're not all going to be different numbers okay. This is about, I think about
12.8 million. Okay so now I need to multiply it by the number of people that have lived.
So a bit of googling, you see what a recent estimate for the number of people that have ever
lived is: it's about 117 billion. Which I thought was quite interesting actually because I've always
heard this- you hear people say, 'oh there's more people alive today than have ever lived' - that's
not true. It's about 117 billion is the most recent ballpark figure. So I'm just
going to multiply this by 117 billion. Okay, now I know what people are going to say now is that
wasn't life expectancy less in the past? Yes it was, of course in the iron age people only lived
until they're about 20 or so- - (They probably were thinking about numbers less.) - Probably they were
thinking a bit less, they certainly weren't thinking about Graham's number. So this is the number. So
I'm going to go with this, so again it's probably an overestimate for that reason. So what is this
number? So this I'm going to call it n. And as a ballpark figure this is about one and a half times
ten to the eighteen which is a quintillion, actually it's one and a half quintillion I think. So what
is? This this is the number of numbers, our estimate, probably an overestimate, the number of numbers
that humanity has ever thought of in the whole of human history. - (But there's lots of duplicates like-)
Totally. - (Like a lot of those will be 1.) - Yeah yeah yeah, completely, yeah yeah, vast majority is
going to be a significant proportion is going to be 1. Okay so we have to now think about how are
we going to guarantee ourselves a number that no one's ever thought of. So what we're going to say
is because- this is the mathematical problem we're interested in. We've got a sample size of size
one and a half quintillion, okay. Now we could say right what's the probability that all of that
sample is less than 10? It's going to be something tiny right? It's going to be really negligible,
a random sample that big is going to be all less than 10. You say, what's the probability that
all of that sample is less than um a trillion, okay? It's still going to be a very low probability. So
there's a 99% probability that a sample this big will be less than this. - (What mathematical tool did)
(you use to come to that?) - Okay let me show you. So we first have to think about the distribution of
numbers, okay, that how often they appear, you know, in human thought I suppose right okay. So let's
take some numbers n and let's think about the probability distribution of those numbers. Clearly
small numbers are going to appear much more regularly than large numbers, this is intuitively
obvious. - (Because of like football scores and house numbers and-) - Yeah, just your small numbers just crop
up in everyday life. Also you know also we need to put aside sort of slight anomalous things like
years are going to appear more regularly, um powers of 10 appear more more than their sort
of nearest neighbours and so on and so forth. So we're really interested in sort of random numbers
and looking over it on on sort of trends. I saw somewhere that the number 666 appears much more
regularly than those numbers around it as you I expect. So there's all that going on but I want to sort of
sweep that under the carpet, that's not what we're interested in, we're interested in general trends.
So it's clear that it's going to fall with the size of the number. Question is how fast does it
fall? Now you might think - I know you've done videos about Benford's law before - the fall off goes
like 1 over n. I'm not going to use that and the reason is is - well there's two reasons actually - if
I assume it's 1 over n and this this I can't normalise this probability, these probabilities
can't be added to 1, they they always hit infinity. So so I don't want that. So it's clear to
me that at some stage with very big numbers it's got to drop off more quickly. And I did find there is a
paper from 2005 I think it is where they look at the distribution of numbers on the internet,
random numbers, and they also look at powers of 10 but they also look at sort of random numbers.
And what they find is the falloff is going like 1 over n to the 1.3. That's actually good, that's
faster than Benford's law so it's dropping quicker, it means we can normalise this as well. So I want
to think about sets of numbers that aren't as big as a certain value okay. So this is telling me the
distribution of all the numbers, but what I need to think about now is what's called the cumulative
distribution. So I want to take a given number n star say, and I want to ask, what is the probability
that a random number is less than this n star? n star might be 10, it might be 20, it might be a
billion - whatever. What's the probability that a randomly picked number is less than that
value? Well how do I calculate that? well I just have to sum up all these probabilities. So that
corresponds to the area under the curve, and I can figure out what that is with a bit of calculus. We
don't need to do it now but it's going to be- the answer I get is 1 minus n star to the minus 0.3.
This tells us the probability that a randomly chosen number is less than n star, okay. Now what's
the probability that a randomly chosen number is less than infinity? Well it should be 1 right?
Yeah? So what if I put n equals 1- n equals infinity in there, I get 1, so that makes sense.
What's the probability that a randomly chosen number is less than 1? Well it can't be less
than 1, we're choosing- we're only interested in sort of you know positive whole numbers. So I put n
equals 1 in there and I get zero. So you can see this sort of makes sense, it's kind of working for
those cases. The probability that a random number is less than 10 is about a half in this. Okay but
this is only one number, right, we've got one and a half quintillion of these guys to worry about,
okay. So let's think about our one and a half quintillion. So let's ask a slightly simpler
question first: what if I take um one and a half quintillion numbers and I demanded they're all
less than 10. What's the probability that they're all less than 10? Very unlikely,
I've just told you that one number has a basically a 50/50 chance that's less than 10, so that's 0.5.
If I'm interested in n numbers, or one and a half quintillion numbers being less than 10, I need
to basically take that to the power of what one and a half quintillion which is essentially zero.
Okay? Right, so this is no good. So obviously the answer is not 10, we're not thinking about 10.
If I want to generalise this now to sort of any number, n star, then I just plug in this guy. So
this is basically the probability that all of our one and a half quintillion numbers are less than
n star. Okay, so you've got huge one and a half quintillion numbers, what's the probability they're
less than some n-star, whatever n-star, is? It's that. So now we're ready to answer our question.
Let me draw this as a function of n-star. Well it looks something like this, this is asymptote going
towards 1 - okay again this makes sense because what this is telling you if I take n star very big,
then it's saying that the probability that my- really really- let's take n star to Graham's
number. The probability that my qu- you know one and a half quintillion numbers are all less than Graham's
number is essentially 1, it's very likely. If I take n star very small the probability that my
one and a half quintillion numbers are all less than n star, and that's a really
small number, it's essentially going to be zero. So it sort of makes sense.
We're interested in the point where this crosses the 99% threshold. We can calculate
what that is, we just have to solve an equation, the equation we solve is this:
Okay, and if we solve that then we get that x is 1.76 times 10 to the 67. So let's think about what
this is telling us; this tells us that if we have a set of numbers, random numbers, of one and a half
quintillion, which is the amount of numbers we think humanity has picked, then there's a 99%
chance they're all going to be less than this okay? Conversely if you go beyond that there's
a 99% chance you're going to find a number that nobody's ever thought of. - (Okay.) - You seem skeptical.
(I'm not skeptical of the mathematics, but I think your assumptions are so wrong. Let's look at)
(numbers that have 66 digits, right? So down- one down from where you are.)
(If you choose all the numbers in that bracket, I think there's)
(probably maybe one human or 10 humans that have ever thought of the numbers of a 66-digit number?)
(I don't know, a thousand humans maybe have thought of a 66-digit number in their life?)
I guess- - (Have you ever thought of a 66 digit number?) - I'm always thinking of 66 digit numbers.
I don't know, I mean a 99% probability; I know what you're saying, it's- I don't know, that's
what the maths comes out as. Okay can we go up can we do a bit more? Can we think about the future of
humanity? Okay, all right. So let's think of the future of humanity- I said this would
be controversial right. So future of humanity now. So now we have to ask the question, how many people
are going to live? So we now want to think about numbers that nobody will ever think of, or ever use;
maybe maybe a better way to ever use, maybe we should talk about numbers that people are using.
So maybe you're using numbers at once every, you know, 30 minutes that could be very big.
(And you don't have to think of a number to use it do you?) - Maybe that's how we should talk talk about this.
(Now you realise!) - Okay fine. - (All right So- so have I used- am I using a 66 digit number?)
(What's an example of me using a really big number but not thinking of it?) - I mean when you're doing it
on a computer somewhere and messing around that kind of thing. So we're going to do future,
okay, so so we have to ask how many people are going to live? Okay so so now now this is another
debate. So this again is controversial, okay, so there's something called the Doomsday Argument
which is not without its controversy, has its critics. It goes back to Brandon Carter, cosmologist
in the 1970s. He was thinking about things like the Anthropic Principle. The general argument is that
we don't have a special place in human history, okay, there's nothing special about where we are.
So what you could say from that is, is that there's a 99% chance that we're in the last 99% of people.
Okay? That seems reasonable. Okay, that means there's a 99% chance that we're past the 1% point.
Okay, so the 1% point is- so it means there's a 99% chance that 117 billion
is past the 1% point. Okay? So that means there's a 99% chance that there's no more
than 100 times 117 billion. So that puts the total number of humans that will ever live at less than
117 times 10 to the 11. So this is based on these Doomsday Arguments and us not being special and
blah blah blah, this would be enough a limit on the number of humans that will ever live.
And 99% confidence. It's quite scary because this actually brings the ends of the world quite quite
rapid, it's like within- I'd have to check it, but it's something like 10 to 100,000 years and then
it's all gone. So if you redo all this analysis again then you get that 8.2 times 10 to the 73.
So it's basically saying if you choose a 74 digit number no one ever, either past future or present,
in the history of humanity will think of that number or use that number. - (If I choose a 74 digit)
(number right now, that's like potentially my number forever, no one else will ever have it.)
I think so, if you believe a lot of these assumptions that got into this which might be dubious. Although
we're saying this is probably- you know, I think we can probably agree that these are probably over
estimates. If anything these are probably over estimates, we could probably bring this down quite
quite considerably right. - (So if anyone watching this chooses a 74-digit number randomly)
(it's probably theirs forever.) - I think so, I think so, and this is- I mean I've been
really conservative in these assumptions and I think you agree with that because
you're like- if I think of a 12-digit number like, no one's ever thought of that before.
Right? That's what you think right? And so so I'm coming up with much bigger numbers
and really conservative assumptions, so yeah I'd agree. I reckon if you think of a 74-digit number,
totally random; don't don't do 1 with 73 zeroes after it. If you do something a bit more exotic
then yeah I don't think anybody's ever going to think of that, that's just yours forever mate. - (Nice.)
If you enjoy musing about all this big number stuff you're sure to love Tony's new
book called 'Fantastic Numbers and Where to Find Them, a cosmic quest from zero to infinity.' I'll
include an order link in the video description. Reading the book is like a conversation with Tony.
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