Hopefully by now you've seen our video
on what Graham's number is. If you
haven't, it's the number of dimensions
that a cube must exist in before a
certain configuration of coloured lines
between its vertices must exist. Now this
number is famously very very big. How big
is it? Well we're going to let the man
who first wrapped his head around it, Ron
Graham, explain it to you.
Well I can't really write it yeh; I have
to explain the notation okay, so we use
some notation. So called arrow notation,
and often attributed to Don Knuth,
computer scientist at Stanford, who's very
creative about notation. And the notation
is this; so suppose you write 3
arrows n, that just stands for 3 to
the power n, ok? So 3 arrows 10 is 3
to the 10th power, that's a pretty big
number, ok, so that's- why do you need
arrows to write 3 to the n? Well, you don't,
ok. Well, what do you mean by 3 two
arrows n? Well what that means is 3 to
the arrow of 3 to the arrow of 3 to the
arrow of - dot dot dot dot - 3 to the arrow of 3; and
there are n 3s. So ok, so what does
this mean? 3 to the arrow of 3 is what?
Well it's 3 to the 3. 3 to that arrow is
3 to the 3, 3 to that arrow is 3 to the 3...so
it's a tower of 3 to the 3 to the 3 to
the 3 - there are n 3s. So for example
3 double arrows 3, how much is
that? Well that's 3 single arrow
3 single arrow 3; well 3 to the 3, this
is 3, 3 to the 3 is 27, 3 to 27 which is 3
to the 27 or 3 to the 3 to the 3. 3 to
the 27th power is a pretty big number-
I think it's worth stopping here for just
a moment to let things start sinking in.
Two 3s with one arrow between it,
that's 27. Put two arrows between them
and things escalate to up over seven
trillion, that's a pretty quick
escalation. What happens if we use three
arrows? - So this is 3 double arrow
of 3 double-arrow of 3. What is
3 double arrows 3?
Well it's 3 to the 3 to the 3, okay? 3 to
the 27. So this is 3 double arrows of
anything is 3 to the 3 to the 3 to the 3
to the 3 - and how many? 3 to the 27. 3 to
the 3 to the 3 to the 3 to the- 3 the
27th power. Well that's getting to be a
pretty big number. - (Yeah I think that's an understatement!)
So three arrows has given us a number so big I can't
even write it on the screen,
except maybe like this. - Okay, so now I
think we ran out of this piece of paper actually.
(You want some more?) - Yeah I think we do.
You ain't seen nothing yet,
okay so here we go: 3 four arrow, one more
arrow. Well, what does this mean? 3
three arrows of 3 three arrows of
3. Well, this is 3 triple arrow of
something, what is this number? You
saw this number right here, right, this is
3 to the 3 to the 3 to the 3 to the 3.
Well okay, so what does 3 triple arrow
of anything mean? It's 3 double
arrows of 3 double arrows of 3
double arrows of - finally the end - 3 double
arrows 3. And how many 3s? This
many 3s. So it's a pretty big number.
This is 3 to the 3, 3 to the 27th. And
this is a tower of that high, 3 to the 3
to the 3,
that's this 3 to that. And then the next
one is a tower of that height, and the
next one is a tower of that height. And
you repeat this for 3 to the 3 to the 3
to the 3 to the 27 times, so there's
that many layers. Each one tells you the
number of 3s in the layer below.
Now four arrows has given us a number so big
that mere mortals like us shouldn't even
bother trying to imagine it. You can see
I've just used the term insane number.
And by the way, we're not even in the
ballpark of Graham's number yet.
Now what is this final number we're looking at?
This so called Graham's number? It's a
3 four arrows 3, that's a big
number, but what it represents is the
number of arrows between these two
3s. Remember each additional arrow
put you into a much bigger world, well
now how many arrows do you have? Well
this many. But that's a big number-
So I hope you grasped that adding just one
arrow takes things up by a crazy amount.
Now we've got two 3s and the number
of arrows between them is the insane
number that I couldn't even write down,
and that's the number of arrows. But
that's not it - once you take that 3
insane arrow 3 number you spit out
another number at the end that- I don't
even want to think about that one. You
then take that number and now that's the
number of arrows between the 3s. And
then you repeat that again and again and
again- - And each number tells you how many
arrows are in the thing just below it.
And you keep this process up, each one
telling you how many arrows - 64 levels.
That's the definition of this grand result.
Already, to understand 3 four
arrows 3 is not so easy. But when you
say that's how many arrows are in this
number, and then that number is how many
arrows in the one and so forth, it's- I
think it's unlikely anyone will know the
leading digit of this number.
The last digit you can actually get, the
last- I think there's a list, the last 500
digits you can- because it turns out
you take a big enough power of 3, the
last digits
and are very predictable. But the first
digit, I don't know and- - (It's your number,)
(what do you want it- the first digit to be?)
Well in binary it's 1 so there you go.
So there we go, Graham's number, and if you
think you understand it you probably
don't. And this is the upper bound on the
number of dimensions that a cube lives
in to force this configuration of lines
between vertices - you can see the other
video to have it explained. It's the
upper bound. Now we know it doesn't work
in twelve dimensions, but it could work
in thirteen dimensions. - Could be, yeah,
could be 13. - But it's definitely less than that?
Definitely less than that, yeah. So the-
it's called a small gap in our knowledge.
Somewhere between 13 and Graham's number.
Now Graham's number was once in the
Guinness Book of Records as the biggest
number ever used in a sort of
constructive proof; it's since been
surpassed by other mathematicians but
Graham's number was the first of these
big boys and it still got that bit of
glamour involved with it. If you'd like
to find out more about the number check
out our original video from a couple of
years ago, Ron Graham explaining what
Graham's number is, and also some extra
footage from Ron Graham that ended up on
my virtual cutting room floor. You can
check them out. There are links on the
screen and underneath and thanks a lot
to Ron Graham for telling us all about
Graham's number.