TONY PADILLA: If you actually tried to picture Graham's
number in your head, then your head would collapse to form a
black hole.
That's not just some sort of crazy sort
of pictorial image--
it would.
It would--
you couldn't store that much information in your head.
MATT PARKER: People think mathematicians just basically
look at bigger and bigger calculations, and bigger and
bigger numbers--
which is not entirely true.
But Graham's number I love, because it's the biggest
number that's been used constructively.
TONY PADILLA: Well, because there's a sort of maximum
amount of what we call entropy that can be
stored in your head.
And the maximum amount of entropy you can store in your
head is related to a black hole the size of your head.
And the entropy of a black hole the size of your head
carries less information than it would take to write out
Graham's number.
So the inevitability is if you started to try to write out
Graham's number in your head, then your head would
eventually have so much information that it would
collapse from a black hole.
MATT PARKER: If you start with a small number-- and 3 is a
small number--
what you can do is you can start adding 3 to itself.
So you could do 3 plus 3 plus 3.
And you can keep going.
In fact, what I've done here is I've
multiplied 3 by 3, right?
So you could just do 3 times 3.
That works just as well.
And if you want, you could do lots of these.
You could do 3 times 3 times 3.
And you could multiply it lots of times.
And that's 3 cubed.
TONY PADILLA: OK, and that's 27, so we're happy with that.
I could write this another way.
The way I would write this down in arrow notation would
be I'd write 3 arrow 3.
And that just means the same thing--
3 multiplied by itself 3 times.
Hopefully you're still with me at this stage.
Now, I say, what's 3 double arrow 3?
MATT PARKER: If you do 3 to the power of 3 to the power of
3, we would write that as 3 to the power of, to
the power of 3.
TONY PADILLA: This means 3 arrow 3 arrow 3.
Well, 3 arrow 3--
well, we've already seen that that's 27, so 3 arrow 27.
OK, and 3 arrow 27--
well that's 3 to the power of 27.
MATT PARKER: And if you actually work that out, it
comes out to be around about 7.6 trillion.
Now at this point, you can go wild.
Right?
How many arrows do you want?
So the next one, let's say we did 3 to the power of,
to the power of--
or arrow, arrow, arrow, or whatever you
want to call this--
3.
Well, that is equivalent to 3 to the double, to the double,
to the 3, to the double, to the double.
That's 3 to the power of 3, to the power 3, to
the power of 3.
And that stack--
that stack is 7.6 trillion 3s high.
And you start from the top and work your way down.
And you get an almighty number.
You get a number that is absolutely off the chart.
You couldn't write these numbers down.
You'd run out of pens in the universe.
Don't forget just three 3s stacked
together was 7.6 trillion.
Now we've got a stack of 3s 7.6 trillion of them high.
And the question is, why would you want to know, right?
And so actually the reason we have arrow notation is to look
at very huge numbers.
The famous, the quintessential never-ending--
well, it does end, it's a finite number--
is Graham's number.
And it's the solution to a mass problem.
So in math we do things called combinatorics, where you look
at big combinations.
And we look at networks, which mathematicians call graphs,
and you look at different ways of coloring in graphs.
And so mathematicians looked at ways to color in,
effectively, graphs that are linked to
higher dimensional cubes.
Bear with me for all this.
You can get cubes in higher dimensions and look at
different ways to color them in.
And they tried to count the number of dimensions--
I've got an analogy.
There's a very famous analogy for how this works.
Imagine you've got a group of people.
So you could have, for example, three people trying
to have a relaxing time drinking champagne.
You can then try and select committees, or subsets, from
that group of peoples.
TONY PADILLA: You could put some people in one committee,
some of the people in another committee, and some people
could be in a few committees, and there's a whole bunch of
committees that you could put together.
And then what you do is, you say, OK, I've got all these
committees.
And I'm going to sort of pick pairs of committees.
So committees can form pairs, and each committee can be in
more than one pair, and so on.
And then you say, OK, I've got all these pairs of committees.
And I'm going to give them a color--
each pair's going to have a color, blue or red.
OK.
Now, I ask the question--
how many people do I need there to be, in the first
place, to guarantee that there are at least four committees
for which--
let's get this right--
MATT PARKER: There are four--
There are four committees--
TONY PADILLA: --each pair, made out of those four
committees, has the same color--
MATT PARKER: --and all people appear in--
I forget.
TONY PADILLA: --and for which each member of that committee
is in an even number of committees?
MATT PARKER: The ultimate question is, if I put these
weird conditions on those links of matching up different
committees, what's the smallest number of people
required for that to be true?
TONY PADILLA: So that's the question that Graham was
trying to answer in a very roundabout sort of way.
So, he said, OK, fine--
BRADY HARAN: But he wasn't applying it to committees.
It was for something--
TONY PADILLA: No, it was to do with hyper cubes in higher
dimensions, but it's the same question, essentially.
MATT PARKER: And they worked out that there is an answer--
it's not infinite.
And the answer is not bigger than Graham's number.
And Graham's number was developed in 1971 as being the
maximum possible number of people you need
for this to be true.
And at the same time they worked out the smallest
number, which was six.
So somewhere between six and Graham's
number is your answer.
However, to actually see Graham's number-- we have some
more paper--
we use arrow notation to get to Graham's number.
We start--
and I used 3s for a reason, because you start with a 3--
arrow, arrow, arrow, arrow.
And you call that your first number, and the notation is to
call that g1.
And already don't forget how mind boggling this
number was last time.
This is already off the chart, right?
TONY PADILLA: Let's call this stupidly big.
OK.
All right.
Now we say, well, it's g2.
Well, g2 is a 3 where we've got a lot of arrows.
How many arrows have we got?
We've got g1 of them.
So this was stupidly big.
This is stupidly, stupidly big.
Right?
And then we carry on.
We do g3.
And we get a whole bunch of arrows.
How many?
Well, you guessed it--
g2 of them.
MATT PARKER: And then, the thing is, you're getting
numbers which are beyond arrow notation, right?
This is just-- ah.
And then you keep going, right?
And Graham's number is if you keep doing this, you keep
doing g's, right?
You go all the way down to g64 equals Graham's number.
TONY PADILLA: So it's just unimaginably big--
I mean literally.
That's Graham's number.
What do we know about Graham's number?
Well, we don't know what its first digit is.
We do know its last digit.
Its last digit is 7.
The part we know about is the last 500.
The last one is 7.
MATT PARKER: People say, how big is it, right?
And you can't even describe how many digits this number--
you can't.
The number you would need to say how many digits there are,
yourself, you couldn't describe how many digits.
And then--
Ah!
And so the answer to this problem is somewhere between 6
and Graham's number.
Recently, though, mathematicians have narrowed
it in even further.
I think it was early 2000, someone pulled in to be
between 11 and Graham's number.
So we're narrowing in, right.
We're gonna get there.
As far as mathematicians are concerned, 11 to the biggest
number ever used
constructively is quite precise.
Because no matter how big a number you think of, right--
and this is just stupid big--
it's smaller than infinity.
There's still an infinite number of numbers that are
bigger the Graham's number.
So, frankly, we've pretty much nailed it,
as far as I'm concerned.
TONY PADILLA: Yeah, I mean it's not the largest number
being used in a mathematical proof.
There's the sort of tree theorems that use larger
numbers, now.
But you know, back in the '70s it was.
Just an interesting anecdote about Graham himself.
He was actually a circus performer as well as a
mathematician.
He certainly did a few circus tricks when he
came up with this.