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.
Mine is Dave's number. It's Graham's number plus 5, so I know it ends with a 2.
through out the entire video I was waiting for them to explain how they got the number 7. but they didnt. 1/10
If you like that you'll love TREE(3) https://www.youtube.com/watch?v=0X9DYRLmTNY
Honestly, the post title is selling it short. Graham's number is so large that if you took its square root, then took the square root of that number one second later, then continued taking the square root of the resulting number every second until the heat death of the universe, the final number still wouldn't fit in the observable universe.
Mine is Jenny's number. We know that it ends in 8675309.
I love numberphile. The whole channel is full of interesting mathy type stuff.
I learned about Graham’s Number from Day9.
https://youtu.be/1N6cOC2P8fQ
That's Numberwang!
That’s Numberwang