Okay, let's have another crack at how big
Graham's number is.
Let's start with
3 arrow 3, this is just 3 to
the power of 3, bang bang bang. That's
27. The arrow meant this number to this
power, simple enough. Now
how about 3 double arrow 3? When
we have a double arrow we're essentially
saying take this leading number and
string it out with single arrows like
this and do it this number of times. So
if the second number had been, say, 10
we'd have had 10 3s like this.
Luckily the second number was just a
3 so we have a manageable three
3s; 3 arrow 3 arrow 3. Now
we know what a single arrow means, so 3
to the power of 3 arrow 3.
Now we know from a moment ago that 3
arrow 3 was 27, so we have 3 to
the 27; bang bang, bang, bang, bang, that's
about 7.6 trillion. Another
way to think about double arrows, which
is very helpful, is that you are building
a tower of this number of this height. So
in our example, a tower of 3s which
is 3 high. And here when you work it
out you go from top to bottom and right
to left. A 27 here, 7.6 trillion
overall. So there's 3 double arrow
3, a pretty big number;
what about 3 triple arrow 3? Well
just like double arrows created a string
of single arrows, triple arrows create a
string of double arrows. Take the lead
number, again 3 in our example, and
string it out like this: 3 double
arrow 3 double arrow 3 double arrow; etc etc.
Again, if the trailing number had been
say 10 it would have looked like this
with ten 3s, but again luckily the
trailing number here was a measly 3
so we only need three 3s. This is
what we're left with: 3 double arrow
3 double arrow 3. Now remember
what I said about double arrows
representing that power tower? So now
we're saying we're towering this number,
3, and the tower is this number high,
3 double arrow 3. Now we already
saw that 3 double arrow 3 is
actually about 7.6 trillion;
so that's our tower height. Imagine that,
a tower of 3s that is
7.6 trillion digits high. Starting at the
very top, working right to left, here we
are at the summit; we have 3 to the
3, that's 27, then down a step
7.6 trillion now. And now we work
all the way down this epic tower,
trillions of times; the number getting
crazily big with each step down. By the
time we get down to base camp here at
the very bottom we will have created a
number so big it doesn't even bear
thinking about.
Isn't this arrow notation amazing? Just
adding single arrows between the 3s took
us from 27 to 7.6 trillion
to a number that seems beyond
comprehension. Now you can probably guess
what a monster 3 four arrow 3 is going to
be. This expands to 3 triple arrow 3
triple arrow 3. Now triple arrows mean
3 double arrow followed by this
chain.
But now the trailing number that tells
us what to put here and how long the
chain is going to be happens to be 3
triple arrow 3; and we know that
number is ridiculous
so this chain is going to be pretty darn
long. And knowing what these double
arrows mean, we have to start
constructing another of these enormous
power towers, but this time the height of
our tower is going to be figured out
using this beast of a number. So let's
see how this plays out; we start here
with 3 to the 27, already a
pretty big number, but that just tells us
the height of the tower below, how many
3s that are here. But this number
just tells us the height of another
tower of 3s below that. And this
continues, each whopper of a number
simply telling us how many 3s there
are in the tower below it; you keep
building down. And how many layers do you
build down? Well this number, 3 three
arrow 3 which is unimaginable in itself;
so can you imagine how big the final
tower of 3s must be after we've
iterated all the way through this
process? Can you imagine that? I think
you're probably can't.
Let's call this monstrosity G1. Is that
Graham's number? Nowhere near it.
What we do is we now take another pair
of 3s and the number of arrows
between them is 3 four arrow 3 -G1. That's
not the number of digits here, that's not
even the number of powers we're talking
about - that is the number of arrows; and
we've shown how quickly arrows can blow
things up beyond belief. Now we can't
even count the arrows! - Boy, that escalated quickly.
Let's call this new number G2.
Now take another pair of 3s; how many
arrows are we going to put between these
ones? You guessed it - G2. This gives us G3.
Then we do it again and we keep
repeating the process; the huge number we
get becomes the number of arrows in the
next number. And you keep doing this
64 times. When you finally finish,
there you go,
you have Graham's number.
This is the upper bound to an answer to
a reasonably simple question involving
coloured lines between vertices on cubes
in higher dimensions. That's explained in
another video by Ron Graham himself.
Check out the links that are on the
screen or in the video description.
But this is such a big number, that 2 to
that power and that power are kind of
hard to distinguish, it's such a big
number.