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