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  • 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.

TONY PADILLA: If you actually tried to picture Graham's

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Graham's Number - Numberphile

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    陳柏良 posted on 2013/11/17
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