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

• black hole.

• That's not just some sort of crazy sort

• of pictorial image--

• it would.

• It would--

• 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

• And the maximum amount of entropy you can store in your

• 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

• 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--

• 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

• 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

• 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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