字幕列表 影片播放 列印英文字幕 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.