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# Graham's Number

Try a kiloGraham.

posted by Eideteker at 4:42 AM on November 9, 2012 [3 favorites]

Up Arrow Notation (aka "Knuth Up Arrow Notation") is to exponentiation as exponentiation is to multiplication and as multiplication is to addition.

2 + 2 + 2 +...10 times => 2 x 10

2 x 2 x 2 x...10 times => 2

2

(what a terrible up arrow--what's the real unicode up arrow notation arrow?)

posted by DU at 4:49 AM on November 9, 2012 [3 favorites]

Not quite. The continuation of that chain from addition through multiplication to exponentiation is tetration. The UAN is more general, and starts earlier in the sequence than you suggest. The single arrow is exponentiation, the double arrow is tetration, and the triple arrow continues from there and so on.

posted by edd at 5:09 AM on November 9, 2012 [2 favorites]

Graham's number is useful! It keeps you regular

posted by GenjiandProust at 6:55 AM on November 9, 2012 [3 favorites]

Mathematicians giving patronizing explanations of abstract concepts to non-mathematicians frightens me. There is no accepted mathematical notation of "stupidly big."

posted by charlie don't surf at 8:19 AM on November 9, 2012

That apparently is actually a thing, Space Kitty.

posted by phl at 9:44 AM on November 9, 2012 [1 favorite]

Not only that, but if you turn it upside down, the last seven digits spell "BOOBIES". Try it!

posted by albrecht at 9:53 AM on November 9, 2012

Yeah, I meant the last seven digits after you flipped it over...

Let me try to make amends for my lulzy comment by saying something that I think is seriously interesting about Graham's Number, which is that I think it shows the difference in the ways that we think we "understand" numbers. That is, we have different regimes for conceptualizing numbers based on their sizes: small numbers (on the order of 10) we can think of in terms of picturing groups of objects; slightly larger numbers (maybe 100-1000) we might think of in terms of their arithmetic properties, whether they're primes or powers or whatever; larger numbers still (the thousands up into the trillions and quadrillions) we might have descriptive names for; and at some point we are only accustomed to really thinking about numbers in their decimal expansions. What Graham's Number shows is that even that fails us at some point when we venture into the realms of numbers with more

So the question is: what does it really mean to know a number that big? The video makes the point that the entropy of Graham's Number is greater than the entropy of a black hole, but that's a really flawed way of thinking about entropy, treating the number as a random sequence of digits (or, more accurately, bits). What's probably more appropriate is to consider its Kolmogorov Complexity, which is something like the length of the shortest computer program that could generate the number. We know that such an algorithm exists that's fairly short, because we can describe the number in terms of constructing it, defining the up-arrow operators and iterating some number of times, etc. So in a sense, we know the number would fit in our heads because it's already there. What's really amazing about it is that it comes about organically as a solution to a fairly simple problem.

posted by albrecht at 12:30 PM on November 9, 2012 [1 favorite]

I think that being able to increase the mass-energy density of my brain by thinking very hard would be a great superpower.

posted by fantabulous timewaster at 2:50 PM on November 9, 2012

OK...

But how many integers are there? Infinity? How many even integers are there? Half that, or the same amount? How many real numbers are there between 0 and 1? More or fewer than the number of integers, or is it all the same? I think you're playing a little free and loose with the word "infinity."

posted by King Bee at 7:20 AM on November 10, 2012

Post

# Graham's Number

November 9, 2012 3:02 AM Subscribe

"If you actually tried to picture Graham's number in your head, then your head would collapse into a black hole." SLYT discussion of a number beyond "stupidly, stupidly big." Previously.

I'd think that's an understatement. You're talking black holes the size of a stupidly big number of universes, I'd have thought.

posted by edd at 4:08 AM on November 9, 2012

posted by edd at 4:08 AM on November 9, 2012

I had the opportunity once to hear Ron Graham give a talk about stuff computers could never ever be big enough to handle. Great fun, I was clearly the dumbest guy in the room but it was cool to listen to the audience very quietly go "ooh, ahhh.... huh" over and over.

posted by sammyo at 4:11 AM on November 9, 2012

posted by sammyo at 4:11 AM on November 9, 2012

If I'd known "= stupid big" was a valid math answer I would have used it on tests.

posted by The Card Cheat at 4:14 AM on November 9, 2012 [8 favorites]

posted by The Card Cheat at 4:14 AM on November 9, 2012 [8 favorites]

OK, not to rain on these guys' parade, but I despise when maths folks replace a perfectly good notation (for us common folk) aka the exponent thing and then just plop an up arrow in it because the up arrow is clearly superior, but they never tell us plebs why they need to do that! Also, the fact that they needed parentheses to specify the order of operations tell me that your up arrow is lacking in information.

Also, screw Graham's number, I much prefer the eminently more useful Graham Cracker. That's just me, however.

posted by InsertNiftyNameHere at 4:32 AM on November 9, 2012 [1 favorite]

Also, screw Graham's number, I much prefer the eminently more useful Graham Cracker. That's just me, however.

posted by InsertNiftyNameHere at 4:32 AM on November 9, 2012 [1 favorite]

*"Graham's number + 1"*

Try a kiloGraham.

posted by Eideteker at 4:42 AM on November 9, 2012 [3 favorites]

The point about replacing the exponent with the arrow is that the arrow notation can be expanded in a way that exponents cannot easily.

3^3

3^^3 = 3^(3^3)

3^^^3 = 3^^(3^^3) = 3^(3^(3^(3^(3^3))))

And you're already going beyond what you can easily write with exponents.

posted by salmacis at 4:45 AM on November 9, 2012 [5 favorites]

3^3

3^^3 = 3^(3^3)

3^^^3 = 3^^(3^^3) = 3^(3^(3^(3^(3^3))))

And you're already going beyond what you can easily write with exponents.

posted by salmacis at 4:45 AM on November 9, 2012 [5 favorites]

*just plop an up arrow in it because the up arrow is clearly superior, but they never tell us plebs why they need to do that!*

Up Arrow Notation (aka "Knuth Up Arrow Notation") is to exponentiation as exponentiation is to multiplication and as multiplication is to addition.

2 + 2 + 2 +...10 times => 2 x 10

2 x 2 x 2 x...10 times => 2

^{10}

2

^{22...10 times}=> 2↑10

(what a terrible up arrow--what's the real unicode up arrow notation arrow?)

posted by DU at 4:49 AM on November 9, 2012 [3 favorites]

I tried to picture it.

I know what you're going to say..."watch out for the black holes!"

Well, I'm just a rebel like that. A big math number imagining rebel.

posted by Reasonably Everything Happens at 5:03 AM on November 9, 2012

I know what you're going to say..."watch out for the black holes!"

Well, I'm just a rebel like that. A big math number imagining rebel.

posted by Reasonably Everything Happens at 5:03 AM on November 9, 2012

*Up Arrow Notation (aka "Knuth Up Arrow Notation") is to exponentiation as exponentiation is to multiplication and as multiplication is to addition.*

Not quite. The continuation of that chain from addition through multiplication to exponentiation is tetration. The UAN is more general, and starts earlier in the sequence than you suggest. The single arrow is exponentiation, the double arrow is tetration, and the triple arrow continues from there and so on.

posted by edd at 5:09 AM on November 9, 2012 [2 favorites]

If you possess a stupidly big quantity of cheese, you should serve it with Graham crackers.

posted by Faint of Butt at 5:15 AM on November 9, 2012 [1 favorite]

posted by Faint of Butt at 5:15 AM on November 9, 2012 [1 favorite]

>

Isn't it almost immediately obvious? I mean, sure, nobody is going to

posted by King Bee at 5:28 AM on November 9, 2012 [9 favorites]

*I despise when maths folks replace a perfectly good notation (for us common folk) aka the exponent thing and then just plop an up arrow in it because the up arrow is clearly superior, but they never tell us plebs why they need to do that!*Isn't it almost immediately obvious? I mean, sure, nobody is going to

*really*use 3↑3 when they can just write 3^{3}. But 3↑↑↑3? That's a tower of 3s which is 3^{33}high. You need 7.6+ trillion symbols for that, whereas up arrow notation gets it done in five.posted by King Bee at 5:28 AM on November 9, 2012 [9 favorites]

*Also, screw Graham's number, I much prefer the eminently more useful Graham Cracker.*

Graham's number is useful! It keeps you regular

*and*deters masturbation, much like the cracker, but it has a more amusing notation system.

posted by GenjiandProust at 6:55 AM on November 9, 2012 [3 favorites]

3^^^3 is an exponent tower of 3s that is 7,625,597,484,987 levels high. If it took you a second to calculate each new level of the stack, it would take more than 240,000 years to finish. Call that number X. This is already face-meltingly big.

3^^^4 is an exponent tower of 3s that is X levels high. 3^^^5 is an exponent tower of 3s that is 3^^^4 levels high. When you get to 3^^^X, that is 3^^^^3. Even God couldn't visualize this number and it took only 4 arrows to get there.

But 3^^^^3 is only G1. G2 is 3^^...^^3 with G1 arrows. G3 is 3^^...^^ with G2 arrows.

And as a final kick in the teeth, that number is (an outer limit for) the

Mathematicians frighten me.

posted by Egg Shen at 7:22 AM on November 9, 2012 [5 favorites]

3^^^4 is an exponent tower of 3s that is X levels high. 3^^^5 is an exponent tower of 3s that is 3^^^4 levels high. When you get to 3^^^X, that is 3^^^^3. Even God couldn't visualize this number and it took only 4 arrows to get there.

But 3^^^^3 is only G1. G2 is 3^^...^^3 with G1 arrows. G3 is 3^^...^^ with G2 arrows.

*G64*is Graham's number.And as a final kick in the teeth, that number is (an outer limit for) the

*dimensions of a hypercube*whose vertices caught your interest.Mathematicians frighten me.

posted by Egg Shen at 7:22 AM on November 9, 2012 [5 favorites]

Question they never answer in the video :

Graham's number is supposed to be the largest number that can be used to satisfy this problem, yes? Why? I don't fully understand what the number is supposed to be doing [and venturing to wikipedia just makes me feel dumber].

posted by FirstMateKate at 7:45 AM on November 9, 2012

Graham's number is supposed to be the largest number that can be used to satisfy this problem, yes? Why? I don't fully understand what the number is supposed to be doing [and venturing to wikipedia just makes me feel dumber].

posted by FirstMateKate at 7:45 AM on November 9, 2012

Based upon a probably badly informed reading - they're dealing with graph theory of some sort. How connections are formed. I'm not sure exactly how this relates to hypercubes, exactly, but I'd imagine it's something to do with the vertices of a hypercube. On preview, as Egg Shen said what they're trying to do is find a way to determine the number of dimensions of a hypercube given the number of vertices. At least, that's how I'm reading Egg...

What I'm curious is why 3, exactly? What happens if you said n=x, let n b the number of arrows. so 2^^2, 3^^^3, 4^^^^4, 5^^^^^5, etc... what's the pattern there.

That said, I love the black hole thing because the concept of entropy and blackholes and the holographic principle is fucking fascinating and it's crazy to see a number that exceeds the entropy of the brain.

They say it's impossible to know what it is. How long would it take to calculate it? I imagine greater than the age of the universe, but still...

posted by symbioid at 8:00 AM on November 9, 2012

What I'm curious is why 3, exactly? What happens if you said n=x, let n b the number of arrows. so 2^^2, 3^^^3, 4^^^^4, 5^^^^^5, etc... what's the pattern there.

That said, I love the black hole thing because the concept of entropy and blackholes and the holographic principle is fucking fascinating and it's crazy to see a number that exceeds the entropy of the brain.

They say it's impossible to know what it is. How long would it take to calculate it? I imagine greater than the age of the universe, but still...

posted by symbioid at 8:00 AM on November 9, 2012

*Mathematicians frighten me.*

Mathematicians giving patronizing explanations of abstract concepts to non-mathematicians frightens me. There is no accepted mathematical notation of "stupidly big."

posted by charlie don't surf at 8:19 AM on November 9, 2012

"Space," it says, "is big. Really big. You just won't believe how vastly hugely mindboggingly big it is. I mean you may think it's a long way down the road to the chemist, but that's just peanuts to space. Listen ..."

posted by symbioid at 8:20 AM on November 9, 2012 [1 favorite]

posted by symbioid at 8:20 AM on November 9, 2012 [1 favorite]

I learned from QI that 7 is the final digit in Graham's number, but I'm entirely too innumerate to know if they were joking.

posted by Space Kitty at 8:34 AM on November 9, 2012

posted by Space Kitty at 8:34 AM on November 9, 2012

^_^

posted by ostranenie at 8:41 AM on November 9, 2012

posted by ostranenie at 8:41 AM on November 9, 2012

*I'm entirely too innumerate to know if they were joking.*

That apparently is actually a thing, Space Kitty.

posted by phl at 9:44 AM on November 9, 2012 [1 favorite]

*I learned from QI that 7 is the final digit in Graham's number, but I'm entirely too innumerate to know if they were joking.*

Not only that, but if you turn it upside down, the last seven digits spell "BOOBIES". Try it!

posted by albrecht at 9:53 AM on November 9, 2012

Busy Beaver vs. Graham's Number

I don't know enough to tell if this is a valid proof, but assuming the Busy Beaver is an uncomputable Honey Badger then I think it has a fair chance.

posted by RobotVoodooPower at 10:33 AM on November 9, 2012

I don't know enough to tell if this is a valid proof, but assuming the Busy Beaver is an uncomputable Honey Badger then I think it has a fair chance.

posted by RobotVoodooPower at 10:33 AM on November 9, 2012

One of the things I like about Graham's Number is that it shows you just how far you are from being able to comprehend "infinity".

Unlike something like a Busy Beaver number, Graham's Number is very straightfoward in how it's calculated. You can comprehend the process. Yes, it doesn't take long to realize the that the numbers are ungodly big. But you can get a feeling for the magnitude of the number at first - and realizing that there aren't enough atoms in the universe to even represent the number of DIGITS in the number informs you of that scale.

Just that little bit of comprehension you get from trying to do this gives you the feel of a number that is likely far, far bigger than you ever thought of when you hear "infinity". And then you realize just how little you can understand that concept.

posted by evilangela at 10:50 AM on November 9, 2012 [1 favorite]

Unlike something like a Busy Beaver number, Graham's Number is very straightfoward in how it's calculated. You can comprehend the process. Yes, it doesn't take long to realize the that the numbers are ungodly big. But you can get a feeling for the magnitude of the number at first - and realizing that there aren't enough atoms in the universe to even represent the number of DIGITS in the number informs you of that scale.

Just that little bit of comprehension you get from trying to do this gives you the feel of a number that is likely far, far bigger than you ever thought of when you hear "infinity". And then you realize just how little you can understand that concept.

posted by evilangela at 10:50 AM on November 9, 2012 [1 favorite]

I sort of lost track of the explanation around the time they started talking about committees, due to the progressively-irritating cutting between the two of them. However, I remain confident in my comprehension of one being the loneliest number that you'll ever do.

posted by Halloween Jack at 12:00 PM on November 9, 2012

posted by Halloween Jack at 12:00 PM on November 9, 2012

*There's no 'L' in 'BOOBIES,' albrecht.*

Yeah, I meant the last seven digits after you flipped it over...

Let me try to make amends for my lulzy comment by saying something that I think is seriously interesting about Graham's Number, which is that I think it shows the difference in the ways that we think we "understand" numbers. That is, we have different regimes for conceptualizing numbers based on their sizes: small numbers (on the order of 10) we can think of in terms of picturing groups of objects; slightly larger numbers (maybe 100-1000) we might think of in terms of their arithmetic properties, whether they're primes or powers or whatever; larger numbers still (the thousands up into the trillions and quadrillions) we might have descriptive names for; and at some point we are only accustomed to really thinking about numbers in their decimal expansions. What Graham's Number shows is that even that fails us at some point when we venture into the realms of numbers with more

*digits*than we can even comprehend, which is rightly to be expected when we start compounding exponential operators together (the number of digits being basically the log of a number base 10). You don't have to go out that far to get numbers with more digits than you can imagine.

So the question is: what does it really mean to know a number that big? The video makes the point that the entropy of Graham's Number is greater than the entropy of a black hole, but that's a really flawed way of thinking about entropy, treating the number as a random sequence of digits (or, more accurately, bits). What's probably more appropriate is to consider its Kolmogorov Complexity, which is something like the length of the shortest computer program that could generate the number. We know that such an algorithm exists that's fairly short, because we can describe the number in terms of constructing it, defining the up-arrow operators and iterating some number of times, etc. So in a sense, we know the number would fit in our heads because it's already there. What's really amazing about it is that it comes about organically as a solution to a fairly simple problem.

posted by albrecht at 12:30 PM on November 9, 2012 [1 favorite]

"If you actually tried to picture Graham's number in your head, then your head would collapse into a black hole."Well, I tried, but I couldn't picture the number, and I haven't collapsed into a black hole (at least as far as I can tell). Does that mean that I actually failed to try?

I think that being able to increase the mass-energy density of my brain by thinking very hard would be a great superpower.

posted by fantabulous timewaster at 2:50 PM on November 9, 2012

Infinity is defined to be greater than all finite numbers. Brains, like computers, can only perform arithmetic on finite numbers, and arithmetic here includes what you do to get an estimation of the number of digits for instance. Right out the gate, infinity is defined in such a way that trying to get an idea of how big it is would be totally pointless.

So I don't quite get the cult about it. Is it just a bunch of numerologists saying "my number is bigger than yours"?

posted by LogicalDash at 11:59 PM on November 9, 2012

So I don't quite get the cult about it. Is it just a bunch of numerologists saying "my number is bigger than yours"?

posted by LogicalDash at 11:59 PM on November 9, 2012

I'll always remember when Ron Graham came to lecture at my university when I was in grad school. We had, of course, read about his number and the whole up-arrow notation. But when he came to spend some "informal time" with faculty and grad students, it erupted into an hour-or-so long juggling fest (he a former president of the International Jugglers' Association). He taught us about site swapping, introduced us to Polster's book, and taught my friend Ander to juggle. That was a really great afternoon, and not what I expected.

As for Graham's number, I agree that it's stupidly big. I was trying to get a feel for how big it is, and I discovered the Ackermann function which is related in that it creates very large numbers by recursion. It's actually a cool exercise (for folks new to recursive functions) to try to follow that definition and see how the function grows. If you get stuck, Wolfram|Alpha will evaluate the function by typing Ackermann(a, b) where a and b are according to the definition in the article. It's a short process, though, as Ackermann(3, 2) is 29, but Ackermann(4, 2) has over 19,000 digits.

posted by klausman at 12:02 AM on November 10, 2012

As for Graham's number, I agree that it's stupidly big. I was trying to get a feel for how big it is, and I discovered the Ackermann function which is related in that it creates very large numbers by recursion. It's actually a cool exercise (for folks new to recursive functions) to try to follow that definition and see how the function grows. If you get stuck, Wolfram|Alpha will evaluate the function by typing Ackermann(a, b) where a and b are according to the definition in the article. It's a short process, though, as Ackermann(3, 2) is 29, but Ackermann(4, 2) has over 19,000 digits.

posted by klausman at 12:02 AM on November 10, 2012

*Infinity is defined to be greater than all finite numbers. Brains, like computers, can only perform arithmetic on finite numbers, and arithmetic here includes what you do to get an estimation of the number of digits for instance.*

OK...

But how many integers are there? Infinity? How many even integers are there? Half that, or the same amount? How many real numbers are there between 0 and 1? More or fewer than the number of integers, or is it all the same? I think you're playing a little free and loose with the word "infinity."

posted by King Bee at 7:20 AM on November 10, 2012

Yes, I'm aware of the different cardinalities of infinity. I even think it's cool to know that you can't construct a well-ordering of the real numbers. But it seems like the expected response on mentioning aleph-one is "WHOA, bigger than infinity??" That, I don't get.

posted by LogicalDash at 12:47 PM on November 11, 2012

posted by LogicalDash at 12:47 PM on November 11, 2012

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