I've been thinking about Graham's number for a little while now, and I've been trying to visualize g₁. (that being the first of 64 layers leading to Graham's number (g₆₄)

I started on paper, but quickly gave up. Then I installed the GMP library, with the hopes of writing a simple program to calculate the value of g₁. I'm sure I don't have to tell you that it's still much, much, much too large. Even just that first tiny layer is incomprehensibly massive.

So I wonder, how do you visualize g₁? Or even Graham's number? What analogy could possibly stand to help someone appreciate the size of this number?

(I apologize to the people who aren't fascinated by large numbers. I'm finding that's it's sort of an on/off switch with people: either it's really interesting or not at all.)

You don't. You simply run out of universe while trying to imagine it visually .

I read somewhere that if all of the material in the universe were turned into a pen and ink, there wouldn't be enough to write the number. How many universes would it take? My bet's on 42.

My bet is on g₆₃ .

My bet is on 2. Our universe is just really, really tiny compared to an average universe.

If all of the material in the universe were turned into a pen and ink, there wouldn't be any paper to write the number on.

Our average universe is very small compared to other sets of universes.

I'm not sure I could bet on 2. Unless the second universe is 10^{476849737894483} times bigger than ours, or something like that. Is it? That would be way cool.

It is.

IIRC, even a google (as opposed to googleplex) is much larger than the number of particles in the known universe.

so each atom would need ~= 10^20 particles to overflow 10^100.

I bet Graham was pompous. "My number is larger than yours."

Bah, there's more than a googleplex in the open interval (0, 1).

Universe? Look to the space between you thumb and index fingers, FOOLS!!!!!

Goddamnit.
Googol: 10¹⁰⁰
Googolplex: 10^googol
Google: search giant.

I was waiting on that.

Let's say that g₆₄ would fit in 4.2 x 10^{googolplexgoogolplexgoogolplex} universes the same size as ours. Given that, I challenge you all to be the first to compute g₁ into it's full integer form.

[edit] In ten lines of code or less. And it has to be able to work on at least one computer in existence today, however long it takes.

Graham's number is way worse than that. Not only aren't there enough atoms in the universe to write the number, ther aren't enough to write the number of digits in the number. If you tried to express it in terms of x^y^z^w... - a power tower - there still would not be enough to write that number. You can't even write the number of powers involved! Or, in terms mentioned earlier...

Unfortunately, you can't even write the number of universes you would need with the atoms in this universe, let alone 2 or 42.

Graham's number is huge.

It's not the size, it's how you use them. Really!

Johan, do you have a lil' one ?

In the same meaning as yours: The size of the wand does not count, only it's power.

_
Edited

Never having had any reason to think about, or even be very much aware of, Graham's number, the best I can do in terms of visualising it is .

But hey, at least we know it's odd and not prime!

The notion hit me earlier today that Graham's number is likely to include the full text of several well-known books, encoded in integers.

And thinking of 'G ~= inf' did more to expand my vision of infinity than it did to expand my vision of g₆₄.

And no one has conquered my challenge yet! Mwahaha!

This is not Graham or Googol, this is our own common numbers:
Million = 10⁶
Billion = 10¹²
Trillion = 10¹⁸
Quadrillion = 10²⁴

some_numberillion = 10^{6*some_number}

Well, Americans don't count like this, but that doesn't count.

Centillion would be 10⁶⁰⁰, which long ago was noted in the Guinnes book as the biggest number word. My suggestion would be millillion = 10⁶⁰⁰⁰. So if one million is 10⁶ and one millillion is 10⁶⁰⁰⁰, for each "ill" you add in the word, you can add three zeroes in the exponent. I know it's far from Graham, but at least it's (vaguely) based on our convention of number words.

[edit]
For real big numbers you just stress every second "ill" syllable. That way you can count how many "ills" there are. Try:
One millillillillillillillillillillillillillillion. How many zeroes?

I adapted the layer definition on the Wikipedia page into a series notation that I like better. Maybe it'll help.





Graham's Number is g₆₄.

Too bad it wasn't Conway, but Graham. We would be talking about C64.

I can't get over the size of this damn number. It's tweaking with my head. According to the Wikipedia page on up-arrow notation, 3³³³ (that is, a three-level power-tower of 3's) would require ~1.37TB of storage to write out in full integer form. 1.37 terabytes of space for that number.

g₁ expands into a power-tower that's trillions and trillions of levels tall.

That number came from 3^^^^3(four arrows). g₂ has g₁ arrows between those two threes.

I wonder why that's so relevant. If someone came up to me a said 'How massive is infinity times 2?! It's crazy!' I wouldn't care at all. Graham's number seems more substantial in some way.

[appendectomy] I think it's because g₆₄ represents the point at which some random thing must - necessarily - occur. In some extrapolated way.

What is the purpose of Graham's Number?

It's an upper bound for some number in a problem in mathematics.

Its a friggin big number, get over it. You're not going to be able to use it anyways

Really...big...number...

Courtesy of "All Knowledge Is Strange":
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To use it for both parameters of Ackermann's function. Not my idea unfortunately.

One, two, many.

One too many? Or are you referring to this?

I was reading about this, and would 3^^^3 = 3^^3^^3 = 3^^(3^3^3) = 3^^(3^27) = a power tower of 3^3, with 27 (^3)?

3^^^3 = 3^^(3^27) = 3^^7625597484987 = 3^3..3 (with 7,625,597,484,987 exponential 3's)

But that's only three arrows, so a) I'm a n00b with arrow notation and didn't see some mistake or b) g₁ is actually 3^^^^3 = 3^^^3^^^3 = 3^^^(3^^3^^3) = 3^^^(3^^(3^3^3)) = 3^^^(3^^(7625597484987)) = 3^^^(giant_tower) = 3^^3^^3...3^^3^^3^^3 (with giant_tower 3's) So on and so on.

[edited] I was right. So even the pure exponential form of g₁ is apparently impossible to express.

Whoops, I meant (3^27) (^3)'s. So yeah, I had it right.

And yes...that is g1. And g2 has 3(g1 ^)3. That many arrows. Good lord I can barely imagine g1. I definitely agree with whoever said g64 ~= infinity, as that is the only thing I can imagine.

I also agree that g64 gave me a better idea of infinity than the other way around.

Only a mathematician...

How about this?


A is the Ackermann function. What is h₆₄?

Oh, oh wait, I got it.

C(n) = A( _{A( A(n,n),A(n,n) ),A( A(n,n),A(n,n) )} )^C(n-1)

C(g₆₄) is bigger. And wavy-er.

C(g₆₄) is bigger than h₆₄, yes. But what is h_g64?

11 < h_g64 < C(g₆₄)?

I don't think so. If I read it correctly, C(g₆₄) is h₃ to the power of C(g₆₄-1). That just works out to be a power tower g₆₄ levels high, which doesn't compare to a recursive function for speed of growth.

My IQ is g₆₄

In that case, it wraps around because of field overflow. Negative IQ, anyone?

Actually it wraps around so far that it becomes positive again, and ends up somewhere around 131. If you were as smart as I am you would have realized this

A negative IQ is not really impossible, it's just very unlikely. IQ is defined as the normalized score in an IQ test, with a mean of 100 and a standard deviation of 15 IIRC, so if you're stupid enough, say, 10 deviations from the mean, you end up with a negative IQ. Same for very high ones, but the probability for this to happen is so low that someone with this kind of IQ should consider him/herself dead or never born.

Maybe they should but they won't. A person that far below the average wouldn't be smart enough to realize the improbability of his/her existence, while a person that far above the average would see the logical impossibility of a person considering him/herself dead or never born.

I don't think you'd get a negative IQ even by giving a wrong answer to every question in a standardized test. As Tobias pointed out, it's theoretically possible, but maybe it's accurate to say it's unmeasurable or something.

You'd probably have to eat the test paper to get a negative IQ..

Worse than that, they'd need to be comatose or worse.

For every person with -10 < IQ <= 0 there are n₁ persons with 0 < IQ <= 10.
For every person with 0 < IQ <= 10 there are n₂ persons with 10 < IQ <= 20.
For every person with 10 < IQ <= 20 there are n₃ persons with 20 < IQ <= 30.
For every person with 20 < IQ <= 30 there are n₄ persons with 30 < IQ <= 40.

etc

When we come to IQ 100, we have 50% of mankind. I don't know what n₁, n₂, n₃ etc are, I guess they can be calculated. But my guess is that if there were one person with -10 < IQ <= 0, then n₁*n₂*n₃...*n₁₀ would be more than the number of humans ever existed.