Which number is larger:

```
87
```

(10 -1)! or Tree(4)

87

(10 -1)! is an estimate of the number of connections if every particle in the known universe were connected to every other particle.

Which number is larger:

```
87
```

(10 -1)! or Tree(4)

87

(10 -1)! is an estimate of the number of connections if every particle in the known universe were connected to every other particle.

I think you’re seriously underestimating the size of TREE(4).

The reason we need special notation for the size of numbers like Graham’s number or TREE(3) is because tower powers (e.g. 10^99^99^99) just don’t cut it, let alone anything with real-world (somewhat) grokable analogs like the number of connections if every particle were connected to every other.

Any number you can write down with the notation you learned at high school, is basically nothing compared to Graham’s number. And Graham’s number (I’m told) is basically nothing compared to Tree(3). Tree(4) is probably much bigger than tree(3), but my ability to comprehend scales is stretched to the limit with an explanation of Graham’s number.

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n! is less than n[sup]n+½[/sup]e[sup]1-n[/sup] and is not a way to produce huge numbers beyond the Grzegorczyk hierarchy.

Here is an interesting website that discusses large numbers. I’ll have to re-read it myself since it was decades ago that I read it and it appears Robert Munafo never stopped working on it.

Anyway, (10[sup]87[/sup])! is just a “class 3” number — it barely qualifies as “Large.” It’s larger than the classic class-3 number (‘googol-plex’ = 10[sup]10[sup]100[/sup][/sup]), but is still in the same ballpark as large numbers go.

10 raised to the power of googol-plex is a “class-4 number” and 10 raised to *that* power would be class-5. Graham’s number is so large that to describe it as class-N, N would have to be huge.

Nitpick: I think ‘tree(k), Tree(k) and TREE(k)’ represent three different functions, with the ALL-CAPS one being larger than the other two.

Not to get off track but I never knew there was a sup tag! Does sub work?

n[sub]n[/sub]

It does!

And how did you do the 1/2?

I typed it in as a single Unicode character; I figure it should display correctly in most setups.

I’m still trying to parse the OP’s notation. The term (10-1) seems to be just a long way of writing 9. Omitting the “-1” entirely I think the notation here would be:

(10[sup]87[/sup])!

The log of n! is on the order of n log n. The log log of the above would be a bit over … 87. So 2^2^2^87 would be much, much bigger. This is a *tiny* number compared to any of the well known really large numbers we’ve discussed here.

Cool! Thanks - time to go play in a test thread…

The OP was trying to type (10[sup]87[/sup]-1)! . But he didn’t know about the sup tags, so he was trying to do it on multiple lines, but also didn’t know that the extra spaces would be stripped.

Though, honestly, there’s no point in the -1. It’s not like we know the number of particles in the Universe to 87 digits of precision. We don’t even know it to one digit of precision, or really, even to an order of magnitude.

Sure, but it’s factorial, so removing the -1 increases the number by 87 orders of magnitude.

But yeah, that miniscule compared to the orders of magnitude of the original estimate, so …

I’m pretty sure the answer to this question is “yes”, but to make sure:

Is Tree(4) larger than

Etc

9

9

9

10

with each 9 “written” on an atom throughout the known universe?

UIAM, the number you cite is smaller than 4↑↑↑3.

That number, in turn, is *much* smaller than 5↑↑↑3, which is *much* smaller than 4↑↑↑4. And those numbers just use three up-arrows. 4↑↑↑↑4 is ridiculously bigger than those numbers. 4↑↑↑↑↑4 ? Don’t even talk about it!

And Graham’s number doesn’t use 64 up-arrows; it uses some utterly preposterous number of arrows.

I expected someone to give a more complete answer to post #12; I’ll do it myself.

The tower of 9’s (**9[sup]9[sup]9[sup]9[sup]9[sup]9[sup]9[sup].[sup].[sup].[/sup][/sup][/sup][/sup][/sup][/sup][/sup][/sup][/sup]**) described here is abbreviated using Knuth’s **↑** notation as **9↑↑(10[sup]87[/sup])**, or **9↑↑(10↑87)**. (One of the 9’s should actually be a 10 but this is completely inconsequential when dealing with numbers this big.)

Now (10↑87) is a largish number by normal standards; so let’s depict it with two up-arrows instead of a single up-arrow. If you type “3^4^4” into Google, Google comes back with 1.39*10^122. As Google also shows, “4^4^4” is somewhat larger. To use the up-arrow notation to re-express a “tower” of exponents, all the numbers must be the same; then **4↑4↑4 = 4↑↑3** by definition, since three 4’s occur on the left side.

Thus the number OP presents in #12 is much smaller than **9↑↑4↑↑3** which is much smaller than **4↑↑4↑↑4**. Again, we’ve made all the numerals the same so that we can move up a step in the up-arrow hierarchy. **4↑↑4↑↑4 = 4↑↑↑3** since, again, there are 3 instances of ‘4’ on the left.

Any number that needs three up-arrows to express in Knuth’s notation is a BIG number so, yes, the number in #12 is a BIG number. Now let’s look at Graham’s number. The link points to a version of Graham’s number that is very convenient for us here, because it is based on Knuth’s up-arrow notation.

Gc(1) = **4↑↑↑↑4**

With four up-arrows instead of three, this number is MUCH bigger than the number in #12. The latter number is a prosaic tiny number in comparison. But Graham doesn’t stop there.

Gc(2) = **4↑↑…↑↑4** where the number of up-arrows is Gc(1), 4↑↑↑↑4. And, as if this weren’t absurd enough they keep going:

Gc(3) = **4↑↑…↑↑4** where the number of up-arrows is Gc(2).

It’s probably an anti-climax to mention that they don’t stop here; they keep going and *Graham’s Number* itself is Gc(64). They say TREE(3) is even bigger than that. “I don’t know, Boss. My watch stopped.”

Am I correct in thinking that Graham’s number, as opposed to the much larger “arrow numbers” is a part of a PROOF, where the"arrow numbers" are not parts of proofs, but merely examples of extremely large numbers).

If so, then it would seem that Grahams’s number is far more impressive

since any “arrow number” can be surpassed by simply adding 1.

No, Graham’s number is an arrow number. In fact, as previously explained, it has a ridiculous number of arrows. The number of arrows is itself another ridiculous arrow number. Graham’s number is the 64th in a recursive chain of arrow numbers.

You are right about Graham’s number being used in a proof though. I believe it is the upper bound to some combinatorial problem.

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Where is the number of atoms in the universe sitting in the scale?

ME: any “arrow number” can be surpassed by simply adding 1.

SAFFTER: No, Graham’s number is an arrow number

ME: 4↑↑↑↑4, for example, is not writable in our usual mathematical notation m in the known universe.

But is’t’ 4↑↑↑↑4t still a number identical to a number not writeable in our known universe, but still a (non-writable)number in our usual mathematical notation(which we’ll call x).

If so, isn’t x+1 larger than 4↑↑↑↑4?

Indeed, for any number x of any magnitude, x + 1 is a bigger number.

Well, yes, for any number *n* anyone can describe, you can always get a larger number by taking *n* + 1 or 2*n* or 10^*n* or whatever. But the point is, those are all really inefficient ways of making bigger numbers, compared to the techniques used to get Graham’s number or the like.

The OP, in an attempt to get a really big number, started with that number, and then applied to it the fastest-growing function he was familiar with. And compared with the other numbers under discussion here, it was a really, really feeble attempt.