# A very large number that has all of these qualities?

1. Jul 28, 2011

### thefinalson

Hello all! I'm new here; I found these forums by way of a google search after I was unable to get my question answered on yahoo answers. In any event, I don't see any place for new members to "introduce themselves" like a lot of forums have, so I guess I figured it would be alright if I just jump right into my question. Hope that's cool with everyone?

Just incidentally, I am working on a science fiction setting (well sort of science fantasy, it's got a mix of hard and soft sci-fi elements, but I want the hard sf parts to stick within real laws of physics and mathematics), so I will likely be asking more questions in the future as far as this type of thing, I hope you guys can help me out, and I'm certainly happy to credit you in the book acknowledgements as science/math consultants!

So, I'm trying to come up with a number that has both of these qualities and matches both of these criteria:

• It must be a very large number, so large that it must use Conway chained arrow notation. It must be larger than Graham's number; significantly so.

• It must be a mathematical anagram of the Golden Ratio (or Golden Mean, or whatever you want to call it.) By which I mean, it must be composed of only the numbers from that fraction (and zeroes). So it could be like 3 → 1 → 6 → 1 → 8 → 3 and so on…

I hope someone can help (and I hope this is categorized in the right forum, I always seem to be a little unsure about that when I first post in a new forum; I'm not sure if someone might consider this to be a "textbook-style problem" or not)!

Last edited: Jul 28, 2011
2. Jul 29, 2011

### thefinalson

:bump:

I know it's a complicated question…should I move this to the Number Theory section?

In case anyone needed to refer to any of the concepts I mentioned in my question (I'm sure you don't, because honestly I believe all of you are probably far better mathematicians than I, but I just thought I would put this, in case anyone does), here are wikipedia articles that describe them:

http://en.wikipedia.org/wiki/Conway_chained_arrow_notation
http://en.wikipedia.org/wiki/Graham's_number
http://en.wikipedia.org/wiki/Golden_Ratio

3. Jul 29, 2011

### chasrob

First impressions-
You say you want to involve the digits of the Golden Ratio--the GR is an irrational number. Therefore its digits can be expanded to infinity. Therefore there's an excellent chance all single digits 0-9 can be found somewhere in its expansion, so...

This appears to be in chained arrow form. However, according to rule 3 in the definition and overview section of the Wikipedia article you linked, the expression is truncated after the first "1". So your example shortens to 3 -> 1, which is equal to 3.

4. Jul 29, 2011

### thefinalson

Ok, right, see, this is why I need help! :shy: Could you please help me get a similar number (involving the digits of the Golden Ratio in roughly the order that they appear within that irrational number's expansion) but one that contains no "1's" and is an ungodly huge number, like much bigger than Graham's number, and way too big to write within the observable universe?

5. Jul 30, 2011

### chasrob

How about the natural base, e, raised to the power 1618033988 (the first ten digits of the Golden Ratio)? That's equal to 3.3136932969011345 × 10702703232. Or 1618033988!!!!!!!!!! (same digits factorial ten times), equal to a power tower of 14,197,751,787 tens with a 5.211971113009518 on top. Both far too big to write within the observable universe. Though neither are not as big as Graham's number.

Last edited: Jul 30, 2011
6. Jul 30, 2011

### Mentallic

That first remark guarantees no digital representation of the number could be contained within our universe

I'm not quite sure of what you're asking. Of course you could create a number bigger than Graham's number, but what is the purpose? Does this number have to represent something?

7. Jul 30, 2011

### chasrob

Yeah, your reasons for needing Graham's number are not clear. The golden ratio is fine. But GN is impossible to represent without special notation. The standard exponents and power towers, or log of the log etc., don't work. Chained arrow notation would approximate GN: its between 3→3→64→2 and 3→3→65→2. So 3→3→66→2 would be larger. But combining chained arrow and the digits of the golden ratio... you could drop all ones and zeros but then it wouldn't be the golden ratio anymore.

I'm writing a sci-fi story myself, and it involves large numbers also--why I'm interested.

8. Jul 30, 2011

### pmsrw3

Using the digits of the decimal expansion of the golden ration seems kind of arbitrary to me. 1, 6, 1, 8, 0, 3, ... are consequences of arbitrary evolutionary accidents that resulted in us having 10 fingers. If we had 12, it would be 1, 7, 4, b, b, 7, ... If you want an approximation scheme (which is all that a decimal expansion is) for the golden ratio that has more universal validity, how about the Fibonacci numbers?

9. Jul 31, 2011

### thefinalson

That's very cool! PM me, we ought to compare notes .

I'll try to explain all my logic as best as I can. It's not that I need something that approximates GN, it's simply that I want this number I'm creating (or really defining; you can't actually create a number, they all exist already lol, but, you know what I mean) to be larger than Graham's number—I'm simply using it as a yardstick. The reason for this is pretty arbitrary really, just that I've seen it mentioned a couple of times as an awfully big number, and I want mine to be even bigger! :tongue2:

Dropping all the 1's and 0's out of the Golden Ratio, I think is what I'm going to have to do. It won't be the GR anymore, but oh well, it'll have to be close enough. So, doing this, we would get:

6 → 8 → 3 → 3 → 9 → 8 → 8 → 7

That number would be much larger than Graham's number, wouldn't it?

Oh, and you were curious what this number was for within my SF story, right? Well, basically, to make a long story (or, in this case, book lol :tongue:) short…this number is used by an immensely powerful, immensely ancient multiverse-spanning civilization for measuring horrendously huge numbers, such as all of the possible configurations of atoms in all of the universes that exist!

10. Jul 31, 2011

### thefinalson

I could use the Fibonacci numbers instead, they seem to basically just be a different "version" (if you will) of the Golden Ratio; wikipedia describes them as, quote, "intimately connected". So, if I were to make my super-number with the Fibonacci numbers, it would be this:

2 → 3 → 5 → 8 → 13 → 21 → 34 → 55 → 89 → 144 → 233 → 377

Am I correct in assuming this would also be much larger than Graham's number?

11. Jul 31, 2011

### Hurkyl

Staff Emeritus
I strongly suspect you are running down a false path -- you may be better served by thinking more upon your actual goal, rather than spending all of your effort upon this idea you had to achieve the goal.

One of the features of a "horrendously large number" is that it's pretty much cannot every be used except for one thing (and even that use might not be very useful). No other number you ever encounter will ever be similar enough to it to make a comparison useful.

(Recall that Graham's number originally came about to prove a certain number N was actually finite -- they did so by proving N had to be less than Graham's number. At the time, I think, people thought that N=6. Today, I think people estimate it to be 13)

I also believe that any number you imagine as being horrendously large is actually far, far, far smaller than Graham's number. If the number in your example is finite, I imagine it's unlikely to be bigger than
$$10^{10^{10^{10^x}}}$$​
for some modest number x.

Have you considered suggesting and hinting at the number you're thinking of? From a story telling perspective, doing so might be far more compelling than trying to come up with an explicit number (which you're going to have to explain anyways, if you want people to feel the way you want them to).

12. Jul 31, 2011

### agentredlum

Why don't you use googol-->GN-->googolplex.

This is certainly greater than Grahams Number, and makes a sandwich between googol-googolplex that Sci-Fi readers may find tasty, I would.

13. Aug 1, 2011

### thefinalson

Thank you both very much for your advice, Hurkyl and Agentredlum! I appreciate the feedback.

Hurkyl, I understand what you're saying, and you're probably right. In this case, the number actually is useful for more than one thing, because it's sort of the "base number" for a new measurement system used to deal with things that exist on a nearly-infinite scale, which this particular fictional civilization encounters on a regular basis! It's probably not that realistic to have any number larger than $$10^{10^{10^{10^x}}}$$ but this book is meant to be over-the-top and "beyond the impossible", so unrealistically huge is how the number should be! :tongue:

Agentredlum, googol → GN → googolplex indeed would make a cool "number sandwich", and one that I think would be more immediately recognizable to a lot of SF readers, so I'm definitely going to consider using that. On the other hand, though, my fictional civilization's whole numerical system is based around natural units, physical constants, and numbers inherent to the universe itself. For example, their units of distance are based around the Planck length. So that kind of makes me lean towards the Fibonacci numbers instead.

Just to check/confirm, 2 → 3 → 5 → 8 → 13 → 21 → 34 → 55 → 89 → 144 → 233 → 377

This would be much larger than Graham's number, right? I'd just like to be clear on that.

Last edited: Aug 1, 2011
14. Aug 1, 2011

### Mentallic

I've yet to even be able to concoct a problem where I need to input fathomably large numbers (powers) to yield an unfathomably large number such as Graham's number. I'm not even remotely close!

For example, these numbers wouldn't even need to be expressed in up-arrow notation.
All you have there is xyz where x is the number of configurations of an atom, y is the number of atoms in the each universe, and z is the number of universes. If we take each as being 10100 which is already more than the estimated number of atoms in the universe by a factor of 1020, but nonetheless, this would produce an answer of (10100)3=10300 possible configurations of all the atoms in all the universes. This is nothing compared to Graham's number, which is why I agree with Hurkyl that such a number he posed would be more than enough for anything you would need.

15. Aug 1, 2011

### thefinalson

Ok, well, still, first off I'd like to just get a straight, yes or no answer to my initial question, if I could please:

And secondly,

Alright well, hold your horses there for one minute, Mentallic. $$10^{100}$$ may be a lot more than the number of atoms in one universe, but how many possible positions and quantum states and combinations of possible positions and quantum states are there for all of those atoms? And then how many universes are there? It could be infinite. Plus you're only counting the observable universe, as the entire universe. There could be a lot more. And remember, this is science fantasy here!

Last edited: Aug 1, 2011
16. Aug 1, 2011

### GenePeer

The problem with what you're aiming for is the number will be so huge, you won't even be able to describe how huge it is. Just from reading the wikipedia page, $g_{1}$, which is the first step for a sequence of 64 terms that are growing unimaginably faster than exponentiation, is already too big to be written in the observable universe. It is simply impossible to fathom how large Graham's number, $g_{64}$, actually is. No sort of comparison system you can ever come up with will help. It will take the fun out of the book, if all the reader knows about that measure is it's "unimaginable size". What's fiction if you can't imagine it?

Last edited: Aug 1, 2011
17. Aug 1, 2011

### chasrob

Yep, that would be true. But its not the Fibonacci sequence, you left off the first three digits- 0,1,1. Which would truncate the notation and it would be equal to zero.

18. Aug 1, 2011

### Mentallic

It still wouldn't be anywhere close. These problems can be solved with manageable numbers (although huge that we can't give a digital representation of the figure).
For example, if we take the approximate number of atoms in the universe as 10100 and assume each atom is unique, and then to find each and every possible position that the atoms can arrange themselves in, it will be 10100! But n!<nn so 10100!<(10100)10100=1010102 which definitely doesn't require up arrow notation to be represented. You can go on and find all the quantum states, all the possibilities, but if you try to describe anything physical in our universe (or all the finite number of universes), it won't come close to Graham's number.

Well I assumed that if they're going to be counting all the universes combined, then it wouldn't be infinite. If it is, then there's no point in talking about large numbers because infinite is just that, infinite.

Which is what we've been trying to do here! We can't even concoct a fictitious scenario that would require us to express the figure in terms of up arrow notation.

I'm actually curious now as to how Graham's number was developed...

19. Aug 2, 2011

### thefinalson

Ok, well firstly, thank you very much, Chasrob, for being the only one to actually answer my question; that was the only thing I really wanted to know! Yeah, I know it's not exactly the Fibonacci sequence without the 0, 1, and 1, but obviously I can't have my super-number be equal to zero, so it'll have to be close enough.

Ahh, well, as I've said previously, I'm not really as good a mathematician as most of you probably are, (I'm more of a writer and a storyteller and world-builder with just a layman's interest in, and knowledge of, math and science) but still, that being said, even with my dilettante level of expertise, I'm still really sure that you have a way low estimate there. If we are to assume that there are 10100 atoms in an "average" universe, then all of the possible arrangements and configurations (10100!) has to be far higher than 10100. I say this with confidence because the number of possible configurations of a set of things must always be exponentially greater than the set of things itself. For example, look at our Latin alphabet. It contains only 26 letters, yet the number of possible words that could be formed is enormous. Just try to imagine how many words we could make if our alphabet had 10100 letters in it! Of course I realize atoms aren't words, but the analogy is illustrative nonetheless.

Graham's number was developed to answer the question: "Consider an n-dimensional hypercube, and connect each pair of vertices to obtain a complete graph on 2n vertices. Then color each of the edges of this graph either red or blue.
What is the smallest value of n for which every such coloring contains at least one single-colored 4-vertex planar complete subgraph?

Graham & Rothschild (1971) proved that this problem has a solution, N*, and gave as a bounding estimate 6 ≤ N* ≤ N, with the upper bound N a particular, explicitly defined, very large number. (In terms of Knuth up-arrow notation, N=F7(12), where F(n)=2 up-arrow n3.) The lower bound of 6 was later improved to 11 by Geoff Exoo of Indiana State University (2003) and even further to 13 by Jerome Barkley in 2008. Thus, the best known explicit bounding estimate for the solution N* is now 13 ≤ N* ≤ N."

Which brings me to my next point, by the way. What if I propose, in my sf setting, that each possible thing (including different laws of physics and logic, different spatial dimensions, and even different laws of math than we enjoy in our universe) must exist in some universe out there? Including every possible solution N* for Graham's number, as well as lots of other such problems as the one GN was produced to solve—then I am quite certain there would be far more than 10100 things for my civilization to measure!

But as I stated before, I'm really not enough of an expert on these sorts of matters to truly know, and as such, I bow out of the debate graciously. If you say there's less, then there's probably less.

Finally, though,

Well, this setting is meant to be "unimaginable"—but you're right, it's no good if my readers don't know what this number really means! What would you suggest would be the best way to succinctly explain how big a Conway chained arrow notation number is in a little story footnote or sidebar to readers who may or may not have any great background in the mathematics of ridiculously large numbers, but assuming at least that they understand how normal exponentiation, like 1020, works? The wikipedia example is a little too long-winded and a little too dependent on links to other articles to try to summarize in my book! :tongue2:

Last edited: Aug 2, 2011
20. Aug 2, 2011