Really, Really Big: Graham's Number

[01101001]

I recently had the pleasure of meeting Graham's Number (http://mathworld.wolfram.com/GrahamsNumber.html), the Guinness Record largest named number -- one with a "practical" puspose, anyway, its being used in a mathematical proof.

It is shown to be an upper bound on a not unreasonably stated word problem (and it also relates to the coloring of the corners of hypercubes of dimension n, but that's so hard to picture):

"Take any number of people, list every possible committee that can be formed from them, and consider every possible pair of committees. How many people must be in the original group so that no matter how the assignments are made, there will be four committees in which all the pairs fall in the same group, and all the people belong to an even number of committees."

Some uncommon notation is need to express it with a formula. Just as multiplication is a generalization of addition, and exponentiation is a generalization of multiplication, the arrow function is a generalization of exponentiation.

For instance:

3↑3 is the ordinary 3-cubed, 27.

3↑↑3 is 3 raised to (3 raised to 3), 3↑27, a good-sized number, 7,625,597,484,987.

3↑↑↑3 is 3↑↑(3↑↑3), or 3↑↑7,625,597,484,987, a very large number. That is: 3 raised to (3 raised to (3 raised to...)))
7,625,597,484,987 times.

3↑↑↑↑3 is 3↑↑↑(3↑↑↑3). Big. Very big.

Still with me? 3↑↑↑↑3 is the starting point for defining Graham's number.

G1 is 3↑↑↑↑3.

G2 is 3↑↑...G1 total arrows...↑↑3. Yikes! But, we've only just begun.

Generally, Gn is 3↑↑...Gn-1 total arrows... ↑↑3

Finally, Graham's Number is G64. That is: 3↑↑...G63 total arrows...↑↑3. It makes me feel woozy. I think I'm gonna hurl.

The tickling thing about the amazingly gigantic Graham's Number is that it is an upper bound for a value that experts in the field believe, but haven't prooved, to be the much, much less astounding number, 6. Six!

Now, next time someone throws a googolplex (http://en.wikipedia.org/wiki/Googolplex) (between 10↑↑3 and 10↑↑4, much smaller than 10↑↑↑2) at you, you can handily trump them.

[frogesque]

Yeah - But does it = 1 ? 8)

(runs and hides under table)

[Brady Yoon]

Is it bigger than a googolplex?

[The Supreme Canuck]

Much.

[TriangleMan]

[old man] You call that a big number? Back in my day we were using numbers like that for long division when I was in Grade 2! [/old man] :wink:

[George]

[old man] You call that a big number? Back in my day we were using numbers like that for long division when I was in Grade 2! [/old man] :wink:
Remember, we were smaller back then, so they just looked bigger. :)

[logicboy]

:lol: :lol: :lol: rofl

[01101001]

Is it bigger than a googolplex?
Well, I mentioned googolplex in my original post, but let's revisit it.

A googolplex is 10↑(10↑(10↑2)).

A googolplex is less than 10↑↑4 = 10↑(10↑(10↑10))

A googolplex is way less than 10↑↑↑2 = 10↑↑10

I think that gives some feeling about how the arrow operator notation can generate gigantic quantities.

A googolplex is way, way less than that 3↑↑↑↑3, the G1 starting point for Graham's number.

If you are standing on the number line at just G2 (3↑↑...G1 total arrows...↑↑3), looking back toward zero, a googolplex might look infinitesimally small!

[hedin]

Is it bigger than a googolplex?
Well, I mentioned googolplex in my original post, but let's revisit it.

A googolplex is 10↑(10↑(10↑2)).

A googolplex is less than 10↑↑4 = 10↑(10↑(10↑10))

A googolplex is way less than 10↑↑↑2 = 10↑↑10

I think that gives some feeling about how the arrow operator notation can generate gigantic quantities.

A googolplex is way, way less than that 3↑↑↑↑3, the G1 starting point for Graham's number.

If you are standing on the number line at just G2 (3↑↑...G1 total arrows...↑↑3), looking back toward zero, a googolplex might look infinitesimally small!

errrrr you lost me I ain´t that smart :oops:

[tofu]

You know, the Guinness people are not mathematicians. How much time do you think they spent researching that? Take the total profits they get from selling their books, which probably isn't a lot in the age of the Internet, and divide by the number of mostly made up world records contained within. That's the amount of money they can on average afford to spend researching any one record. Multiply that by minimum wage and you probably get something like a minute or two of time.

So, I don't put much faith in what they say with regard to the largest named number. In fact, I can think of another named number that is probably as large, SuperK. It was used by Donald Knuth in one of his books. For those of you that don't know, he is an imminent computer scientist, a real smart guy. Chances are, we wouldn't have forums like this without him. If the Guinness people didn't at least acknowledge that in their listing of the record, then I don't think they did a good job.

[01101001]

In fact, I can think of another named number that is probably as large, SuperK. It was used by Donald Knuth in one of his books. For those of you that don't know, he is an imminent computer scientist, a real smart guy. Chances are, we wouldn't have forums like this without him. If the Guinness people didn't at least acknowledge that in their listing of the record, then I don't think they did a good job.

Guinness (http://www.math.ucsd.edu/~fan/ron/images/guiness.html) qualified it by saying Graham's number is the largest named number used in a proof.

I'm quite unconfident that Guinness did a lot fo research to discover this fact, but I do have a lot of confidence that if some mathematician had used a larger number in a proof, he/she would have let the Guinness folks know about it. Anyone who did the simplest search on the topic of large numbers would rather quickly run into the Guinness claim.

Obviously, I could name a number, oh, Phil, that is Graham's number + 1, but that is not quite so interesting.

What do you have for Knuth's Super K? With a quick Google search (http://pub21.ezboard.com/fopenitforumfrm7.showMessage?topicID=97.topic&inde x=5), all I saw for it was 10↑↑↑↑3. That's 10↑↑↑(10↑↑↑10).

That is big, bigger than G1, 3↑↑↑↑3, even, but must be less than G2, and just pathetically less than G64.

By the way, this arrow notation is due to Knuth.

[tofu]

What do you have for Knuth's Super K?

I couldn't find anything with a google search either. I loaned my dad the book. If you're interested, I'll get it back from him this weekend and transcribe a couple of paragraphs.

By the way, this arrow notation is due to Knuth.

That's interesting. I didn't know he had invented it.

[frogesque]

I wouldn't think that Guinness need to make a lot of profit from the annual Book of Records. It's sound advertising and gets the company an awful lot of publicity both from the book and the stunts that folk do to get into the records. It's in Guinness' interest that those records are as accurate as they can possibly make them. Guinness would be virtually unknown in the US without media attention on the book. Smart move for a huge market - CocaCola and Pepsi must be kicking themselves for not coming up with the idea first.

Daily Mirror article (http://www.mirror.co.uk/news/allnews/content_objectid=13619029_method=full_siteid=50143 _headline=-THAT-S-A-NEW-WEIRD-RECORD-name_page.html)(a national UK daily tabloid)

THAT'S A NEW WEIRD RECORD Nov 13 2003

By Andrea Henry

FORGET the tallest, the smallest, the fastest or even the slowest, it's time to brace yourself for the grossest records in history.

The Guinness Book Of World Records celebrated its 100 millionth copy with a big bash in London on Tuesday and a new pocket-sized book - The Best Gross World Records.

History (http://freespace.virgin.net/james.robertson/history2.htm)

On 12 September 1954, Sir Hugh invited Norris and Ross McWhirter to see if their fact and figure agency in London could help. An office was set up at 107 Fleet Street and intense work began on the first slim 198 page edition. The printers bound the first copy on 27 August 1955. Well before Christmas the Guinness Book was No. 1 on the bestsellers list.

[Fortis]

Of course you can name and register your own number here. ;) :)

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http://home.columbus.rr.com/realnumbers/index.html

[Bad jcsd]

As the mathwold link states Graham's number is widely recognized to be the largest number with practical use.

Actually wait I've just discovered a new number!!!!! g_64 + 1 !!!!!

[jfribrg]

As the mathwold link states Graham's number is widely recognized to be the largest number with practical use.

Actually wait I've just discovered a new number!!!!! g_64 + 1 !!!!!

Assuming that those trailing exclamation points are factorials, that is quite a large number, but I have a better number. (g_64+1)!!!!!!+1 I'll call it jfribrg's number. This has the property that all prime factors are > bad jcsd's number.

[mickal555]

I think I'll expand the number and post it my web-site for fun.....
I just want to ask... is this possible
oh and whats it used for anyway?

[jfribrg]

I think I'll expand the number and post it my web-site for fun.....
I just want to ask... is this possible
oh and whats it used for anyway?

If every particle in the universe were a bit, it would not be possible to store this number in the typical binary format. Don't even think of trying to print it out.

Another question is whether the number could be stored if every particle in the universe were a qbit. I'll ponder this a while and post my conclusion. Or I could simply wait for toSeek to read the post and do the pondering for me.

[mickal555]

Awww fiddle sticks

but
I don't want to post in in binary I want to post it in Decimal is that what a qbit is.