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#165824 - 02/13/07 01:02 PM Re: Large numbers revisited

Pooh-Bah

Registered: 07/10/05

Posts: 1773

Loc: Apple Valley, CA, USA Alex: I have this instinctive feeling that there should be a way to express an unimaginatively large finite number in a minimal space. It's been suggested that G(G(10^303!)!)!, is a pretty good one. However, G(G(10^999!)!)!, is obviously bigger and so I suppose one might start by attempting to express this number in even fewer characters. However, if an unimaginatively larger number could be expressed in just a few more characters that would be all right too

It's like defining Type-2 and -3 words. Hard to express though a few liberal thinkers--eta and zm for instance--seem to catch on. As the Zen master might say, allow your intuition freedom to roam without being excessively judgmental_________________________

dalehileman

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#165825 - 02/13/07 02:54 PM Re: Large numbers revisited

Carpal Tunnel

Registered: 06/24/02

Posts: 7210

Loc: Vermont big-ass

etc.

almostifinity_________________________

formerly known as etaoin...

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#165826 - 02/13/07 03:25 PM Re: Large numbers revisited

Pooh-Bah

Registered: 01/05/01

Posts: 1814

Loc: Spam Factory Quote:

As the Zen master might say, allow your intuition freedom to roam without being excessively judgmental

Anyway, for your purpose of expressing a vast but finite number to a non-scientific audience I would actually not use the factorial ! or any other operands and stick to simply writing the number using strictly numerals. For any given number of numerals used, a power tower or tetration as it is called will be the highest number.

9^9 is greater than 99

9^(9^9) is greater than 999 and is in fact an incredibly large number. If you were to actually evaluate that number, it would have 369,693,100 digits! Read more about it here. Excerpt follows:9^9^9

Date: 09/10/97 at 18:53:43

From: Sean H

Subject: 9 ^ 9 ^ 9

Dr. Math,

In my Algebra II class, my teacher asked us what the largest number we

can write is, only with a limit of 3 digits. I answered 9 ^ 9 ^ 9

(nine to the ninth to the ninth) and was correct.

Getting home, and on my computer however, I set out to find the answer

to that question. 9 ^ 9 was easy: 387420489. But now, how in the

world to do 9 ^ 387420489?

I tried writing a C++ program to do it, only the largest data type you

can use, an unsigned long, only supports up to 10 digits. That wasn't

getting me anywhere. On the Internet, I saw one person say that the

answer would have 300,000,000 digits! Is this true?

So my question: Is it really that large an answer? And is there a way

I could find the answer? If it really is that big, how long would it

take to answer on a large mainframe? (I'm talking Cray here). Would it

be hours, days, months, or years?

Thanks a lot,

Sean

--------------------------------------------------------------------------------

Date: 09/25/97 at 15:47:28

From: Doctor Ken

Subject: Re: 9 ^ 9 ^ 9

Hi!

To find out how many digits some number has, the best way is to take

the log base 10 of that number and round down to the nearest integer,

and add 1.

For instance, 42 has 2 digits, and the log base 10 of 42 is about

1.62325: round that down and you get 1, add 1 and you get 2. 34578 has

5 digits, and its log base 10 is 4.5388: round it down and you get 4,

add 1 to get 5. 1000 has 4 digits and its log base 10 is 3. Round down

to get 3, add 1 to get 4.

So we want to find the log base 10 of your number, 9^387420489. Let's

do it:

Log (9^387420489) = 387420489 * Log(9) (Pull out the power)

= 387420489 * 0.954243 (do the log)

= 3.6969309963 x 10^8 (by calculator)

= 369693099.63

So your number has 369,693,100 digits. That's a big number!

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#165827 - 02/15/07 10:12 AM Re: Large numbers revisited

Pooh-Bah

Registered: 07/10/05

Posts: 1773

Loc: Apple Valley, CA, USA What's the macro for the upside down v--thanks guys_________________________

dalehileman

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#165828 - 02/15/07 10:35 AM Re: Large numbers revisited

Pooh-Bah

Registered: 07/10/05

Posts: 1773

Loc: Apple Valley, CA, USA Fal: Thank you exceedingly for your participation, but I am confused about Graham's Number. I have heard it expressed as G(63) or as just plain G, but I don't understand how it can still be Guiness' winner if G(65) is bigger and could just as easily be used in a new problem

http://thesaurus.maths.org/mmkb/entry.html?action=entryById&id=2525

Not good at math_________________________

dalehileman

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#165829 - 02/15/07 11:52 AM Re: Large numbers revisited

Carpal Tunnel

Registered: 04/03/00

Posts: 10523

Loc: this too shall pass Quote:

What's the macro for the upside down v--thanks guys

d'oh.

SHIFT-6: ^

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#165830 - 02/15/07 12:05 PM Re: Macro ... Pollo ...

Carpal Tunnel

Registered: 08/13/05

Posts: 3290

Loc: R'lyehWhat's the macro for the upside down v?

What's a macro macro macro, dada? The word,macro, as are all other words, is synonymous withdrive. Dada!_________________________

Ceci n'est pas un seing.

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#165831 - 02/15/07 12:54 PM Re: Macro ... Pollo ...

veteran

Registered: 01/23/02

Posts: 1523

Loc: Virginia, USA

Dale, no matter what specific number one is given, she could easily construct a bigger number by adding 1, or a much bigger number by taking the factorial or the graham's function of that number.

The interesting thing about Graham's number is that it's the biggest number that was actually used non-parenthetically. http://www-users.cs.york.ac.uk/susan/cyc/g/graham.htm

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#165832 - 02/15/07 01:07 PM Re: Macro ... Pollo ...

Pooh-Bah

Registered: 07/10/05

Posts: 1773

Loc: Apple Valley, CA, USA Fal, thank you for that link, and you have to forgive my abysmal mathematical ignorance, but I don't understand it. Is Graham's number G or G(63); and if the latter, how is 63 the upper limit? And does it somehow depend upon the number 3? And why would Guiness cite it if it isn't, inasmuch as upon acceptance all the math whizzes in the world would immediately set about devising a problem using a bigger number

tsu: Thank you for ^. That in using a keyboard some 72 years I had never found occasion for it, is testimony to my mathematical vincibility. For my benefit (and others of my ilk) you might explain its use and limitations; that is, where parens are called for, when not, etc

zm: Not yet anyhow

Myr: Trouble with parallel universes is that each one would have to be serial also. at risk of offending some present, this seems unmanageable even if She is omnipotent_________________________

dalehileman

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#165833 - 02/15/07 01:24 PM Re: Macro ... Pollo ...

Carpal Tunnel

Registered: 04/03/00

Posts: 10523

Loc: this too shall passThank you for ^. That in using a keyboard some 72 years I had never found occasion for it, is testimony to my mathematical vincibility. For my benefit (and others of my ilk) you might explain its use and limitations; that is, where parens are called for, when not, etc

not me, bro. I don' be no steenken mathematician for about 6^2 years now.

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