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andrewdt85
08-22-2005, 12:27 PM
I did Google this- and to prove it, I'll say that the name Google comes from googol meaning 1 followed by 100 zeros :)

I saw a scientific list of big number names that didn't say. I just want the answer of the highest number name, not the name of the highest mathematical term or something I won't understand.

CookingWithGas
08-22-2005, 12:33 PM
Never heard one higher than googolplex, which is 10googol

http://en.wikipedia.org/wiki/Googolplex

Q.E.D.
08-22-2005, 12:33 PM
Depending upon your definition of the term "name", the Googolplex (a one followed by a googol zeros) seems a likely contender. Of course, one can also construct compound names such as a "googolplex of googolplexes" to name arbitrary numbers as large as necessary.

andrewdt85
08-22-2005, 12:41 PM
Nope- no combo of names.

That's what numbers are- combos- and they are infinite.

leandroc76
08-22-2005, 12:47 PM
sooo... wouldn't it be like:

Nine Hundred and Ninety Nine Googolplex, Nine hundred and Ninety Nine....(whatever the name is before googolplex...) ?

tdn
08-22-2005, 12:47 PM
Nope- no combo of names.

That's what numbers are- combos- and they are infinite.
So are you looking for something like Nine hundred ninety-nine google, nine hundred ninety-nine bazillion, nine hundred ninety-nine quintillion, four hundred seventy three?

leandroc76
08-22-2005, 12:49 PM
or even and googolplex and one... sort of

leandroc76
08-22-2005, 12:51 PM
Scratch it...

http://mathforum.org/library/drmath/view/59154.html

av8rmike
08-22-2005, 12:55 PM
This page (http://home.earthlink.net/~mrob/pub/math/largenum.html) may answer some of your questions about number names, or it may raise more questions. I can't say, because I guess I really don't understand the boundaries of your OP. Note that that author also claims a "googolplex" as the largest number with a name, but suggests other constructs as "googolplexplex" as possible.

Plynck
08-22-2005, 01:02 PM
Three.

No, wait... Four!

No, wait....

More on googol and gazillion (http://boards.straightdope.com/sdmb/showthread.php?t=2581)

Bippy the Beardless
08-22-2005, 01:40 PM
Not sure if you would say this fits the bill, but Graham's Number (http://www.absoluteastronomy.com/encyclopedia/G/Gr/Grahams_number.htm) is 'somewhat' larger than a googleplex and has at least one serious use in a maths proof.

ultrafilter
08-22-2005, 01:59 PM
As large as it seems, a googolplex is meaningfully expressible in exponential notation, which means that it's a mere pittance. There are various special notations for very large integers, and it's easy to come up with finite numbers that are beyond human comprehension. The problem is that it's almost impossible to compare numbers in one notation with those in another.

I'm going to pick the Moser (http://home.earthlink.net/~mrob/pub/math/largenum-3.html) as the largest number that has a "name" in the sense of the OP. If you're allowing any finite sequence of characters as a name, though, we should talk about the Busy Beaver function....

ultrafilter
08-22-2005, 02:00 PM
Not sure if you would say this fits the bill, but Graham's Number (http://www.absoluteastronomy.com/encyclopedia/G/Gr/Grahams_number.htm) is 'somewhat' larger than a googleplex and has at least one serious use in a maths proof.

The upper bound for Graham's number, you mean. As far as we know it could be 11.

AHunter3
08-22-2005, 02:17 PM
sooo... wouldn't it be like:

Nine Hundred and Ninety Nine Googolplex, Nine hundred and Ninety Nine....(whatever the name is before googolplex...) ?
Well, it would be except that there are huge swaths of numbers in there that have no names.

The longest name that has a near-absolutely official name would be 999 vigintillion 999 novemdecillion 999 octodecillion 999 septemdecillion 999 sexdecillion 999 quintdecillion 999 quattuordecillion 999 tredecillion 999 duodecillion 999 decillion 999 nonillion 999 octillion 999 septillion 999 sextillion 999 quintillion 999 quadrillion 999 trillion 999 billion 999 million 999 thousand 999.

(You'll pardon me if I don't spell out the "nine hundred ninety nine"s, yes?)

I say "near-absolutely official" because once upon a time there was an alternative (chiefly British) naming convention that would render the number 1,000,000,000 as "one milliard" rather than "one billion", and a thousand milliards was a billion. But that system has effectively fallen by the wayside. Milliards there may rarely be, but billiards are almost exclusively played on gaming tables and of trilliard and quadrilliards no one has heard at all.

I also say "near-absolutely" because there is a less official nomenclatural system extending upwards (mostly just by common prefix extension): a thousand vigintillions would be an unvigintillion, a thousand of those would be a duovigintillion. Makes sense enough, right?

After escating through uno, duo, tre, quattuor, quint, sex, septem, octo, and novemvigintillions, we would come (by extenion) to the trigintillions. An even thousand trigintillions would be an unotrigintillion, a thousand of those would be a duotrigintillion, and a mere 10 of those suckers would be not only 10 duotrigintillion but also your basic googol. (And now you see why "googol" isn't incorporated into a normal numbering convention. It has a nice profound-sounding number of zeros after it, but those zeros don't come in full packs of threes like millions, billions, and duotrigintillions).

Going on upwards,duotrigintillions would yield to trevigintillions once you got a thousand of them together, and 1000 of those = quattuortrigintillions...may I skip on, assuming youv'e got the pattern, and note that novemtrigintillions in the thousands are quadragintillions [123 zeros], followed (one full tier later) by qunquagintillions [153 zeros], sexagintillions [183 zeros], septuagintillions [213 zeros], octogintillions [243 zeros], nonagintillions [273 zeros], and finally centillions [303 zeros]?

And there I think you have it. I did not mention the googol plex because I did not come to it and pass it. That's because I didn't come to it, and shall not. I'm only up to 303 zeros. You can, if you wish, extend the naming convention using the same logic that got me to a centillion but you've got a ways to go before you get a googol zeros behind a 1.

And that prevents you from saying "999 googol plex, 999 <something>, 999 <something> <... very long elllipsis here...>, 999 thousand, 999" is the longest number. So many unnamed numbers, so little time :)

pulykamell
08-22-2005, 02:33 PM
The upper bound for Graham's number, you mean. As far as we know it could be 11.

You're the math dude, but as I understand the entry given, I thought it said that Graham's number is the lowest-known upper bound for Ramsey's number, which is somewhere between 11 and Graham's number, inclusive.

borschevsky
08-22-2005, 02:33 PM
You could also just try to be tricky, and do something like: Define borschevsky's number as one more than the largest named number (other than itself).

Probably works fine until somebody defines their number as bor's number + 1. :)

ultrafilter
08-22-2005, 02:40 PM
You're the math dude, but as I understand the entry given, I thought it said that Graham's number is the lowest-known upper bound for Ramsey's number, which is somewhere between 11 and Graham's number, inclusive.

The Ramsey's number thing is almost certainly a mistake, but Mathworld (http://mathworld.wolfram.com/GrahamsNumber.html) seems to think that Graham's number is the upper bound.

Bippy the Beardless
08-22-2005, 03:22 PM
Does Graham's number refer to g64 or to N*
In N*<=g64

pulykamell
08-22-2005, 04:14 PM
Does Graham's number refer to g64 or to N*
In N*<=g64

g64 , according to the Mathworld link.

where Graham's number g64 is recursively defined by

Thudlow Boink
08-22-2005, 04:27 PM
Nine Hundred and Ninety Nine Googolplex...bottles of beer on the wall,
Nine hundred ninety-nine googolplex bottles of beer,
Take one down, pass it around,
Uh...
a whole heckuva lot of bottles of beer on the wall.

andrewdt85
08-22-2005, 09:18 PM
That was a good explanation, AHunter3. For Graham's number- whoo boy, I don't have those years of mathematical education yet.

DarrenS
08-22-2005, 11:44 PM
You could also just try to be tricky, and do something like: Define borschevsky's number as one more than the largest named number (other than itself)I think you've opened yourself up to a Berry Paradox (http://en.wikipedia.org/wiki/Berry_paradox) there.

Mathochist
08-22-2005, 11:53 PM
I did Google this- and to prove it, I'll say that the name Google comes from googol meaning 1 followed by 100 zeros :)

I saw a scientific list of big number names that didn't say. I just want the answer of the highest number name, not the name of the highest mathematical term or something I won't understand.

The number thing others have answered more or less accurately. I just want to pick a bit of a nit about the source of the name "google". Yes, the gloss of googol as a large number is intended (with respect to the amount of results returned), but it also refers to google, a deprecated form of goggle: "To turn the eyes to one side or other, to look obliquely, to squint". That is, it looks at a lot of different things in all sorts of directions.

David Simmons
08-23-2005, 01:20 AM
How about the second Skewes Number (http://mathworld.wolfram.com/SkewesNumber.html) which equals 101010103?

David Simmons
08-23-2005, 01:28 AM
Compared to Skewes Number2 a googolplex (1010102) is virtually zero.

AHunter3
08-23-2005, 10:07 AM
I have way too much time on my hands, it would seem. ;)

...there is a less official nomenclatural system extending upwards (mostly just by common prefix extension)...and finally centillions [303 zeros]...

OK, here's one you could count to without hitting unnamed numbers (if it weren't for mortality and the heat death of the universe and stuff like that):

ultrafilter
08-23-2005, 10:11 AM
How about the second Skewes Number (http://mathworld.wolfram.com/SkewesNumber.html) which equals 101010103?

Still expressible in exponential notation. Not a contender.

Chronos
08-23-2005, 03:44 PM
...bottles of beer on the wall,
Nine hundred ninety-nine googolplex bottles of beer,
Take one down, pass it around,
Uh...
a whole heckuva lot of bottles of beer on the wall.It's easier if you use bigger numbers. Try this on for size:

Aleph-null bottles of beer on the wall,
Aleph-null bottles of beer
Take one down and pass it around
Aleph-null bottles of beer on the wall

At least, if you accept "aleph-null" as a number. But if you do, then it's still not the largest number, since you'd then also have to accept aleph-one, aleph-two, aleph-centillion, etc., and there are other "numbers" which are bigger than any aleph, and so on.

NurseCarmen
08-23-2005, 03:56 PM
Probably works fine until somebody defines their number as bor's number + 1. :)Nobody would do that. Because NurseCarmen's number is bor's number times two. :D

Plynck
08-23-2005, 04:48 PM
Well, as we know, if we divide 1 by smaller and smaller numbers, we get larger and larger numbers. So we divide one by zero for a very, very, very large number.

"But that is mathematically impossible!", you would say.

Ahhhhh, but here's the beauty of it: We take the result and subtract 1.

((1/0) - 1)

And we name it "plynck's constant".

Yep, got it all figured out, and before dinner, too :D

furt
08-23-2005, 06:04 PM
It's easier if you use bigger numbers. Try this on for size:

Aleph-null bottles of beer on the wall,
Aleph-null bottles of beer
Take one down and pass it around
Aleph-null bottles of beer on the wall

At least, if you accept "aleph-null" as a number. But if you do, then it's still not the largest number, since you'd then also have to accept aleph-one, aleph-two, aleph-centillion, etc., and there are other "numbers" which are bigger than any aleph, and so on.
Meh. After the first duodecillion beers or so, who gives a crap anymore?

David Simmons
08-23-2005, 08:23 PM
Still expressible in exponential notation. Not a contender.Are there any rational numbers not expressible in exponential notation? Irrational numbers being not expressible at all in digits, only their approximations.

And we really need to find help for AHunter3 ;)

ultrafilter
08-23-2005, 08:42 PM
Are there any rational numbers not expressible in exponential notation?

When you're dealing with large numbers, "expressible" seems to be taken as a synonym for "intelligibly expressible". Graham's number, for instance, is expressible in exponential notation, but not in any meaningful way.

Little Nemo
08-23-2005, 08:58 PM
Infinity is certainly an established name.

A Googleplex is probably the highest commonly known number.

Graham's Number appears to be the highest number that's ever been actually used.

The Moser appears to be the highest definable number.

999 vigintillion 999 novemdecillion 999 octodecillion 999 septemdecillion 999 sexdecillion 999 quintdecillion 999 quattuordecillion 999 tredecillion 999 duodecillion 999 decillion 999 nonillion 999 octillion 999 septillion 999 sextillion 999 quintillion 999 quadrillion 999 trillion 999 billion 999 million 999 thousand 999 is the highest number that could theoretically be counted to (as in all numbers less than it would have a name).

David Simmons
08-23-2005, 09:03 PM
Infinity is certainly an established name.But infinity isn't a number.

pulykamell
08-23-2005, 09:05 PM
Infinity is certainly an established name.

A Googleplex is probably the highest commonly known number.

Graham's Number appears to be the highest number that's ever been actually used.

The Moser appears to be the highest definable number.

999 vigintillion 999 novemdecillion 999 octodecillion 999 septemdecillion 999 sexdecillion 999 quintdecillion 999 quattuordecillion 999 tredecillion 999 duodecillion 999 decillion 999 nonillion 999 octillion 999 septillion 999 sextillion 999 quintillion 999 quadrillion 999 trillion 999 billion 999 million 999 thousand 999 is the highest number that could theoretically be counted to (as in all numbers less than it would have a name).

How do you figure? It looks like AHunter3 has counted quite a bit higher than you...

Thudlow Boink
08-23-2005, 09:21 PM
But infinity isn't a number.Correct me if I'm wrong, but I'm pretty sure the unmodified word "number" has no formal definition in mathematics, so you can't really pin down whether something is or is not a "number"—it depends on what kind of numbers or what set of numbers you're talking about.

I think most of the posts so far have assumed the OP was asking about the biggest real number—a natural assumption, since the set of real numbers (or perhaps some subset thereof) is what most of us think of when we think of "numbers." And yes, infinity is not a real number. But if you allow things like transfinite numbers (as Chronos mentioned), that's a whole new ball game, and you can easily get "bigger" numbers than any of the real numbers mentioned.

AHunter3
08-23-2005, 09:26 PM
Little Nemo is correct if we confine ourselves to officially accepted names. The other names are considered logical extensions of the official naming convention ("considered" meaning "by people other than just me, i.e., I didn't just make them up or anything ;)

I have no idea why the official named series stops at the vigintillion mark.

ultrafilter
08-23-2005, 10:16 PM
Infinity is certainly an established name.

But it doesn't refer to anything specific; in fact, I don't think it means anything out of context.

The Moser appears to be the highest definable number.

It may the highest number that anyone's bothered to define, but simply definining a k-Moser as (Moser)k gives you a lot of larger, defineable numbers.

David Simmons
08-23-2005, 11:50 PM
Correct me if I'm wrong, but I'm pretty sure the unmodified word "number" has no formal definition in mathematics, so you can't really pin down whether something is or is not a "number"—it depends on what kind of numbers or what set of numbers you're talking about.

I think most of the posts so far have assumed the OP was asking about the biggest real number—a natural assumption, since the set of real numbers (or perhaps some subset thereof) is what most of us think of when we think of "numbers." And yes, infinity is not a real number. But if you allow things like transfinite numbers (as Chronos mentioned), that's a whole new ball game, and you can easily get "bigger" numbers than any of the real numbers mentioned.Chronos did add the proviso that you need to accept transfinite "numbers" as numbers - I don't. The don't obey rules of addition, subtraction etc. as far as I can tell or the rule of succession that tells us how to form members of the set.

I wouldn't be surprised to be wrong about this but that's my story until a mathematician comes along to show that such entities do follow such rules.

sturmhauke
08-24-2005, 02:04 AM
When you're dealing with large numbers, "expressible" seems to be taken as a synonym for "intelligibly expressible". Graham's number, for instance, is expressible in exponential notation, but not in any meaningful way.
I only have an extremely tenuous grasp of this stuff after reading some of the links in this thread, but maybe I can shed some more light on this concept. (Not for you, ultrafilter, but for we who do not possess math superpowers.) Consider a googol. In exponential notation, this is written as 10^100 (I'm avoiding the superscript notation for a reason.) Exponentiation is shorthand for repeated multiplication, so while you could express a googol thusly -

10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10

- this method is not exactly conducive to ease of use. Simillarly, since multiplication is repeated addition, you could write it as 10 + 10 + 10 etc. until you get an even more unwieldy block of numbers and operators 10 times the size of the one above.

Now we come to arrow notation, which is a way to express repeated exponentiation. A googolplex is 10^googol, which is the same as 10^10^100, or 10^10^10^10. In other words, you perform exponentiation 4 times. This is can be expressed as 10^^4. 10^^100, written out as exponentiation, would look like the multiplication block above, replacing * with ^. Similarly, 10^^^100 would be 10^^10 repeated 100 times, and 10^^^^100 would be 10^^^10 repeated 100 times.

What happens when you have too many ^s to easily work with? You use chained arrow notation. 10->100->100 is the same as 10, then 100 ^s, then 100. So if you took the multiplication block above and replaced each * with 100 ^s, you would have some inkling of how big this number is. If I tried to write that out as 10 * 10 * 10 etc. it would break the SDMB and I would die of old age long before I made the barest dent in the task.

You can chain more arrows together for even more unfathomably large numbers, but that's beyond my comprehension to work with.

Malacandra
08-24-2005, 03:12 AM
In all the talk of Graham's Number and moser, it's worth mentioning that there is a proof somewhere out there that Graham's is substantially the bigger. But as I don't understand it, I'll not bother to look it up and quote it.

moser3 is barely a blip on the radar compared to "2 in the moser-gon", though if I need to explain that you should probably look up the definition of mega and moser for a start.

naita
08-24-2005, 03:51 AM
I say "near-absolutely official" because once upon a time there was an alternative (chiefly British) naming convention that would render the number 1,000,000,000 as "one milliard" rather than "one billion", and a thousand milliards was a billion. But that system has effectively fallen by the wayside. Milliards there may rarely be, but billiards are almost exclusively played on gaming tables and of trilliard and quadrilliards no one has heard at all.

It's still the standard in much of the non-English speaking world, although in Norwegian at least the only 10^3 intermediate level with name separately is the milliard. A billiard is a thousand billions, a trilliard a thousand trillions. It's a far more superior system.

Malacandra
08-24-2005, 05:28 AM
It's still the standard in much of the non-English speaking world, although in Norwegian at least the only 10^3 intermediate level with name separately is the milliard. A billiard is a thousand billions, a trilliard a thousand trillions. It's a far more superior system.

...on the comparatively rare occasions you actually need a colloquial name for numbers that size. :)

(And I'm not going to pick nits over "far more superior" until my Norwegian improves a lot.)

naita
08-24-2005, 06:07 AM
...on the comparatively rare occasions you actually need a colloquial name for numbers that size. :)

(And I'm not going to pick nits over "far more superior" until my Norwegian improves a lot.)
Hey, don't you recognise hyperbole when you see it? ;) Thank you for not picking nits though, I see I wrote "with name separately" instead of "we name separately". :smack:

Malacandra
08-24-2005, 07:16 AM
Hey, don't you recognise hyperbole when you see it? ;) Thank you for not picking nits though, I see I wrote "with name separately" instead of "we name separately". :smack:

It's cool. I only correct people's grammar when I can do so in their native tongue. Which, as I'm English, means... well, you can figure it out. :cool:

08-24-2005, 07:28 AM
...on the comparatively rare occasions you actually need a colloquial name for numbers that size. :)Hey, inflation'll get us there, just you wait. From my favourite film (http://uk.imdb.com/title/tt0064116/combined):

Cheyenne: Harmonica, a town built around a railroad.
[laughs]
Cheyenne: You could make a fortune. Hundreds of thousands of dollars. Hey, more than that. Thousands of thousands.
Harmonica: They call them "millions."
Cheyenne: "Millions." Hmm.

David Simmons
08-24-2005, 08:23 AM
How about the second Skewes Number which equals 101010103?Still expressible in exponential notation. Not a contender.I finally tumbled to wahat you are talking about. :smack: No, Skewe's Number isn't a name like one quintillion, but then neither is Googol or Googolplex. Skewe's Number identifies by a name a particular number and is just as specific to that number as any other name.

So there, too.

CynicalGabe
08-24-2005, 09:25 AM
Any of those could be the biggest number, but surely 1 is the loneliest number.

08-24-2005, 09:30 AM
Three. It's name is Brian Carruthers of 22 Railway Terrace.

Bytegeist
08-24-2005, 10:51 AM
OK, here's one you could count to without hitting unnamed numbers (if it weren't for mortality and the heat death of the universe and stuff like that):

Nine hundred ninety nine centillion nine hundred ninety nine novemnonagintillion [...] nine hundred ninety nine thousand nine hundred ninety nine.

Hmmm. So if I made two 500-centillion purchases on my credit card, I wouldn't be able to write a valid check at the end of the month, to pay off the balance. The amount would have no name.

I suppose I could go to the bank and get a 10306 dollar bill to mail, but I hate trusting that much cash to the post office, let alone walking around town with it.

David Simmons
08-24-2005, 11:11 AM
Hmmm. So if I made two 500-centillion purchases on my credit card, I wouldn't be able to write a valid check at the end of the month, to pay off the balance. The amount would have no name.

I suppose I could go to the bank and get a 10306 dollar bill to mail, but I hate trusting that much cash to the post office, let alone walking around town with it.This puts me in mind of Robert Benchley's Understanding International Finance. His contention, humorous of course, was that in high finance money doesn't really exisit. All that is happening is that someone subtracts a couple of billion from one account and adds it to a different account.

Checks are sort of like that. No actual money changes hands for each individual transaction. Every once in while the banks readjust their reserves, or whatever it is they do to keep things straight, but when I send a check to someone most times what happens is that a number is subtracted from my account and added to theirs. Now and then a bank will hand a customer a paltry few dollars in cash for a check but that was the bank's money. The customer's money in the bank is just numbers in a computer memory somewhere.

AHunter3
08-24-2005, 11:49 AM
Hmmm. So if I made two 500-centillion purchases on my credit card, I wouldn't be able to write a valid check at the end of the month, to pay off the balance. The amount would have no name.

Welllllll....................

I suppose we could extend the naming convention a tiny bit farther: 1000 centillions is plausibly an "uncentillion" by the same logic that 1000 decillions is an undecillion. And a thousand uncentillions is plausibly a duocentillion, and so forth up until we're sitting there looking at a thousand novemcentillions, at which point we legitimately have an unprecedented naming problem. Is it a "centidecillion"? A "dekacentillion"? An "eleventurytillion"? Something else?

I stopped at "centillion" because

a) it's a nice round stopping point, nomenclaturally

b) the remaining tiny handful of extensible names depends entirely on vocabulary already listed; "centillion" is the last new word in the series

c) ummm, it's a very high opalescent kind of number?

sturmhauke
08-24-2005, 03:54 PM
I finally tumbled to wahat you are talking about. :smack: No, Skewe's Number isn't a name like one quintillion, but then neither is Googol or Googolplex. Skewe's Number identifies by a name a particular number and is just as specific to that number as any other name.

So there, too.
I don't think you get it. ultrafilter is saying that there are some numbers that are so large, they cannot be readily expressed with exponents, and so other types of notation were developed. This is what I tried to explain in my earlier post. So I will try to demonstrate just how huge Graham's number is. First we start with

g1 = 3 ^^^^ 3
= 3 ^^^ 3 ^^^ 3
= 3 ^^^ (3 ^^ 3 ^^ 3) (the parentheses aren't really necessary, I'm just adding them for clarification)
= 3 ^^^ (3 ^^ (3 ^ 3 ^ 3))
= 3 ^^^ (3 ^^ 27)
= 3 ^^^ (3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3)

Now, it should be clear that this is already a big number. I tried to calculate 3 ^^ 27, so I found this Big Number Calculator (http://world.std.com/~reinhold/BigNumCalc.html) It won't calculate 3 ^^ 27 directly, so I entered 3 [x**y] 3, then took the answer and entered [x**y] 3 again and so on. I only got as far as 3 ^^ 9 before it started really slowing down. So I went a step further to 3 ^^ 10 - that is, 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 - and got a result of 1.50541641451 * 109391. This calculator uses your computer to do the calculations, and I have a pretty fast computer, but 3 ^^ 10 still took about 15 minutes to calculate. I estimate 3 ^^ 27 to be at least 101,212,755,270,733. Getting back to the number g1, we now have

g1 ~ 3 ^^^ 101,212,755,270,733

Already this is extremely large, and I'm not going to try to go further with the calculation, but this is still not Graham's number. Next we define g2:

g2 = 3 (g1 of ^ in a row) 3

And then generalize for gn:

gn = 3 (g(n - 1) of ^ in a row) 3

Graham's number is g64. Hmm, it occurs to me that I have been evaluating expressions left to right instead of right to left, which means that my numbers are far smaller than they should be. Anyway, I hope it's apparent that these numbers are nearly indescribably vast. The second Skewes number is only 10 ^ 10 ^ 10 ^ 10 ^ 3, which is not even as big as g1.

Bippy the Beardless
08-24-2005, 05:13 PM
Just seeing the above made me wonder is g[sub]0 propperly defined as 4, or is that fancyful thinking. Working backwards from the general formula for working out g numbers.

Also does g in g numbers stand for Graham or for something else.

sturmhauke
08-24-2005, 05:27 PM
I dunno, I was just going off the Mathworld articles. I haven't studied this stuff before.

Mathochist
08-24-2005, 05:44 PM
It's easier if you use bigger numbers. Try this on for size:

Aleph-null bottles of beer on the wall,
Aleph-null bottles of beer
Take one down and pass it around
Aleph-null bottles of beer on the wall

At least, if you accept "aleph-null" as a number. But if you do, then it's still not the largest number, since you'd then also have to accept aleph-one, aleph-two, aleph-centillion, etc., and there are other "numbers" which are bigger than any aleph, and so on.

I have trouble accessing some of them.

:D

ultrafilter
08-24-2005, 06:13 PM
Just seeing the above made me wonder is g[sub]0 propperly defined as 4, or is that fancyful thinking. Working backwards from the general formula for working out g numbers.

The question doesn't even make sense to me. g0 is whatever Graham chose for it to be.

David Simmons
08-24-2005, 09:23 PM
I don't think you get it. ultrafilter is saying that there are some numbers that are so large, they cannot be readily expressed with exponents, and so other types of notation were developed. This is what I tried to explain in my earlier post. So I will try to demonstrate just how huge Graham's number is. First we start with

g1 = 3 ^^^^ 3
= 3 ^^^ 3 ^^^ 3
= 3 ^^^ (3 ^^ 3 ^^ 3) (the parentheses aren't really necessary, I'm just adding them for clarification)
= 3 ^^^ (3 ^^ (3 ^ 3 ^ 3))
= 3 ^^^ (3 ^^ 27)
= 3 ^^^ (3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3)

Now, it should be clear that this is already a big number. I tried to calculate 3 ^^ 27, so I found this Big Number Calculator (http://world.std.com/~reinhold/BigNumCalc.html) It won't calculate 3 ^^ 27 directly, so I entered 3 [x**y] 3, then took the answer and entered [x**y] 3 again and so on. I only got as far as 3 ^^ 9 before it started really slowing down. So I went a step further to 3 ^^ 10 - that is, 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 ^ 3 - and got a result of 1.50541641451 * 109391. This calculator uses your computer to do the calculations, and I have a pretty fast computer, but 3 ^^ 10 still took about 15 minutes to calculate. I estimate 3 ^^ 27 to be at least 101,212,755,270,733. Getting back to the number g1, we now have

g1 ~ 3 ^^^ 101,212,755,270,733

Already this is extremely large, and I'm not going to try to go further with the calculation, but this is still not Graham's number. Next we define g2:

g2 = 3 (g1 of ^ in a row) 3

And then generalize for gn:

gn = 3 (g(n - 1) of ^ in a row) 3

Graham's number is g64. Hmm, it occurs to me that I have been evaluating expressions left to right instead of right to left, which means that my numbers are far smaller than they should be. Anyway, I hope it's apparent that these numbers are nearly indescribably vast. The second Skewes number is only 10 ^ 10 ^ 10 ^ 10 ^ 3, which is not even as big as g1.I think I'm beginning to get the idea of the notation. I think Asimov did something along this line in an essay that included what he called T numbers. T stood for a trillion and he then did the same things as you did above with a similar notation.

Yeah, Skewes Number2 is merely big but not big enough. I think the claim is that it is the largest number that had appeared in a mathematical proof up to that time.