What's the biggest number that has a name?

[sturmhauke]

@ultrafilter

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]

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.

moser[sup]3[/sup] 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]

@AHunter3

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]

@naita

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]

@Malacandra

…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]

@naita

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.

[Dead_Badger]

@Malacandra

…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:

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]

@ultrafilter

@David Simmons

How about the second Skewes Number which equals 10[sup]10[sup]10[sup]10[sup]3[/sup][/sup][/sup][/sup]?

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]

Any of those could be the biggest number, but surely 1 is the loneliest number.

[don_t_ask]

Three. It’s name is Brian Carruthers of 22 Railway Terrace.

[Bytegeist]

@AHunter3

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 10[sup]306[/sup] dollar bill to mail, but I hate trusting that much cash to the post office, let alone walking around town with it.

[David_Simmons]

@Bytegeist

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 10[sup]306[/sup] 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]

@Bytegeist

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]

@David Simmons

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

g[sub]1[/sub] = 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 It won’t calculate 3 ^^ 27 directly, so I entered 3 [xy] 3, then took the answer and entered [xy] 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 * 10[sup]9391[/sup]. 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 10[sup]1,212,755,270,733[/sup]. Getting back to the number g[sub]1[/sub], we now have

g[sub]1[/sub] ~ 3 ^^^ 10[sup]1,212,755,270,733[/sup]

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 g[sub]2[/sub]:

g[sub]2[/sub] = 3 (g[sub]1[/sub] of ^ in a row) 3

And then generalize for g[sub]n[/sub]:

g[sub]n[/sub] = 3 (g[sub](n - 1)[/sub] of ^ in a row) 3

Graham’s number is g[sub]64[/sub]. 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 g[sub]1[/sub].

[Bippy_the_Beardless]

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]

I dunno, I was just going off the Mathworld articles. I haven’t studied this stuff before.

[Mathochist]

@Chronos

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.

[ultrafilter]

@Bippy the Beardless

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. g[sub]0[/sub] is whatever Graham chose for it to be.

[David_Simmons]

@sturmhauke

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

g[sub]1[/sub] = 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 It won’t calculate 3 ^^ 27 directly, so I entered 3 [xy] 3, then took the answer and entered [xy] 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 * 10[sup]9391[/sup]. 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 10[sup]1,212,755,270,733[/sup]. Getting back to the number g[sub]1[/sub], we now have

g[sub]1[/sub] ~ 3 ^^^ 10[sup]1,212,755,270,733[/sup]

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 g[sub]2[/sub]:

g[sub]2[/sub] = 3 (g[sub]1[/sub] of ^ in a row) 3

And then generalize for g[sub]n[/sub]:

g[sub]n[/sub] = 3 (g[sub](n - 1)[/sub] of ^ in a row) 3

Graham’s number is g[sub]64[/sub]. 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 g[sub]1[/sub].

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 Number[sub]2[/sub] 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.