What is the largest possible number you can write using only 2 digits?

[MadHatter]

This trivia question was thrown at me tonight, so I responded with the obvious remark of "99".

Wrong.

I got this question in a trivia chat room, so the answer was never revealed.

Does anyone here know the answer?

[Una Persson]

FF? (Hexadecimal)

[GKittridge]

9 raised to the 9th power (which is written with a regular 9 followed by a superscript 9). This is the number 9 multiplied by itself 9 times, which is 387,420,489.

[Joe_Cool]

Originally posted by GKittridge
9 raised to the 9th power (which is written with a regular 9 followed by a superscript 9). This is the number 9 multiplied by itself 9 times, which is 387,420,489.

I can beat it (provided you can use other symbols):

9!9!

9! is the same as writing 1*2*3*4*5*6*7*8*9, which is 362880.

So 9!9! is the same as 362880362880, which is some huge number I don't even want to think about. It should have something close to 2 million digits. If my computer ever gets done running the calculation I'll give you the answer to 10 sigfigs or so.

[whitetho]

How about 1/0 (one divided by zero), which is infinity?

[Scarlett67]

Originally posted by whitetho
How about 1/0 (one divided by zero), which is infinity?

1/0 is undefined, because there is no number you can multiply by 0 to get 1.

(BTW, 0/0 is called indeterminate, because you can multiply any number by 0 to get 0.)

[neuroman]

Originally posted by Anthracite
FF? (Hexadecimal)

I think the use of the term "digits" means only 0-9 (ie, two base 10 characters.) Otherwise you could use an arbitrarily large base.

[yabob]

And you have to disallow other punctuation marks as well, or you can apply factorials to factorials:

99!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

would be quite large, indeed. If you limit to the same number of operator symbols as digits, 99!! beats Joe_cool's example - 99! should be in the vicinity of 10120 (very rough estimate). The factorial of that ...

My first thought was 99. I would guess that this was the "trick" the chat room was looking for - no punctuation marks involved.

[Joe_Cool]

Originally posted by yabob
And you have to disallow other punctuation marks as well, or you can apply factorials to factorials:

99!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

would be quite large, indeed. If you limit to the same number of operator symbols as digits, 99!! beats Joe_cool's example - 99! should be in the vicinity of 10120 (very rough estimate). The factorial of that ...

My first thought was 99. I would guess that this was the "trick" the chat room was looking for - no punctuation marks involved.

uh, good point. I didn't think of that. haha

[yabob]

Sorry, that estimate's way low ... I meant to say 10140 the way I was thinking about it, and it's still a very rough estimate.

(my calculator shows 69! has an exponent of 98, and won't show a higher factorial because the exponent exceeds 100. There's 30 more factors all between 70 and 99. 100's would add 60 to the exponent, so I just guess that I wind up adding about 40. I told you it was a rough estimate. Anybody that cares to can easily produce a better estimate with a bit of calculation.)

[The Ryan]

Originally posted by yabob
My first thought was 99. I would guess that this was the "trick" the chat room was looking for - no punctuation marks involved.
Although in a chat room, it would require punctuation marks. Or at least, special coding.
What about aleph 9? Or aleph aleph? :)

[Chronos]

The Windows calculator can handle up to 10499, I believe, and it says that 99! is 9.332621544394e+155 . After that, there's some sort of formula for approximating large factorials (think we're justified in calling that large?), but I can't remember what it is... I know it has an xx term in it, but other than that...

Also, using conventional notation, if you want to apply multiple factorials, you need parentheses. A double exclamation point is usually interpreted as a symbol in itself, and signifies what's known as the "double factorial", where you skip every other factor. For instance, 99!! would be 99*97*95*93*...*5*3*1 . Of course, this is smaller than 9! . I think that using only two nines and two other characters, Joe_Cool's 9!9!, at 6.44*102017526 (no programming required, just a little 8-digit scientific calculator and some algebra) is the best we can do.

[Chronos]

*slaps The Ryan with a transfinitely large trout* Silly, Aleph isn't a digit, and by Russel's Paradox, AlephAleph is undefined.

[yabob]

OK. I'd never seen that convention before. Just out of curiosity - is there any combinatoric problem you know of where it turns out to be a useful construct?

[aseymayo]

I can write any large number you like, if the two digits are my two fingers and I'm writing in dust (or sand - but dust is a lot more common at my house).

[black rabbit]

damn. aseymayo got to it before me.

[Sofa King]

2^n, where n represents any number proposed in this thread.

[Sofa King]

I should have said, "the largest number".

[Chronos]

The double factorial isn't used much in combinatorials (I think), but it can be found in many infinite-series representations of functions. When I get to work I'll see if I can look any up for you.

[lawoot]

Originally posted by aseymayo
I can write any large number you like, if the two digits are my two fingers and I'm writing in dust (or sand - but dust is a lot more common at my house).

Knowing how alot of these trivia/brain teaser tings work, this probably IS the right answer

[CC]

re: the above: let's agree that when it comes to silly questions or trivia or puzzles like this, that there exist a large number of good answers. "The answer..." ? I don't think so. Maybe "the answer that the guy making it up was thinking of," ...ya know? I mean, don't sell yourself short - NO one really has THE answer.

[JasonFin]

Don't forget about tetration! Tetration, for those not in the know, is iterated exponentiation, in the same sense that exponentiation is iterated multiplication and multiplication is iterated addition. Tetration is normally represented by a superscript to the left of a number. yx can be defined to mean "an exponentiated stack of y many x's."

99 represents 999999999 (note the nine nines in the series; that's the way tetration works). This number is so unimaginably huge that scientific notation fails completely in expressing it. Even a googelplex is only 1010102, or less than 410 — enormously smaller than 99.

I doubt this is the answer that the ask er of the original question had in mind, but it's probably larger than whatever he or she was thinking of.

[wolfman]

how about 11? (If I screwed that up its supposed to be 1 in base 1). I'm not entirely sure what the mathematical consensus on the value of that is, but it could be viewed as an infinity.

[Chronos]

Well, I found a real example that uses the double factorial, but it wasn't the example I was thinking of. I noticed it in a table of integrals, so I stopped looking. To wit:
/ .5Pi2n+1(2n)!!
|(sinx) dx = --------
/ 0(2n+1)!!(The Handbook of Chemistry and Physics, CRC Press, integral #650)
It also occurs to me that we can do even better than 9!9!... We could take 99!! . Or, if we want to be non-mathematical smart aleks, we can go

/¯¯\/¯¯\
\__/\__/
//
//which is a very large number, indeed.

[Cabbage]

Chronos:

Aleph isn't a digit, and by Russel's Paradox, AlephAleph is undefined.

Are you sure about the AlephAleph part? It seems like an OK cardinal to me. I don't think Russel's Paradox negated the existence of any particular cardinals, rather, it just said that it was contradictory to speak of such things as the "set of all sets" or, more specifically, "the set of all sets that do not contain themselves".

[APB9999]

How about limx->0 1/x

There are two digits in the above expression. Can't get much bigger than that!

Silly Nitpick:
Another answer is, you can make ANY number with two digits, if they are one and zero and are used in a binary system. You want to say using two numerals.

[Fernmeldetruppe]

Just turn an "8" on it's side. (That seems to be what we're reaching for.)

If we must use a second digit, I would add a "+0".

(Of course, we could take two sideways "8"'s and sum them, multiply them, make them and exponent or make them a tetration, but I'm not sure that those numbers would be defined, meaningful or even bigger.)

[Una Persson]

Originally posted by neuroman
Originally posted by Anthracite
FF? (Hexadecimal)

I think the use of the term "digits" means only 0-9 (ie, two base 10 characters.) Otherwise you could use an arbitrarily large base.
It's evident from the posts immediately after mine that I missed the point entirely. There's no need to pick my example out.

Plus, others are using an awful lot of flexibility in defining "digits", or adding other items (!, lim, etc.). I've always heard of hex numbers described as hex "digits". But it's irrelevant, since I obviously missed the answer.

[Chronos]

Well, 'infinity', itself, isn't a number, but there are numbers which are infinitely large-- Those are the alephs that me, Cabbage, and The Ryan mentioned. They're called transfinite cardinals. Aleph0 is the cardinality of the set of integers (the amount of integers), aleph1 is the cardinality of the set of all subsets of the integers, and also the cardinality of the set of all real numbers, aleph2 is the cardinality of the set of all subsets of a set of cardinality aleph1, etc. Interestingly, most operations on them are defined, but even tetration will just get you back the same number.

And no, Cabbage, I'm not certain about the alephaleph business, especially not if a real mathematician says otherwise. That was just a wild guess from a non-mathematician.

[Cabbage]

...aleph1 is the cardinality of the set of all subsets of the integers, and also the cardinality of the set of all real numbers, aleph2 is the cardinality of the set of all subsets of a set of cardinality aleph1, etc.

Actually, that's the (generalized) continuum hypothesis. Aleph0 (the first transfinite cardinal) is the cardinality of the integers, and 2Aleph0 is the cardinality of the set of all subsets of the integers (and the cardinality of the set of real numbers), but it isn't necessarily the case that 2Aleph0 = Aleph1 (the second transfinite cardinal). 2Aleph0 is at least as big as Aleph1, but it could be much larger. The continuum hypothesis says that they are, in fact, equal, but the continuum hypothesis is independent of the standard set theory axioms. (The generalized continuum hypthesis says that 2Alephn = Alephn+1 for all n).

Some set theorists have been working toward a "natural" extension of the standard axioms which would decide the truth/falsehood of the continuum hypothesis. The tendency has been to reject the continuum hypothesis; in fact, a new axiom has been proposed (Woodin's axiom, which I know nothing about) which implies that the continuum hypothesis is false--the cardinality of the reals is strictly greater than Aleph1.

[yabob]

I'll put two cents also, though my knowledge of foundations is VERY rusty:

I'd like you to tell me what alephaleph means. I can't attach a meaning to it, as alephs with integer subscripts are defined orders of cardinality, aleph not being a symbol with meaning in its own right.

Chronos, you are accepting something called the continuum hypothesis when you say that aleph0 = cardinality of the real numbers. Like the axiom of choice, this is something you can be consistent both with and without. Most people accept "with".

The generalized continuum hypothesis suggests that you get all orders of cardinality by marching through "power set of power set of ..." to obtain the sequence of alephn's.

I poked around for a background document. I'm not happy with this, as it's a bit heavy going for the casual reader, but anyway:

http://www.ii.com/math/ch

And I saw your use of !!. Thanks.

[Chronos]

Ok, I'll defer to the mathematicians on that one... I must have mis-learned the continuum hypothesis. The way I was taught, was that it was proven using the standard axioms that the power set of the integers had the same cardinality as the reals. What was presented in the (admittedly introductory-level) class I had as the Continuum Hypothesis was that there were other transfinite cardinals in between aleph0 and aleph1. Apparently, I was taught wrong, or at least, learned wrong.

[yabob]

I see my reply crossed Cabbage's. Cabbage is probably more up to date than I am when he suggests current trends to reject the continuum hypothesis. Believe him.

[Cabbage]

yabob:

I'd like you to tell me what alephaleph means. I can't attach a meaning to it, as alephs with integer subscripts are defined orders of cardinality, aleph not being a symbol with meaning in its own right.

I read it as meaning alephaleph0, i.e., the smallest transfinite cardinal that is greater than infinitely many transfinite cardinals (aleph0, aleph1, aleph2, aleph3,...(running through all the natural numbers),...,alephaleph0).

Actually, however, I think it would be more proper to index the aleph's with ordinals rather than cardinals, so instead of alephaleph0, I would write it as alephomega (omega is the first infinite ordinal, just as aleph0 is the first infinite cardinal).