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

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?

FF? (Hexadecimal)

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![sup]9![/sup]

9! is the same as writing 123456789, which is 362880.

So 9![sup]9![/sup] is the same as 362880[sup]362880[/sup], 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.

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.)

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.

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 10[sup]120[/sup] (very rough estimate). The factorial of that …

My first thought was 9[sup]9[/sup]. 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

Sorry, that estimate’s way low … I meant to say 10[sup]140[/sup] 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.)

Although in a chat room, it would require punctuation marks. Or at least, special coding.
What about aleph 9? Or aleph aleph? :slight_smile:

The Windows calculator can handle up to 10[sup]499[/sup], 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 x[sup]x[/sup] 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 99979593531 . Of course, this is smaller than 9! . I think that using only two nines and two other characters, Joe_Cool’s 9![sup]9![/sup], at 6.4410[sup]2017526[/sup] (no programming required, just a little 8-digit scientific calculator and some algebra) is the best we can do.

slaps The Ryan with a transfinitely large trout Silly, Aleph isn’t a digit, and by Russel’s Paradox, Aleph[sub]Aleph[/sub] is undefined.

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?

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).

damn. aseymayo got to it before me.

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

I should have said, “the largest number”.

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.

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