zlacker

Why not mention factorials? 9!! is way bigger than 9^9^9. And you can write many exclamation marks in 15 seconds.

>>crayol+(OP)
Wow, 9!! is so large that the exponent almost needs it's own exponent.

    9^9^9 =~ 2e77
    9!! =~ 1.6e1859933

>>crayol+(OP)
Good one! I was only thinking of e^e^9 which is way smaller than 9!!

>>vacri+L
Correct me if I'm wrong, but aren't you supposed to evaluate stacked exponents from the top down? That is, 9^(9^9) instead of (9^9)^9. If that's the case, 9^(9^9) is much larger than 9!!. Though I'm not sure how much larger, since I couldn't find any big integer calculators online that would give me an actual result for 9^(9^9).

>>crayol+(OP)
As peterjmag indicated in his comment, 9^9^9 looks like 9^(9^9) which is actually greater than 9!!. A different way of looking at it is n! < n^n. We can see from here that factorization isn't really the new paradigm that the author is looking for; it's just a part of the exponentiation paradigm.

Furthermore, factorials don't really scale or stack easily. What the author is getting at in the relevant location is stacking the same concept: 1. Multiplying is just adding the same number several times. 2. Exponentiation is just multiplying the same number several times. 3. Tetration is just exponentiation several times. 4. Etc.

This allows us to generate the infinite hierarchy easily expressible by the ackerman numbers (which is basically A(i) = f_i(i,i)), which doesn't generate itself as easily with factorialization in place of exponentiation.

>>peterj+w5
9^(9^9) is ... wait for it... 4e369693099 bigger than 9!!

http://www.wolframalpha.com/input/?i=9%5E9%5E9+-+9%21%21

>>cradle+C6
Ah thanks! How could I forget Wolfram Alpha? =)

>>crayol+(OP)
When writing factorials you would want to write (9!)! since 9!! is actually a different operation (the double factorial). 9!! = 9 x 7 x 5 x 3 x 1, so 9!! is less than 9!.

>>cradle+C6
Shouldn't it be (9!)!? http://www.wolframalpha.com/input/?i=9%5E9%5E9+-+%289%21%29%...

>>peterj+w5
It seems you're correct - I was just pounding the values in sequence into a calculator, much like a determined monkey would.