# Which number is bigger?

David113
Posers and Puzzles 30 Jun '11 13:28
1. 30 Jun '11 13:281 edit
9^(9^(9^(9^...))), with 100 exponentiations, or

(...(((9!)!)!)!...), with 100 factorials?

Putting it another way, what is bigger - a(100) where a(0)=9 and a(n+1)=9^a(n), or b(100) where b(0)=9 and b(n+1)=b(n)! ?
2. 30 Jun '11 15:50
Originally posted by David113
9^(9^(9^(9^...))), with 100 exponentiations, or

(...(((9!)!)!)!...), with 100 factorials?

Putting it another way, what is bigger - a(100) where a(0)=9 and a(n+1)=9^a(n), or b(100) where b(0)=9 and b(n+1)=b(n)! ?
Question: Taking the general case where "100" is replaced by m, wouldn't the same inequality relationship hold for m=100 as for m=1 and m=2? Or is there a "crossover point" where the inequality reverses? Why would there ever be a crossover point?

Obviously with m = 1, 9^9 >> 9!. So, calling 9^9 "g", and 9! "h", then just as obviously, g^9 >>> h!. And so on.

Or so it seems at first glance.
3. 30 Jun '11 16:04
Originally posted by JS357
Question: Taking the general case where "100" is replaced by m, wouldn't the same inequality relationship hold for m=100 as for m=1 and m=2? Or is there a "crossover point" where the inequality reverses? Why would there ever be a crossover point?

Obviously with m = 1, 9^9 >> 9!. So, calling 9^9 "g", and 9! "h", then just as obviously, g^9 >>> h!. And so on.

Or so it seems at first glance.
Edit: BIG OOPS. Too early in the day.
4. 30 Jun '11 18:13
Also, it is not g^9, but 9^g.
5. 01 Jul '11 17:23
Originally posted by David113
9^(9^(9^(9^...))), with 100 exponentiations, or

(...(((9!)!)!)!...), with 100 factorials?

Putting it another way, what is bigger - a(100) where a(0)=9 and a(n+1)=9^a(n), or b(100) where b(0)=9 and b(n+1)=b(n)! ?
OK I'll start over. Taks the first 9^9 compared to 9

9*9*9*9*9*9*9*9*9 = 387 420 489

9*8*7*6*5*4*3*2*1 = 362 880

(The above can be done using Google)

Now compare 387 420 489^9 to 362 880!

That's 1.9662705 e77 which is BIG, and ...

See http://en.wikipedia.org/wiki/Stirling%27s_approximation

... a number somewhat larger than (n/e)^n where n = 362,880 and e ~2.71828183

which tacks on "00000" to "1" n+1 times.

Meaning the scientific notation is about 1 e(10000*362,880)

which is HUGE.

And it keeps going that way.

But maybe there's another OOPS coming.
6. AThousandYoung
01 Jul '11 18:23
So...the powers function beats the factorial function right?
7. 01 Jul '11 19:09
NOT 387 420 489^9 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

9^387 420 489 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

(The exclamation marks are not factorials)
8. 02 Jul '11 00:124 edits
lets do it with tens

10^10 = 1e10
10! = 3 628 800

10^(10^10) = 1 followed by 10000000000 zeroes
at 64 characters per line and 200 lines per page that number would take 781250 pages to write down.

from stirling's apporoximation
n! ~ sqrt(2*pi*n)(n/e)^n

so 10!! ~ 4775*1334960^3628000

log10(10!!) ~ 3.6 + 3628000*6 ~ 21768003.6

so 10!! ~ 1 followed by 21768003 zeros
At 64 characters per line and 200 lines per page that number would be only 1700 pages.

So it looks like the nested factorial grows a lot slower than the nested power
9. 02 Jul '11 16:47
Originally posted by David113
NOT 387 420 489^9 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

9^387 420 489 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

(The exclamation marks are not factorials)
Oops again.

((2^2)^2)^2 = 256

and

2^(2^(2^2)) = 65 536
10. 03 Jul '11 22:22
Originally posted by iamatiger
lets do it with tens

10^10 = 1e10
10! = 3 628 800

10^(10^10) = 1 followed by 10000000000 zeroes
at 64 characters per line and 200 lines per page that number would take 781250 pages to write down.

from stirling's apporoximation
n! ~ sqrt(2*pi*n)(n/e)^n

so 10!! ~ 4775*1334960^3628000

log10(10!!) ~ 3.6 + 3628000*6 ~ 21768003.6

so 10!! ...[text shortened]... ges.

So it looks like the nested factorial grows a [b] lot
slower than the nested power[/b]
Now trying 9s

9^9 = 387420489
9! = 362880

9^(9^9) = 9^387420489
log10(9^(9^9)) = 387420489*log10(9)
log10(9^(9^9)) = 346275.522
so 9^(9^9) ~ 1 followed by 346275 zeroes

9!! ~ sqrt(2*pi*362880)(362880/e)^9 {stirling's approximation}
9!! ~ 1510*133496^9
log10(9!!) ~ log10(1510) + 9*log10(133496)
log10(9!!) ~ 49
so 9!! ~ 1 followed by 49 zeroes

Nested powers win hands down