Googol Factorial

Date: 11/09/96 at 10:48:18
From: Chris Myrick
Subject: Googol Factorial

What is googol factorial?  I saw a web page that gave instructions for 
computing the factorials of very large numbers, but I couldn't figure 
out how to compute googol!.  The web page only gave the first few 
hundred digits of the factorial.

Date: 12/11/96 at 17:28:11
From: Doctor Daniel
Subject: Re: Googol Factorial

Wow!  A googol is a pretty impressively sized number to begin with 
since it is a 1 followed by 100 zeros.  Taking its factorial, which 
means multiplying it by all the whole numbers greater than zero which 
are less than it, results in an immense number!  Let's think about how 
big it would be:

10! is about 3.6 million, which is roughly 3 followed by six digits.

30! is roughly 3 followed by 32 zeros (do you know scientific 
notation? It's 3 x 10^32).  

60! is roughly 1 followed by 80 zeros.  

From this you can begin to see a pattern: the number of digits in the 
factorial of a number is a bit bigger than the number itself.  (Like 
how 30! is roughly 30 digits long). This is true, in particular, for 
large numbers.  

So, since it is true that this pattern continues as the number gets 
bigger, googol! will be more than a googol digits long. 
(I guess it'll actually be a few hundred googol, but that's not 
really a big deal at that point since the numbers are already so 
large.)

That's a really big number.  How big is it?  Well, suppose you somehow 
computed it on a computer.  How long would it take a computer to print 
the answer?

The answer is, more or less, forever.  In fact, the number of digits 
in googol! is VASTLY greater than the number of particles in the 
universe, so even if each particle were to represent a digit, we'd 
STILL be stuck.  Another way of thinking about its size is to let each 
digit represent one year. But that's billions of billions of billions 
of billions of... of billions of years.  That's a while!

I suspect that the method you saw on the Web was some trick formula 
used to approximate factorials.  Since they only needed to print the 
first couple of hundred digits, they didn't bother to compute their 
answer fully (because they couldn't).  

A lot of formulas like this exist: one is called Stirling's 
approximation.  If you don't know some algebra, this might confuse 
you, but if you do, it might amuse you to see the formula I usually 
wind up using, which comes from Stirling's approximation:

  n! > sqrt (2*pi*n) * (n/e)^n 
  n! < sqrt (2*pi*n) * (n/e)^(n+1/12n)

If we set n = googol, you can see just how huge this is.  But it 
might be possible somehow to get the first several digits of googol! 
this way. There's also a few other things that are like factorial that 
we can approximate.  So that's possible too.

So you can't easily compute googol! after all.  But this should give 
you some idea of why not.

-Dr. Daniel,  The Math Forum
 Check out our web site  http://mathforum.org/dr.math/   

Date: Tue, 8 Dec 1998 21:34:51
From: Jason Rodgers
Subject: Re: Googol Factorial

A good while back the question was raised on how to compute (10^100)!

To figure out the first digits use this method.
It's good for x => 10 and only to about 12 decimal digits

An=[(x+.5)*ln(x)-x + 1/(12x)-1/(360x^3)+1/(1260x^5)-1/(1680x^7)
+.918938533205]/ln[10]

The power of ten of the answer is int(An).
The first few digits of x! are 10^(An-int(An)).

The reason I wrote is because the problem just looked interesting.

Date: Sun, 10 Jun 2001 01:43:26
From: Robert Munafo
Subject: Re: Googol Factorial

The factorial of googol is given by a series expansion of the Gamma
function, and it is approximately:

  (10^100) ! = 10 ^ ( 9.9565705518097 x 10 ^ 101 )