Googol FactorialDate: 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 ) |
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