How Big is a Googolplex?

Date: 05/29/2003 at 08:59:21
From: Jim
Subject: Googolplex

Is there anything on Earth that is as big as a googolplex?

Date: 05/29/2003 at 10:16:17
From: Doctor Ian
Subject: Re: Googolplex

Hi Jim,

It's pretty easy to set up a situation in which the number of possible
_arrangements_ of a set of objects reaches a googol:

   How Big is a Googol?
   http://mathforum.org/library/drmath/view/62494.html 

And that's with only 70 objects!  

Suppose you have something like a salt crystal, with 10^23 sodium and
chlorine atoms in it. You could, in theory, dissolve the crystal in
water and then evaporate the water to get another crystal, in which
the atoms would be arranged differently.  

How many possible arrangements are there?  That would be

  (10^23)!  = 10^23 * (10^23 - 1) * (10^23 - 2) * ... * 3 * 2 * 1

Would that be as big as a googolplex?  We can use Stirling's
approximation,

            ~
         n! =  sqrt(2pi*n) * (n^n) * (e^-n)

to estimate the size of this factorial.

Our n is 10^23, but this will work out nicer if we use e^23 instead
(which makes our crystal about 1/4 of its original size):

     sqrt(2pi*e^23) * ((e^23)^(e^23)) * (e^-(e^23))

   = sqrt(2pi*e^23) * (e^(23*e^23)) * (e^-(e^23))

   = sqrt(2pi*e^23) * (e^(23*e^23 - e^23))

   = sqrt(2pi*e^23) * (e^(22*e^23))

which is a big number, but nowhere near a googolplex, which is

  googolplex = 10^googol = 10^(10^100)
   
Can we think of anything else that grows really quickly?  How about
the number of subsets of a set?  If a set has n elements, the number
of subsets of that set is 2^n.

So suppose we write the numbers 1-70 on a set of cards.  The number of
arrangements of those cards will be about a googol.  And the number of
subsets of those arrangements will be about 2^googol.  Now we're 
within striking distance of 10^googol, and it's just a matter of
playing around to find the right number of cards.  

So if you consider possible subsets of possible arrangements of cards
to be 'things', then we can pretty easily create a situation in which
there are a googolplex of those. 

Also, note that (googol!) > googolplex:

   Googol Factorial
   http://mathforum.org/library/drmath/view/57903.html 

So the number of possible arrangements OF the number of possible
arrangements of 70 cards is at least a googolplex. 

But the key word in these descriptions is 'possible'. 

See also, from our archives:

   More 0's in a Googolplex than Atoms on Earth? 
   http://mathforum.org/library/drmath/view/59178.html 

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/