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/ |
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