View Full Version : Application of Googolplex.
12-04-2006, 06:12 PM
As you all probably know, a googol is 1 followed by 100 zeros. Googolplex is 1 followed by googol zeros. My question is simply this: where in the universe do you find something that numbers googolplex? As I understand it, even the total number of atoms in the universe is under a googol. Where do you find a googolplex number of things? Give at least one example please.
Thank you in advance to all who reply :)
12-04-2006, 06:18 PM
it is found in mathematics only, I think. there aren't that many fundamental particles in the universe according to the most popular models
12-04-2006, 06:46 PM
Apparently the biggest number ever proposed as part of something that is useful is Graham's number (http://en.wikipedia.org/wiki/Graham%27s_number) .
It's the lowest known upper bound of something really freaky, and can't be expressed using our normal 10^power notation even if we assigned all the particles in the universe to it. Still there is hope for us mortals because the lower bound is 6, or possibly 11. Now that's a margin of error! :p
12-04-2006, 06:50 PM
Combinations can easily reach over 10100. 70! (70*69*68*...*3*2*1) is just over it, and factorials are how you calculate combinations and permutations. For example, if you combine two decks of cards and regard each card as distinct, there are 104!/(n!*104-n)! ways you could choose n cards.
12-04-2006, 07:02 PM
The number of universes that are expected to obey the various rules of string theory is estimated to be far larger than a googol: 10500.
In 2003, cosmologist Andrei Linde of Stanford University and his collaborators showed that string theory allows for the existence of dark energy, but without specifying the value of the cosmological constant. String theory, they found, produces a mathematical graph shaped like a mountainous landscape, where altitude represents the value of the cosmological constant. After the big bang, the value would settle on a low point somewhere between the peaks and valleys of the landscape. But there could be on the order of 10500 possible low points -- with different corresponding values for the cosmological constant -- and no obvious reason for the universe to pick the one we observe in nature.
Combinations and permutations of a large number of items can easily exceed a googol.
But no physical thing, not even the number of zeroes to fill up the universe at 1000 to the cubic inch, is even a googol. A googolplex is not remotely imaginable or expressible. There can't be any examples.
12-04-2006, 08:20 PM
Not to be confused with the Googleplex (http://maps.google.com/maps?f=q&hl=en&q=googleplex&ie=UTF8&filter=0&sll=46.800059,-95.625&sspn=62.007115,114.257812&z=18&ll=37.422828,-122.084804&spn=0.002203,0.003487&t=h&om=1&iwloc=B).
12-04-2006, 09:21 PM
Sure theres an application for the googleplex. Assume there are 10^80 particles in the universe and 10^20 seconds in the age of the universe. If we treat each particle as a 10 sided die and roll it once per second, the chances of it coming up 1 every time is 10^10^20. The chances of every die coming up 1 every second is 1 in (10^10^20)^(10^80) which is 10^(10^20*10^80) which is 10^10^100. So on average, for every googleplex galaxies, you will have one in which every particle comes up as all 1 every second of it's existance.
If we assume that there are 10^60 plank time units in the age of the galaxy, and we can roll it every plank time, then theres a 1 in 10^10^140 chance. Given that this is roughly what a particle does at every plank time unit (although I'm not sure how many discrete "decisions" each particle has per plank time unit), this can be thought of as the probability of a random universe existing. In other words, there are about 10^10^140 potential universes, give or take a few orders of orders of magnitude.
Another meaningful instance of a googleplex can be in the measuring of statistical distributions. I'm not quite certain but I think the number of standard deviations from the norm required to get a probability of 1/(1 googleplex) is a reasonable number. Thus, if we assumed that the quantum probability curve of a particle is normal, then the probability of it being on the other side of the universe should be somewhere around the vicinity of a 1 googleplexth. If anybody is willing to solve for the cdf of the normal curve to get an analytical answer would be much appreciated.
12-04-2006, 09:45 PM
Ok, so I've played around with some statistical packages and it seems like the the chances of something exceeding roughly 10^50 standard deviations from the norm is about one googleplexth. Given one meter is 10^35 planck lengths and 1 light year is 10^15 meters, this means that if you had a normal probability distribution for a particle with mean at one spot and a standard deviation of 1 planck length, the chances of it being 1 light year away is 1 googleplexth.
I don't think anything in the real world follows the normal distribution that far before getting quantized but it works as a theoretical construct.
12-04-2006, 10:19 PM
All the remarks relating to the "universe" should have said "observable universe", as (in my understanding anyway) the universe is infinite and it is only its extent that is observable from a single location that is finite.
I think University of Pennsylvania physicist Max Tegmark has demonstrated that the average distance from here to the closest "Hubble volume" (observable universe centered on another point in which every observable detail would have to match because packing the space with protons in random quantum states would have reproduced our pattern by then is 10^10^118 m. That is not very different from a "googleplex" (as these things go, eh?)
Anybody more familiar with Tegmark's work please correct me - this is from memory.
12-04-2006, 10:48 PM
All the remarks relating to the "universe" should have said "observable universe", as (in my understanding anyway) the universe is infinite and it is only its extent that is observable from a single location that is finite.Well, the Universe might be infinite, and that's the standard default assumption given the curvature (or rather, lack thereof) we observe, but if so, it's unprovable. So I wouldn't say you're wrong, but I wouldn't say you're right, either.
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