Sunday, November 4, 2012

Fun with Exorbitantly Large Numbers

I came up with a concept in the shower this morning called the Turing Constant, which I defined as the number of "operations" done by a system in a single step. In a four core computer, this would be four. I chose to represent the Turing Constant with the Greek letter Xi, which is surprisingly hard to type, so I won't.

Then, I had an idea. From a purely mathematical perspective, it's possible to calculate the Turing Constant of any system - like the universe. I went a head and made a few assumptions: I assumed the mass of the universe to be constant to within a reasonable degree, thinking that the matter-energy conversion produces a net change of zero. Mass and energy are equivalent in physics, so this is not a huge problem. Once you know the mass of the universe, all you need to know is how many moles of stuff are in the universe.

That's right, moles. One mole is 6.02x1026 of a thing. It's equivalent to the mean molecular weight of an object. Using fancy (and fuzzy) maths involving the expanse of the known universe, its suspected density, and the mean molecular weight of the universe's five most common elements, I came up with a very large number.

1.77 × 1081

As it happens, this represents more or less the number of atoms in the universe. It is a very, very large number, but certainly not the largest number ever derived. It's not even the largest number I know. It is, however, very, very big. It's 177 million, trillion, trillion, trillion, trillion, trillion, trillion atoms. It's so large that it's enough to construct, well, a universe.

And that's how many permutations happen within the universe in a single step. But what do I mean by a step? Well, that's fairly elementary. A "step" in the universe should be the smallest meaningful period of time. An instant.

We have such a time. See, the universe has a finite resolution in terms of distance - the Planck Length. The time it takes a photon to traverse one Planck Length is the Planck Time, which is the smallest meaningful unit of time.

It's about 5.4 x 10-44 seconds.

As it happens, the age of the universe is known! That means, we can figure out how many steps it's taken to get to the precise time: 7.90x1060 steps.

How many individual things have happened in that time?

1.40 × 10142

This is an incredibly large number. 

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