If you know the diameter of the observable Universe and you want to calculate its circumference with the accuracy of the diameter of a proton, the number of digits of pi that you need is 43.
The smallest possible distance is the Plank lenght, 1,610^-35 (110^-15 is the diameter of a proton). And for that you only need around 60 digits of pi to calculate the circumference of the universe.
Of course, that is just for the simple operation of calculate the circumference given the diameter, more complex operations with pi may require more precision.
> According to the generalized uncertainty principle (a concept from speculative models of quantum gravity), the Planck length is, in principle, within a factor of 10, the shortest measurable length – and no theoretically known improvement in measurement instruments could change that.
At least according to Wikipedia, it seems it is indeed the smallest observable distance. Although it has never been proven and follows from the theoretical generalized uncertainty principle.
https://en.m.wikipedia.org/wiki/Planck_length#Theoretical_si...
From what I understand the particles you need to probe distances on the order of the Planck length are so energetic that their own gravity would start interfering. Their Swarzschild radius would become bigger than the distance you're trying to measure.
Does that mean the rest of the digits of pi are not "real," at least according to a realist rather than a Platonic philosophical position on the meaning and nature of mathematics? Seems like you could argue that digits beyond what are needed to render measurement to within one Planck length are meaningless and therefore a kind of fiction... at least if you take that philosophical position.
It's been estimated that if the universe were a computer, it could have performed no more than 10^120 operations on 10^90 bits of data so far (based on the size, age, and total energy of the known universe). http://arxiv.org/abs/quant-ph/0110141 I think the number of physically relevant bits of pi would be represented in there somewhere. But there's a long road ahead. If the universe keeps "computing" forever, the precision of numbers involved could also keep growing.
If the universe were a computer, it would not have to compute π to "simulate" physical processes. Mechanical processes don't depend on the value of π directly.
Pi appears in many more places than in ratio of circumferences to their diameter. For example, if you flip a coin n times, the probability of getting exactly n heads and n tails asymptotically tends towards 1 / sqrt(π*n). The probability that two randomly chosen integers are coprime is 6/π^2. Etc.
I think this assumes the order of magnitude of the size of the observable universe.
Clearly if the diameter of the universe is smaller than a proton then you don't need 43 digits of pi to calculate its circumference to smaller than the diameter of a proton. So if the diameter of the universe is 10^1000000^100000000 proton wide the precision you need for pi would be way higher than 43?
The post i replied to started off with "If you know the diameter of the observable Universe" but if the result it depends on the size of the observable universe then you ought to say "given the size of the observable universe".
Starting with 'if' would imply that the result only requires you 'knowing' the size of the obsv. unv. but does not depend on its actual size.
There's a difference between "the size of the universe" and "the size of the observable universe", see also "cosmic microwave background", "dark energy", and "inflation."
