Enumerate all the fractions with at most 3-digit denominator (0, 1, 1/2, 1/3, 2/3, 1/4, ..., 998/999) between 0 and 1. Now what is the probability that amongst them we find an approximation to a random real that is at least as good as 355/113 is to pi?
Seems to be about 15%, so not much mystery here. But we allow denominators as large as 999 because we use the decimal system. In hexadecimal it would be at most 2-digit denominators, up to ff = 255.
In this case, the probability is about 1%. Much more mystery. So it comes down to, how large a set of denominators we think 113 is drawn from, and whether living in a 1 in 100 universe is enough to be mysterious in this case.
So just a note, you've got a great deal of repetition in your search. 0.5, for instance, turns up once for every even denominator. That's going to skew your percentages a bit.
Yes, the constructed list contains a lot of duplicates (removing duplicates with the python set operations shortens it by 40% - would be a worthwhile optimization for the runtime, too), but my percentages are not based on that.
But: I sampled a random real, and looked at whether any fraction in the list is close enough to this real. And repeated the sampling 1000 times. This is how the percentages are calculated, and the duplicates in the list don't affect this.
Enumerate all the fractions with at most 3-digit denominator (0, 1, 1/2, 1/3, 2/3, 1/4, ..., 998/999) between 0 and 1. Now what is the probability that amongst them we find an approximation to a random real that is at least as good as 355/113 is to pi?
Seems to be about 15%, so not much mystery here. But we allow denominators as large as 999 because we use the decimal system. In hexadecimal it would be at most 2-digit denominators, up to ff = 255.
In this case, the probability is about 1%. Much more mystery. So it comes down to, how large a set of denominators we think 113 is drawn from, and whether living in a 1 in 100 universe is enough to be mysterious in this case.
Sampled by this python code (takes minutes to run): https://gist.github.com/3172978