If you throw a dart to uniformly hit the interval [-1, 1] x [-1, 1], what is the probability that it will hit within the unit disk?
Real numbers are complete under all sorts of important properties such as integration of continuous functions over compact areas. If you don’t have this closure, the math becomes impractical.
Cool. So in principle we could just as well use the rationals from the standpoint of scientific inference. But we use the reals because it makes the math easier. Thank you.
I should probably also mention that if you actually used rationals, the way you would do it (when running into the tough integrals) would be by just phrasing everything in terms of bounded approximations, which is basically just unrolling a construction of the real numbers. So you might as well just use real numbers.
If you throw a dart to uniformly hit the interval [-1, 1] x [-1, 1], what is the probability that it will hit within the unit disk?
Real numbers are complete under all sorts of important properties such as integration of continuous functions over compact areas. If you don’t have this closure, the math becomes impractical.
Cool. So in principle we could just as well use the rationals from the standpoint of scientific inference. But we use the reals because it makes the math easier. Thank you.
I should probably also mention that if you actually used rationals, the way you would do it (when running into the tough integrals) would be by just phrasing everything in terms of bounded approximations, which is basically just unrolling a construction of the real numbers. So you might as well just use real numbers.