This might not be the best written or best sourced article, but I would start here: https://en.wikipedia.org/wiki/Computable_analysis https://en.wikipedia.org/w

This might not be the best written or best sourced article, but I would start here: https://en.wikipedia.org/wiki/Computable_analysis https://en.wikipedia.org/wiki/Computable_analysis Most operations for defining real numbers are computable. The reason for this is that, in practice, continuous functions and functionals are the same thing as computable functions and functionals. Continuous functions which aren't computable are generally pathological curiosities. One consequence of this heuristic is that since the Riemann integral is a continuous functional, one might guess that it is also computable. Indeed it is (but the proof isn't obvious). This then implies the computability of the constant pi because pi is equal to the integral 1 ⌠ ⎮ ________ ⎮ ╱ 2 ⎮ 2⋅╲╱ 1 - x dx ⌡ -1 The computability of integration implies the computability of root-finding because one can use the argument principle to do it.