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On the contrary - computers certainly can represent irrational numbers - but not with floating points.
by daniel-levin 10y ago
On the contrary - computers certainly can represent irrational numbers - but not with floating points.
- jacobolus 10y agoComputers, like humans, can only represent a vanishingly tiny proportion of “real numbers”, either by giving specific ones a name, or some kind of symbolic formula by which to compute with them. To do practical computations we need to break everything down into basic arithmetic operations, which ultimately means we have a concrete multiplication/addition(/other operation) table encoded somewhere, e.g. in silicon. One that I like for doing exact arithmetic on numbers represented by black-box functions is a continued fraction representation, see http://perl.plover.com/yak/cftalk/INFO/gosper.txt http://perl.plover.com/yak/cftalk/INFO/gosper.txt (Note: floating point is successful because trying to use irrational numbers in computations is most of the time a huge waste of computational resources compared to approximating with a binary fraction.)
- evincarofautumn 10y agoHere’s an interesting precise representation of algebraic numbers as roots of integer polynomials: http://twistedoakstudios.com/blog/Post6871_impractical-experiments-1-representing-numbers-as-polynomials http://twistedoakstudios.com/blog/Post6871_impractical-exper...
- conistonwater 10y agoThat's pretty much how a typical CAS represents algebraic numbers, I don't think there's much impractical about that idea.
- tgb 10y agoNo, but algebraic numbers are only a tiny portion of all irrational numbers and importantly misses key ones like e and pi, so the original point still stands.