Comment Re:Computers do what they are told to (Score 4, Informative) 250
A system using proper fractions can actually get exactly the right answer every time OR it will overflow and we will know for a fact the answer isn't exact.
What theory of numeration are you using, that has all numbers rational? I'm sorry, but even the humble square root is something I don't want to give up, to say nothing of transcendental functions. The theory of exact arithmetic on the reals is not all that well developed. Bill Gosper makes a start, and a handful of researchers take it somewhat further, but actually using exact arithmetic for everything you'd want to do remains a mirage.