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So, TL;DR:, one cannot build a meaningful calculus using the rational numbers alone. Not unexpected, judging by its history... A more interesting (to me, anyway
by Koshkin 7y ago
So, TL;DR:, one cannot build a meaningful calculus using the rational numbers alone. Not unexpected, judging by its history... A more interesting (to me, anyway) fact is that one can build (or "model") any number system - all integers, rationals, reals, etc. - using just the set of natural numbers (i.e. positive integers).
- Jaxan 7y agoWell building the reals from rationals is not easy at all. It took a few decades to get a reasonable definiton. And in constructive mathematics, there are several competing (non equivalent) definitions of the reals. It's a very deep topic, yet we seem to understand (at least in calculus) what real numbers are.