0

Irrationals and transcendentals are interesting and useful numbers, to be sure, but I think the importance of rationals can be forgotten when we get caught up in the mystique of other numbers. They seem, in some sense, to be the only numbers with which we can do completely precise computations.

To this end, I've been wondering if there is a way to restrict calculus to rational numbers and still obtain similar results.

Immediately there are problems with the exponential and trigonometric functions (since $e$ and $\pi$ are certainly not rational). But theories of rational trigonometry exist to handle the latter point.

Anyway, is this something people have looked into before? My apologies if my question doesn't make sense.

Ducky
  • 2,116
  • 1
    Well you have to abandon the usual notion of continuity. So you're going to lose the intermediate value theorem right off the bat, and you need a new definition for differentiability. – dalastboss Feb 01 '15 at 22:58
  • 2
    You can see at the question http://math.stackexchange.com/questions/1078593/do-we-really-need-reals. – Emilio Novati Feb 01 '15 at 22:58
  • Some completely precise computations can be done with irrational numbers, for example: $\sqrt2\sqrt8=4$. – John Bentin Feb 01 '15 at 23:00
  • Sorry - seems my question has already been asked. And to answer the question of "why?": plainly I like rationals better, and also, no construction of the real numbers that I've seen has particularly resonated with me. – Ducky Feb 01 '15 at 23:03
  • If $\sqrt 2\cdot\sqrt 3=\sqrt 6$ is inaccurate, what do you mean accurate? Something that can be done by hand, or with a computer? How will you calculate something on numbers which are larger than the number of atoms in the universe? – Asaf Karagila Feb 01 '15 at 23:03
  • By "accurate" or "precise", I guess I would mean that the answer is something whose decimal representation can be written down in finite steps.

    Not that this is the only, or best, definition, but it's what I like. It's why rationals please me.

     – Ducky Feb 01 '15 at 23:06
  • Please write for me the result of $A(5,5)$, the Ackermann function with $5$ in both arguments. – Asaf Karagila Feb 01 '15 at 23:10
  • Sorry, but I don't know how to compute it. From my understanding, though, it's possible to do so in finite steps since it is a terminating recursive process. – Ducky Feb 02 '15 at 03:05

0 Answers0