Why is physical distance not modelled on rational numbers? Physicists use real numbers to model distance or physical line. My intuition is completely neutral on whether to use rational numbers or reals for physical quantities like distance, or temperature etc. Is there any historical or psychological reasons do prefer reals? Or there is some mathematical advantages working with real. Is calculus on rational numbers harder than calculus on reals? 
 A: Calculus over the rational numbers is really not so pleasant. For example, the Intermediate Value Theorem, The Maximum Value Theorem, and The Mean Value Theorem all fail for functions continuous on the rationals.
Edit: A better alternative for $\mathbb{Q}$ would be the completion of it with respect to a $p$-adic metric (or the real numbers, of course). Calculus over $\mathbb{Q}_p$ has been studied, see here. See also for $p$-adic quantum mechanics.
A: Mathematically formal considerations of measuring starts with Greeks, as far as I know. Let us first consider Euclidean geometry, where we can construct right triangle with leg lengths $1$ and $1$. What is the length of hypotenuse? It is (the positive) solution of equation $x^2 = 2$ by Pythagoras theorem, easily shown not to be rational number. Furthermore, $\pi$ is by definition such number that circumference of a circle is given by $C = 2r\pi$, also not rational. Skipping some time ahead, yet another important mathematical constant is given as limit of sequence of rational numbers: $e = \lim_n(1+\frac 1n)^n$ which is not rational. Point being, reals emerge naturally even while we do elementary constructions.
A: Of course, the rationals are used for all experimental measurements. But as others are trying to tell you, it makes a lot of sense to do theoretical work and formula development in the idealized world of reals, and then map back to rational approximations when actual measurements and calculations are needed. There really is no harm in doing this given that the reals and rationals are dense in each other. And while there are not perfect squares or circles, it really is not better to use rational approximations of these objects. The completeness of the reals is used all over the place in calculus, and the rationals simply are not complete. Do theory in the reals and map back to rationals when you need a measurement.
A: The very definition of "physical quantities" is that they are "measurable".  Irrational numbers cannot, by definition, be measurable.  So either our definition of physical quantities is wrong, or incomplete, or all operations on physical quantities must be represented by operations only on rational numbers which must reduce to operations on the reals in the limit of idealistic models of relationships between physical quantities, such as the use of calculus of variations to solve Lagrangians and Hamiltonians.
