Are certain measurements always irrational numbers?

Question

In statistics we distinguish between discrete and continous random variables. In my course we were given examples for continous random variables like the mass of a baby and speed of a car since both can take every value in between a certain intervall. The definition for discrete random variables was that you can actually count them.

And here my question raises: Cantor proved that rational numbers are countable. Only the amount of real numbers and imaginary numbers are not countable.

So is the idea of providing such examples like mass and speed reasoned in the fact that they are basically irrational numbers?

Like theoretically you could give endless significant numbers for the mass of an object with an infinitely exact scale.

Answer 1

Mass and speed are classically assumed to be part of $\mathbb{R}_{\ge0}$. They could be irrational, rational, or integral - what matters is that they are real. In either of these three cases, you would not know the value of your quantity a priori, so the better your measurement device, the more significant figures you would determine.

Answer 2

Another set of thoughts on your question: Measurement is not predicated on the idea that the values are irrational—as lionsuneater says in their answer, they're just assumed to be real values—but it is true that at any point in time, the value is irrational with probability $1$.

This does not mean that they can't be rational, however. If we assume an entirely classical world, then as an object accelerates from $0$ m/s to (say) $30$ m/s, its velocity must pass through rational values—infinitely often, in fact. It's just that probability is "funny" on spaces with infinite possibilities, and real values are "almost all" irrational in a precise sense that we don't need to get into here.