What is the 'complete' number system to measure objects we see in reality?

Question

It is clear that we need at least the real numbers to measure lengths as it is very easy to construct a length of $\sqrt{2}$ geometrically. However, consider every physics situation where we want to measure some thing (eg: length , mass time etc ) are real numbers enough to measure any physical system?

Answer 1

Are real numbers enough to mathematically model physical system ?

Not really. You need complex numbers to model the behaviour of quantum wave functions. Of course, you could replace complex numbers by pairs of real numbers with appropriate rules for addition and multiplication. But then you could replace real numbers with limits of sequences of rational numbers with appropriate rules for addition and multiplication. And you could replace rational numbers with pairs of integers with appropriate rules for addition and multiplication. So if you go down this route, you could model every physical system using only integers - plus a lot of increasingly complicated rules for modelling other number systems using only integers. Just like you could work out the cost of your weekly shopping trip using only Roman numerals - but why would you want to ?

Of course, we don't actually know for certain that space and time are continuous. If they are discrete then maybe real numbers are unnecessary - you could model the whole of reality using only (very large) integers (and, in the complex realm, Gaussian integers).

Answer 2

consider every physics situation where we want to measure some thing

Since you are talking about measurements there is no need to go to the reals. All measurements have some uncertainty. And regardless of how small that uncertainty is, there are an infinite number of rational numbers that are consistent with the measurement to within that uncertainty. So you need rationals to represent measurements, but not reals.

real numbers enough to mathematically model the physical system?

Now, this is a different question. Just because the measurements only need rationals does not mean that the models will work with only the rationals. However, it is difficult to say that there are no alternative models that might work with a different set of numbers. I think that all we can do is say what types of numbers our current models do actually use.

Our current models use real numbers, complex numbers, and tensors formed with real and complex numbers. Our current models also would work well or even better with the hyperreals. And some of the statements that we make about our models are actually incorrect with the reals but correct with hyperreals. So I think that there are good arguments that hyperreals may be even more natural than reals in our existing models.

But again, I think it is wise to recognize that measurements don’t require anything beyond the rationals.

Answer 3

It is clear that we need at least the real numbers to measure lengths as it is very easy to construct a length of $\sqrt2$ geometrically.

Is it, now? As I understand it, the traditional, geometric way to construct a length of $\sqrt2$, from a length of $1$, is to draw a certain sequence of perfectly straight lines and perfect circles. While this is nice and well-defined in the mathematical world of geometry, it is also unphysical. In physics, all measurements have nonzero uncertainties, and since the rationals are dense in the reals, it suffices to represent measured quantities with rational numbers. Of course, since $\mathbb{R}$ is an extension field of $\mathbb{Q}$, rationals can be treated as elements of $\mathbb{R}$, but the key property of $\mathbb{R}$ that distinguishes it from $\mathbb{Q}$ (completeness) is not that important in this context, so it's fine to treat measured quantities as elements of $\mathbb{Q}$.