Disclaimer: not a professional physicist or mathematician, so (deserved) tomato-throwing is welcome.
I've been pondering the "naturalness" of real numbers for some time now, in the sense of the degree to which they could faithfully describe measurable quantities in the physical world as we know it. One observation people make is that a number of infinite magnitude might be "unphysical", assuming the Universe is some finite multiple of Hubble volumes large, but as far as I'm aware there are viable theories that the Universe might be infinite. From that POV, I'm not concerned about infinite magnitude, but I do have a question about infinite decimal precision.
Unless I misunderstand things, quantized space implies that physical objects and distances can't be arbitrarily small - at some point, as in other quantum theories, you reach a "resolution" that's not divisible any further. So I was thinking - if both the fabric of the Universe and the fundamental interactions in it are quantized, wouldn't that imply that any number tied to a measurable physical quantity would have to have a finite decimal precision? Furthermore, would the uncertainty principle affect the extent to which we can know the least significant digits after the decimal point?
To give an example, in the real number system, $\frac{1}{3} = 0.3333...$. In a putative number system that restricted itself to "physicalness" in the sense of this post, would it rather be the case that e.g. $\frac{1}{3} = 0.3333...328475$ with a finite number of decimal digits, where the last $n$ ones are "fuzzy"? Does such a number system (or a generalization thereof) already exist? And yes, I'm aware that the computer implementation of floating-point numbers has limited precision, but I'm interested more broadly in whether that's just a limitation of computers or a feature of the universe.
Apologies if my question is ill-posed or an apple-orange blend.
