You are getting into some VERY deep waters here. Let me put this to rest in the best (Physical) way I know how: I know of no combination of "quantum" units that makes everything an integer. As the number pi alone is irrational I suspect that such a unit scheme is impossible. (This doesn't even count the possibility that constants such as Planck's constant, the mass of an electron, etc. aren't also irrational. As far as I know there is no way to prove or disprove the rationality of such numbers at this time.)
-Dan
Any kind of number can exist in nature. You're using numbers to quantify something, so really you could call anything "1 thing." For example, I have 1 shirt. Within the shirt are thousands of threads. Within the threads are a much larger amount of fibres, etc.
When looking at the number of shirts, I see 1. If I were to look at the number of (2 shirts), I would have 0.5.
Really any number is in nature, it just depends on what you're trying to quantify.
This is how I think of numbers:
I think of natural numbers as simply the intersection of all the inductive sets, so natural numbers are simply sets. I think of integers are the equvilance classes of N x {0,1}, where the 0 represents the positive and 1 represents the negative. I think of the rationals as simply the field of quotients of the integers. I think of the reals are the completion of the rationals under Dedekind cuts. And I think of the complex numbers as simply the Cartesian product of the reals with themselves.
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It depends how you want to think of numbers. If you are a mathematician you think of them like I described them above. Numbers are simply sets (there are a certain elegance to that). Nothing more. But the majority of people do not view numbers in such an way. Most people associate numbers with certain physical means. So naturals are used for counting things. Rationals are used for doing parts of a whole. Reals can be used to measure distance. And complex can be used to represents points on a plane.