Comment by jiggawatts
1 month ago
Numbers interpretable as ratios are the Rational numbers, by definition, not the Reals.
This entire discussion is about mathematical concepts, not physical ones!
Sure, yes, in physics you never "need" to go past a certain number of digits, but that has nothing to do with mathematical abstractions such as the types numbers. They're very specifically and strictly defined, starting from certain axioms. Quantum mechanics and the measurability of particles has nothing to do with it!
It's also an open question how much precision the Universe actually has, such as whether things occur at a higher precision than can be practically measured, or whether the ultimate limit of measurement capability is the precision that the Universe "keeps" in its microscopic states.
For example, let's assume that physics occurs with some finite precision -- so not the infinite precision reals -- and that this precision is exactly the maximum possible measurable precision for any conceivable experiment. That is: Information is matter. Okay... which number space is this? Booleans? Integers? Rationals? In what space? A 3D grid? Waves in some phase space? Subdivided... how?
Figure that out, and your Nobel prize awaits!
Rational numbers are ratios of integers.
There are plenty of ratios that are ratios of other things than integers, so they are not rational numbers.
Ratios of numbers that are not integers or Rationals are... the Reals. I mean sure, you could get pedantic and talk about ratios of complex integers or whatever, but that's missing the point: The Rationals are closed under division, which means the ratio of any two Rationals is a Rational. To "escape" the Rationals, the next step up is Irrational numbers. Square roots, and the like. The instant you mix in Pi or anything similar, you're firmly in the Reals and they're like a tarpit, there's no escape once you've stepped off the infinitesimal island of the Rationals.
There are many other kinds of ratios.
Ratios of collinear vectors are scalars a.k.a. "real" numbers, ratios of other kinds of vectors are matrices, ratios of 2D-vectors are "complex" numbers, ratios of 2 voltages are scalars a.k.a. "real" numbers, and so on.
In general, for both multiplication and division operations, the 3 sets corresponding to the 2 operands and to the result are not the same.
Only for a few kinds of multiplications and of divisions the 3 sets are the same. This strongly differs from addition operations, which are normally defined on a single set to which both the operands and the result belong.
In practice, multiplications and divisions where at least one operand or the result belong to another set than the remaining operands or result are extremely frequent. Any problem of physics contains such multiplications and divisions.
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