Comment by adrian_b
1 month ago
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.
1 month ago
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.
> Ratios of collinear vectors are scalars a.k.a. "real" numbers, ratios of other kinds of vectors are matrices,
Wat? Vectors do not form a division algebra. You can't "divide" vectors! Perhaps you're thinking of the dot product, which returns a scalar.
Also, you're conflating physical/engineering concepts (such as units of measure) with mathematical abstractions such as number spaces, which don't have units.
Physical measurements exist in the real world, with its limitations, units, and practicalities.
Mathematical numbers exist in a pure theory space that is completely and totally independent of any physical reality. They're just axioms and rules. Definitions and logical conclusions.
Don't mix up the two!
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