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for number systems in which "base" is meaningful (i'm not sure that Roman numerals, for example, really have a base) base refers to the number of distinct symbols, and so can only have a positive integer value. for example, base ten allows ten distinct symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 base two allows 2 symbols: 0, 1 and base 1 allows 1 symbol: 1
The symbols themselves are unimportant, so i could express 5 in base 1 like this: LSD LSD LSD LSD LSD or in base 2 like this: LSD shrooms LSD, where shrooms = 0 and LSD = 1
So, you can't really have a non-integer number of different symbols.
Using the typical definition of a base, counting in base pi would be: 1 2 3 10, with (as you noted) 3 - 2 > 10 - 3. This is pretty inconvenient.
If you had a special reason to count in fractions of pi, you could apply pi as a scalar to some other base. For example, count in base 10, and set 1 = pi/10, so that 10 = pi. Basically, to convert from this system to base 10, just multiply by pi/10.
In any case, i don't think "any number is supposed to be able to support it's own scale for mathematics" in the sense you mean: I think only positive integer bases are useful or meaningful.
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