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steven787 said:
Menago, I understand.

It really comes down to this.

If we want to use the representation of .(3) = 1/3, then
.(3) + .(3) + .(3) = 1 and not .(9)

It can't be both. In the real world a number can't have two values.

We have to pick the set of rules ahead of time (in this case: .(3) = 1/3), and justify the rules with other rules (in this case, the value of n).

It doesn't have two values, it has one value which can be represented in two different ways because 0.(9)=1.