Right, a radix-based system can't directly represent an infinite number of repeating digits.

But a rational number model can hold this value quite easily. It would hold 9/9 (to represent infinite repeating, use 9 in the denominator, instead of 10). You can see that this is another representation of 1, just like 2/4 and 1/2 are the same value.

It's not wierd that two distinct numbers, .9999.... and 1, happen to be equal to each other. Rather, they are the same number, just expressed in two different representations in our Aribic positional number notation. In Roman number notation, X is the same number as VV. Big deal.

—John