well, not entirely sure i agree. for example, how would you denote 1/13?
you need a consistent way of denoting any rational number, not just ones that repeat the last digit.
personally, i like the bar method, where the repeated values have a bar over the top over the length of the repeating part.
so_ 0.9 = 1
would be more accurate (although more cumbersome.)
1/13 = 0.076923
The bar covers the part that repeats.
here's my challenge for you.
accurately add 1/13 to 45/89 in decimal representation.
This can be done, but would be rather tedious (particularly as the decimal expansion for 45/89 has a relatively long repeating digit string). The sum will also have a repeating pattern, of some size less than or equal to the LCM of the repeating digit string lengths of the two operands.