> But 0.999 repeating is a rational number, no need for repeating > decimals at all in the naturals. Repeating zeros is okay I guess, > but why use them in the naturals. In the rationals and reals, > repeating zeros are called 'terminating' decimal expansions and > the trailing zeros are elided.
Infinite decimals represent real number "fractions". The need for 0.999..., such as it is, is for _some real number_ to be assigned appropriately to every infinite decimal expansion. It's the same reason we assign the infinite decimal 0.1234000... to the terminating decimal 0.1234. We want 0.1234000... to be _something_ . What else would it be?