Jim Burns submitted this idea : > On 10/3/2017 8:20 AM, FromTheRafters wrote: > >> 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.
I agree, if you meant infinite 'repeating' decimal expansions, but netzweltler wrote about little n in N (n an element of the naturals) and was using decimal expansions which aren't needed in whole numbers. Sure, in the rationals and the reals decimal expansions certainly make sense.
> 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?
I think you missed my point. Netzweltler should have had little r in R or at least little q in Q to use decimal expansions to illustrate his point. He should use real numbers to illustrate real number ideas. It is plain to see that in the reals there are always other numbers between any two other reals which are 'different' from each other due to the density. The same works for irrationals (fractions) -- but not so much for the naturals.
Why did he specify the naturals when he was trying to say something (wrong) about the reals?