Quote by Hurkyl
No you don't. You only run into a problem if you decide you want to write the answer in the form of a terminating decimal numeral.

I think you do. 2/3 doesn't mean anything because you haven't performed the division yet, You don't know the answer to 2/3, no one does until they perform the division. Might as well just call it x and manipulate it using the rules of algebra.
If you are not interested in numerical values of numbers but interested in their abstract representation as quotients of 2 integers then i guess you don't see a problem.
I ask you... what is the difference between .9999.../2 (infinite 9's) and 1/2?
or 1/1.99999...?
Do you understand my point?
1/2 = 1/1.9999... = .9999.../2 = .9999.../1.9999... = 1.000.../2 = 1.000.../2.000... = .9999.../2.000...
= 1/2.000... = 1.000.../1.9999...
Now you have at least 9 different representations of the same UNIQUE value so looking at the fraction as a solution to the problem has not SOLVED the problem but instead has made it more difficult and more aggravating.
See how easy it is to shoot down the fractional abstractions?
Any expression involving any rational number has now become suspect just because you ran into a problem trying to divide 1 by 3
My last comment is related to my previous post. As far as positive integers with the operation of division are concerned, 1 divided by 3 is the first time you run into a problem that forces you to make a correction. The correction is that we must now accept that 1 = .9999... = 1.000...
This correction was not needed for 1 divided 1, 1 divided by 2, 2 divided by 1
This is the 'spirit' of my argument. I am not arguing that the results are incorrect. If someone want's to choose 1/2 as the representation of 1 divided by 2, that's fine by me but it doesn't change the fact that other representations are possible and are a consequence of the necessary 'correction'.