By the Cantor slash argument, as explained for example here (at about 4:00), a new real number can always be generated out of any list of real number decimal expansions by taking the digits along the diagonal of the list and calling those digits a new number.
I am having a hard time understanding why Cantor's slash argument does not also apply to the rationals.
The rationals do not have infinite non-repeating decimal expansions, but they can be made arbitrarily large. Therefore we can always make list of rationals like the one you make for reals in the slash argument, with the one difference that the "..." at the end of each number is interpreted as "goes on to become arbitrarily large" instead of "goes on forever." On any such list, it will always be possible to generate a new number that is not on the list by taking the diagonals in the same way you do for reals. The rationals are thus unlistable or uncountable in the same way as the reals are. (Please point out the flaw in this reasoning.)
If I may also pose a very closely related follow-up:
If the Ford Circle algorithm is allowed to run infinitely, it will, in that limit, completely fill up every possible spot on the number line with rational numbers. Where on the number line, then, in that limit, is there any space for irrational and transcendental numbers? Or what am I missing here?
Actually, to expound on T. Andrews' comment, by applying the Cantor diagonal procedure to a complete listing of rational numbers (which exists, by Cantor's "anti-diagonal" or "short diagonal" enumeration of $N$x$N$) you will obtain a non-rational number. Guaranteed.
The original listing of rationals was complete (by assumption), and the construction builds a decimal expansion different from all others in the list. Thus the number obtained is irrational.
Your error is that you start with a very small (negligible, cardinal wise) subset of the reals, and your procedure takes you out of this subset into the much larger (humongous really) set of irrational reals. This happens no matter what the exact starting point and the details of your procedure (what enumeration of rationals is used, the digit modification procedure).
edit add-on In contradistinction the result of Cantor's diagonal procedure was in the same set/class, namely the non-terminating decimals, as the original objects of the procedure. end add-on
Actually I bet one can obtain any one real number from doing the Cantor diagonal procedure on an adequate listing of rationals.