The extended reals can also arise directly from the definition of reals either as Cauchy sequences of rationals (as unbounded increasing sequences, etc) or Dede

The extended reals can also arise directly from the definition of reals either as Cauchy sequences of rationals (as unbounded increasing sequences, etc) or Dedekind cuts (let the lower cut contain all rationals) :) for reals IIRC the main reason to exclude the extended reals is to make it a field. However I don't think any of this matters for the point the article made. The problem is not that 1:N/0:N has "nothing sensible" to return, the problem is that if you define it in the way most mathematicians do, a lot of your proofs will be littered with (inline) lemmas showing that the result of such-and-such a division is a finite natural / finite real. This is analogous the problem brought up in the article of having proofs littered with lemmas showing that the result of such-and-such a square root was real.