Well, it's a design choice that you can make for rational numbers and the restricted rationals with error values that we typically use in computer arithmetic. T

Well, it's a design choice that you can make for rational numbers and the restricted rationals with error values that we typically use in computer arithmetic. There's nothing wrong with it and as the article points out, even most proof assistants use this definition for rational division. For real numbers though, division by zero will always diverge or be undefined, no matter how you represent them. All computations on real numbers are continuous and there is no continuous extension of division which includes 0 in the domain of the denominator. One can get around this by switching to projective reals (essentially reals extended with a new element that's both +infinity and -infinity). This is no longer totally ordered and has some other problems, but it's actually a great choice for a lot of numerical computations. Yet, I've never seen something like this implemented in hardware, aside from John Gustafson's unum concept.