But why would we go from what obviously should be a very large boundless number and just replace it with 0. Our few comment discussion is why it’s undefined in a nutshell.
The main issue lies in weakening the field axioms to accommodate any strange new numbers. Instead, defining division by 0 to 0 adds no new numbers, so the field axioms don't change (x/x=1 still requires x≠0). I hope you see the value in extending field theory instead of changing field theory.
If we add new numbers like ∞, -∞, and NaN (as the neighbor comment suggests with IEEE754-like arithmetic), now x/x=1 requires x≠0, x≠∞, x≠-∞, and x≠NaN. Adding more conditions changes the multiplicative inverse field axiom, and thus doesn't extend field theory. Also, now x*0=0 requires x≠∞, x≠-∞, and x≠NaN. What a mess.
The problem is simply that the definition is a lie.
I’m not suggesting that we add numbers or change the definition from undefined. I think undefined is a more accurate description of x/0, because x/0 is clearly far greater than 0.
