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.

that's largely solved problem. ieee758 defines consistent rules for dealing with infinities. even if don't use the floating-point parts and made a new integer format, it almost certainly would make sense to lift ieee754 rules as-is.

A IEEE754-like arithmetic (transrational arithmetic, or transreal arithmetic) creates new problems due to adding new values. 0*x=0 now requires x≠∞, x≠-∞, and x≠NaN. (x/x)=1 now requires x≠0, x≠∞, x≠-∞, and x≠NaN, so this system doesn't satisfy the field axioms. NaN lacks ordering, so we lose a total order relation.

However, you get cool new results, like x/x=1+(0/x). Definitely some upsides.