[topaz0]

I didn't realise this, but apparently it is also possible to do good algebra on this kind of structure by adding an element 0/0: https://en.wikipedia.org/wiki/Wheel_theory (which someone pointed to in one of the discussions -- I forget which one).

[syzarian]

Mathematics is a vast subject and I can’t keep track of all developments. In 2010 there was a paper on meadows. I’ve never heard the term before. In that paper it is written:

As usual in field theory, the convention to consider p / q as an abbreviation for p · q−1 was used in subsequent work on meadows (see e.g. [2,5]). This convention is no longer satisfactory if partial variants of meadows are con- sidered too, as is demonstrated in [3].

So, as I’ve stated many times, I talked about convention and indicated you can use whatever terms you want. In the paper quoted above they acknowledge what the convention is. That is that division is multiplication by the inverse. They are arguing that it is worthwhile in this new algebraic object to change the usual notion a bit. If people agree to a new usage of the word division then definitions will change accordingly. None of this is pertinent to the spirit of the original question given the context under which it was asked. All of this is highly technical.

Definitions and notions change as new mathematics is created (discovered?). This happens all the time. All you have to do is convince other mathematicians to go along with it.

https://arxiv.org/pdf/0909.2088.pdf

EDIT: Regarding what you wrote in your other comment: The analogy is not apt in my opinion. It’s hard to say zero can’t exist because the nonzero…. The moment you say nonzero means it does exist. I think a better way to look at the situation is:

I have an object that is a group under a binary operation f. There is another natural binary operation on that object that operates with f in a consistent way. That operation doesn’t form a group but if I add a symbol to my set and give these rules then both operations interact in a consistent, natural way. I get a group under the new symbol with the second operation while preserving the group under the first operation minus the new symbol.

With extended complex numbers you don’t quite preserve the structures or properties that one normally wants so I’d say it isn’t true division. It is division like.

[topaz0]

I'm happy to agree to disagree about where the line between "division" and "division like" should be placed. As you say, it is a question of convention and not really a question of math. But I don't agree that a student with the curiosity to ask about extending the numbers in various ways would not find something "division like" with the properties they're interested in to be relevant to the question (even if it is missing some other properties that most mathematicians consider to be essential to the notion of division).

[topaz0]

Honestly saying you can't have a number 1/0 because it breaks the ring axioms seems exactly analogous to saying 0 can't exist because it breaks the group axioms for multiplication on the non-zero reals. Is ring multiplication "not really multiplication" because it doesn't satisfy group axioms? That doesn't seem consistent with normal usage to me, but you could imagine a pedantic student coming out of their first group theory course and trying to make that argument.