I don’t really understand the main argument given in the article; since zero has no multiplicative inverse, you can’t prove anything involving a division by zer

I don’t really understand the main argument given in the article; since zero has no multiplicative inverse, you can’t prove anything involving a division by zero. (So why talk about field axioms?) However, to extend an operation usefully to a larger set of numbers, we can isolate some properties that we want to keep, for example: 1. Division is linear in the first argument: (ab + c)/x = a(b/x) + c/x, for all a,b,c,x. 2. If x has a multiplicative inverse, then x * (1/x) = 1. 3. Taking reciprocals twice gets back to the original number: 1/(1/x) = x for all x. The choice 1/x = 0 is consistent with all the above, and so probably still “useful”. What properties do you want out of a division function?