I however have already done as you asked. I posed 1 simple field axiom to be taken as an addition to all current field axiom. The nature of which will allow for division by zero (among other things). It does so with out changing ANY OTHER axiom. Therefore I have altered the definition of a "number" with-in the current system without changing the current system. It can only be said that there is a "failure" in the current system with division by zero. It can not be said that it is "ITERGRAL to the proper functioning of the system". It's that the system can not do it. Except the Riemann Sphere, where division by zero finds meaning.
Also I would disagree with your very first statement. Numbers are already defined exactly as I am posing. It is only that I have made a clear distinction between a "numbers" pieces, where mathematics has currently not.
I look forward do more of your thoughts.