All the articles I've read regarding "Division by Zero" the main argument for it being an undefined operation, because all proofs lead to contradictions.



```
iff (0 / x) = (x / 0) = (0 / 0) = (0)

```

Irrespective of a proof, if the above rules were observed what field axioms of the real numbers would be violated, and how?


In regards to the multiplicative inverse of zero:



```
(0 * x) = (0) == (0 * y) = (0)

(0 * x) = (0 * y)  // dividing both sides by 0 using the rules above results in

(0) = (0)

```

*Quick Reasoning:  

Graphing Division by Zero shows two limits, as the graph tends towards these limits their combined projected values at these two limits negate each other. 
ie: -infinity (as x tends towards 0 from below) and +infinity (as x tends towards 0 from above) either summate or negate each other to Zero.*



Fundamental: Division by Zero
There's a thing called a *meadow* which is a (successful) attempt to make multiplicative inverses globally defined. What it does is instead of defining multiplicative inverses, it defines an operation $M \to M$, $x \mapsto x^{-1}$ with the property **not** that $xx^{-1} = 1$ but that $xx^{-1}x = x$. For any non-zero element then this agrees with the usual inverse but one can extend the inverse operation by defining $0^{-1} = 0$ and it works. I may be wrong, but I think that the result is that every field embeds in a meadow.


So providing you don't claim that $xx^{-1} = 1$ but rather $xx^{-1}x = x$ then you are absolutely fine with $0^{-1} = 0$.


