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.*