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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.

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Math Overflow is not the right place for your question. There are other sites more appropriate for such questions, like Ask Dr. Math (mathforum.org/dr.math). In fact, that site has a whole discussion on this topic: mathforum.org/dr.math/faq/faq.divideby0.html –  Anton Geraschenko Oct 10 '09 at 15:54
    
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closed as off topic by Anton Geraschenko Oct 10 '09 at 15:48

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1 Answer

up vote 1 down vote accepted

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 → M, x → x-1 with the property not that xx-1 = 1 but that xx-1x = 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-1x = x then you are absolutely fine with 0-1 = 0.

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I like this. It seems completely analogous to pseudoinverses in linear algebra; in fact, I think it's just the special case in which you (1) consider a one-dimensional vector space over some field, (2) then consider that field as the ring of linear transformations on that vector space. –  Darsh Ranjan Nov 5 '09 at 16:23
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