Does dividing by zero ever make sense? Good afternoon,
The square root of $-1$, AKA $i$, seemed a crazy number allowing  contradictions as $1=-1$ by the usual rules of the real numbers. However, it proved to be useful and non-self-contradicting, after removing identities like $\sqrt{ab}=\sqrt a\sqrt b$.
Could dividing by $0$ be subject to removing stuff like $a\times\frac1a= 1$?
A way I can think of defining this is to have $\varepsilon=\frac10$. Remeber it may be that $\varepsilon\neq\varepsilon^2$, we don't know whether $\left(\frac ab\right)^n=\frac{a^n}{b^n}$, and other things alike; so they can't disprove anything.
This seems fun! What about defining a set $\mathbb E$ of numbers of form $a + b\varepsilon$? There would be interesting results as $(a+b\varepsilon)\times0 =b$.
Are these rules self consistent? Can such a number exist? If not, what rules can we change to make it exist? If yes, what can we find about it? How can it help us? 
If there is no utter way of dividing by zero without getting a contradiction no matter the rules used, how can that be proven?

If $\,0\,$ has an inverse then $\ \color{#c00}1 = 0\cdot  0^{-1} = \color{#c00}0\,\Rightarrow\ a = a\cdot \color{#c00}1 = a\cdot \color{#c00}0 = 0\,$ so every element $= 0,\,$ i.e. the ring is the trivial one element ring.
So you need to drop some ring axiom(s) if you wish to divide by zero with nontrivial consequences. For one way to do so see Jesper Carlström's theory of wheels, which includes, e.g. the Riemann sphere.
