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?

In short, there is no way to define $\frac{1}{0}$ without getting nothing interesting, as user121926 has already mentioned in his comment.
However, in measure theory we can define $0 \times \infty = 0$ and it keeps certain theorems simple. Nevertheless division by zero still cannot be allowed.
