Division by zero restores values First of all, I am aware of all the laws of algebra , so my question is about intuition .
We know that  $\frac{x}{y}$ undoes the operation $x \times y$ .
Shoudn't  $(x/0)$  undo $x \times 0$  ?
That means , if $x \times 0$ destroys x , maybe $\frac{x}{0}$  should restore it.
A kind of math memory if you like.
To be more clear , I know it's impossible in our current number system as it is , so I am asking , How can we tweak the laws ?
Maybe something like this :
$x \times 0$ = 0 [x]  (destroying)
$\frac{0  [x]}{0}$  = x  [0]  (restoring)
 A: No It should not. First of all, $x\cdot 0=0$, that would mean $\frac{x\cdot0}{0}\cdot0=\frac{0}{0}\cdot0=x$ for EVERY x.
We have this time and time again. There is no way of defining $\frac{0}{0}$ without adding additional information.
About this "additional information": Non-Standard analysis does exactly this. It basically "saves" the original values. Numbers are stored as sequences instead of values and therefore carry additional information that can be used to restore the original number. However, $\frac{1}{0}$ is not well-defined even in NSA, as there is no additional information given.
A: You are absolutely right! Let $R$ be a set of numbers with addition and multiplication where division by $0$ is well-defined.
Suppose that we have unity elemenet $1 \in R$ (if not there is no point of talking about division, at least in my opinion). Then $0 = 0 \times (1/0) = 1$. Therefore, for any $x \in R$, we have $0 = 0 \times x = 1 \times x = x$. Thus $R = \{0\}$.
What we showed is that if division by $0$ is well-defined for a reasonable algebraic structure $R$ with $+, \times$, then such $R$ can contain only one element, which we usually call $0$.

Again, I don't think division by $0$ is impossible, but it forces our number system to become so uninteresting. You can also consider defining a new number system very carefully, but unless you have a good reason and intuition, not many people will be interested in studying it.
A: One I was thinking about numbers as smooth functions on neighborhood of $0$. So $0$ could by written as $x$, $x^2$ .. etc.(but they are different zeroes). They "primary value" would be limit in zero. 
Example: $1/0$ could be written as $1/x^2$. And $\frac{1}{x^2} x^2 = 1$. 
There are problems as what would $1/x$ mean etc. Probably it would take a lot of time to make definitions precise to get a field. And I guess it wouldn't be very usefull theory but you could divide by zero in some sence.
