Dividing a number by zero

Question

Why can't you divide a number by zero? It is possible to say $\sqrt{-1}$ is an imaginary number $i$, but why can't you say $\frac{1}{0}$ is also an imaginary number $z$ (for example)?

Answer 1

Suppose $x=\frac{1}{0}$. Then we should have $x\cdot0=1$, this is not possible since $0$ times any number will give $0$.

In contrast, the idea to let $\sqrt{-1}=i$ is extending the number system which could satisfy all the calculation rules by itself.

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Maybe it will be easier to understand this by taking an example: Let $x=\frac{1}{0}$. Then $$1=x\cdot0=x(1-1)=x\cdot1-x\cdot1=x-x=0$$LHS$\neq$RHS, a contradiction.

Answer 2

Another way to see the problem is by looking at limits: $$\displaystyle \lim_{x->0^+} \frac{1}{x} = \infty$$ But: $$\displaystyle \lim_{x->0^-} \frac{1}{x} = -\infty$$

And since there's no preference to any one-sided limit, the limit should not exist and there isn't a specific value to the function $\frac1x$ at 0.