Why Not Define $0/0$ To Be $0$? For every number $x$, $x\times 0=0$, hence $\dfrac{0}{0}$ can be any number!  So $\dfrac{0}{0}$ "is knows as indeterminate" [1]. 
But what if we define it to be $0$? I already have an answer, but don't know how convincing it is:

$1=\dfrac{1}{1}=\dfrac{1}{1}+0=\dfrac{1}{1}+\dfrac{0}{0}=\dfrac{1\times 0}{1\times 0}+\dfrac{0\times 1}{0\times 1}=\dfrac{0}{0}+\dfrac{0}{0}=0+0=0$, a contradiction.

Is there any better explanation why not to define $\dfrac{0}{0}$ to be $0$ (or any other number)?

Thanks.

[1] http://en.wikipedia.org/wiki/Division_by_zero#In_algebra
You can define $0/0$ if you want, but that would be missing the point. One considers quotients as an operation. So what you want to be able is to have the multiplicative inverse of $0$, which can be easily seen to not make sense.

The idea of considering the fraction $p/q$ is to be able to think about it as $p\times1/q$. But, what arithmetic can you do with $0/0$?
Informally, $0/0$ is indeterminate, so it would be wrong to give it a value. It would be like o items out of 0 things. Is it none of the divisor or all of it, or maybe halfway? Since there's nothing to compare, all of these could be seen as valid viewpoints, so you cannot assign it a specific value.