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