Why Rational Numbers do not include pairs $(a,b)$ with $b=0$? Let $X=Z\times Z$
If we have the relation $R$ on $X$ defined by $(a,b)R(c,d)$ if and only if $ad=bc$. Then, what is the problem if $b=0$? 
Obviously, I'm not looking for the answer that we cannot divide by 0, but rather something more fundamental. I thought that perhaps it violates the reflexivity of the equivalence relation.
Can I have a hint?

As anorton said, this fails to be an equivalence relation when you include $(0,0)$. 
You don't need to throw away all ordered pairs with $b=0$, though; you could just remove $(0,0)$, then get a valid equivalence relation on $\mathbb Z\times\mathbb Z\setminus\{(0,0)\}$. You could think of these as the "extended rationals," $\mathbb Q\cup\{\infty\}$.
