Isn't zero an operator that transform every thing to himself so division by this operator is null, which is why division by zero is undefined?
Something like derivative but special one. $$0=\frac {d}{d(\text{something})}$$ $$0*1=\frac {d 1}{d(\text{something})}=0$$ $$\frac {1}{0}=\frac{1}{\frac {d}{d(\text{something})}}=\text{null.}$$