The null product $0*0,\, 0^0$, and $0^x$ An empty product, or nullary product, is the result of multiplying no numbers. Its numerical value is 1, the multiplicative identity. Common examples are $0!$ and $x^0$.
So $0 \cdot 0  = 1$ therefore $1/0 = 0$.
But that means that $0^2 = 1$ which implies that $0^x = 1$
Look at it this way. $1/$positive number = positive. $1/$negtive number $= $ negative. $1/0$ can be neither positive nor negative. That only leaves one possibility
Also the function $1/x$ goes to infinity from one direction and to negative infinity from the other. 
Half way between infinity and negative infinity is zero
Edit: doesnt work because if $0^x = 1$ for all $x$ then $0^{-1}$ cant be $0$
I am probably just wrong but continuing my train of thought:
$1/0 = 0$
therefore $0^2 = 1$ (this is impossible because $0 \cdot (0+0) = 0 \cdot 0 + 0 \cdot 0$)
So we have:
$0^{-4} = \infty$
$0^{-3} = 0$
$0^{-2} = \infty$
$0^{-1} = 0$
$0^0 = 1$
$0^1 = 0$
$0^2 = 1$
Couldnt be periodic could it?
I dont see how or why it would be.


An empty product, or nullary product, is the result of multiplying no numbers. Its numerical value is $1$ [. . .]  So $0\cdot 0 = 1$.

Firstly, $0\cdot0$ is not an empty product. It is the product of two elements, not the product of zero elements. So, it is a mistake to use this reasoning to arrive at the conclusion $0\cdot0 = 1$. This is the main reason why your attempt at showing $1/0 = 1$ is incorrect.
But okay, let's say we just want to $\textit{define}$ $0 \cdot 0 = 1$. Certainly if we are going to do this, we'd better make sure that it is consistent with the rules we already have for arithmetic on the integers. We quickly run into issues. For one thing:
$$
1 = 0\cdot 0 = 0\cdot(0 + 0) = 0 \cdot 0 + 0\cdot 0 = 2
$$
So something is wrong here. It seems like we (at least) either have to throw away the fact that $0 + 0 = 0$ or we have to throw away the distributive property. Either option doesn't seem very good. So, we see without much difficulty that your definition creates inconsistencies, which is why it is not a valid definition.
