indeterminate forms - Can we say that $frac{0}{0}$ is every number?

Suppose we have an equation $ab=0$. This equation is true when statements $a=0$ or $b=0$ are true.

If $a=0$, then $b=\frac{0}{0}$. That means $b$ could be any number for $ab=0$ to be true. If the set which groups all the numbers is the complex set, then $b$ will every number within $\mathbb{C}$, so $\forall z\in\mathbb{C}:b=\frac{0}{0}=z$. Therefore, $\frac{0}{0}$ is every number.

I know it really is not defined as number but conceptually it is every number, right?

Is this right, or am I missing something?

Answer

When we say that $ab=0$ we define $a$ and $b$ as being unique numbers. So when saying $b$ is all the numbers we are saying that $b$ is not unique. That leads to a contradiction so $b$ is undefined.