On the existence of number systems, and the extent to which we can extend them The more I think about math, the less I realize I know. 
Learning about complex numbers has called me to re-evaluate how I think of negative numbers, or even natural numbers. I have to say the experience has been frustrating and has caused me to be skeptical about literally everything I think I know about math. Nevertheless, is pondering questions about the existence of these number systems useful? Can there be some reward in this? Do we even know that we're right in extending our number system the way we have? 
It seems we extend whenever we can't solve;


*

*To solve $0=x+1 \hspace{5mm}$ we extend to the negatives

*To solve $1 = 2x \hspace{10mm}$ we extend to the rationals

*To solve $x = \sqrt{2}\hspace{9mm}$ we extend to the reals

*To solve $x = \sqrt{-1}\hspace{6mm}$ we extend to the complex


Whats stopping me from extending to solve


*

*$x = \frac{1}{0}\hspace{4mm}$   ?


Just as $x^2 = -1$ seemed meaningless pre-complex numbers, so does $x = \frac{1}{0}$. It seems this is the last type of 'equation' to solve for which we havn't invented some number system. So why can we extend to solve the previous equations, but not this one. Why do we think what we've done is even right?
I know this is a really soft question. But at the same time, the philosophy tag doesn't exist (lol) for no reason. Thanks,
 A: When we adjoin a number satisfying $x^2 = -1$ to the real numbers, the real numbers embed into the result (namely the complex numbers). In other words, the natural map $\mathbb{R} \to \mathbb{C}$ is injective. This allows us to usefully treat real numbers as a special case of complex numbers.
When we adjoin a number satisfying $x = \frac{1}{0}$ to the real numbers, the real numbers do not embed into the result, which is the trivial ring (execise). The trivial ring has only one element and is useless for actually doing any of the things that we use real numbers and complex numbers to do. 
The process of going from the rationals to the reals to the complexes preserves some structure; more precisely, it preserves commutative ring structure. This structure can only be preserved when adjoining an element called $\frac{1}{0}$ by passing to the trivial ring. If we want a nontrivial result, we need to get rid of some of the structure. This is sometimes a useful thing to do; for example, one sensible way to adjoin an element called $\frac{1}{0}$ gives an object called the projective line. The price to pay for doing this is to drop both addition and multiplication as structures, but we get new geometric structure given by the action of a certain group. 
Alternatively, we can get rid of both subtraction and division as structures and work only with the non-negative reals $\mathbb{R}_{\ge 0}$ under addition and multiplication; this gives a semiring rather than a ring. We can adjoin an element called $\infty$ to this semiring satisfying $\infty + x = \infty$ and $\infty \cdot x = \infty$ (for $x \neq 0$) and the non-negative reals embeds into the result. This is sometimes useful to do, for example, in measure theory. 
In short, we can do whatever we want, but we need to be aware of the consequences. 
A: There's really nothing stopping us. And it has been done. See extended real numbers, hyperreal numbers and the surreal numbers.
