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,

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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.

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I'm honoured to have you weigh-in Qiaochu. That being said, the line "we can do whatever we want" made me extremely uneasy. I'm not even sure how to digest it, nor how I'll sleep tonight. –  user45814 Nov 19 '12 at 4:50
@user: why? To me it is extremely freeing. You can think of it as meaning something like the following. As mathematicians, we can play whatever games we want, and some of the games we play are extremely interesting. But we still need to follow the rules of those games, and if we aren't careful choosing the rules, we'll end up with a boring game. –  Qiaochu Yuan Nov 19 '12 at 4:50
@user: by the way, don't be misled into thinking that the only reason to invent new number systems is to solve new kinds of equations. This is a relatively shallow reason. For example, we can extend the complex numbers to the quaternions (en.wikipedia.org/wiki/Quaternion). This doesn't let us solve, say, any polynomial equations we couldn't already solve, but it allows us to do several much more interesting things: describe rotations in $3$ dimensions, describe rotations in $4$ dimensions... –  Qiaochu Yuan Nov 19 '12 at 4:53
I agree it is extremely freeing, thats the problem. What can't we do then? Are there rules we can't bend or can't break? (Sounding more and more like The Matrix) –  user45814 Nov 19 '12 at 5:00
@user: well, if at any point you prove a statement and its negation, that would probably be bad. I still don't see what the problem is. –  Qiaochu Yuan Nov 19 '12 at 5:02