Are there any number systems with practical applications in which division by zero is defined and isn't always one single number?

Question

From my (scarce) observations of "exotic" number systems with practical applications, such as projective geometry, it seems to me like most of those that define division by zero all assign one single number (or two, taking into account a negative) to any $\frac{a}{0}$, usually represented by the symbol $+\infty$ and $-\infty$, so that $\infty + \infty = \infty$. It's somewhat striking to me that they don't usually associate different numbers with different fractions, as in maybe $\frac{2}{0}=2 \infty$, but $\frac{5}{0}=5 \infty$, so that $2\infty + 5\infty = 7\infty$, which intuitively feels like it could offer some advantage. Admittedly I don't have more than intuition to back this up. But the curiosity remains: are there any number systems with practical applications in which division by zero is defined and isn't always either $+\infty$ or $-\infty$, but potentially different infinities?

Answer 1

I am not aware there are other numbers system, apart from the context of non standard analysis referred by Qiaochu Yuan in the comments, which deal with a sort of division by "something similar to 0".

But you can also be interested in surreal numbers. I think they are related.

In another context, you have "different infinities" when you work with projective geometry. You get a different "infinity" for every direction of a vector space. But a projective space doesn't have an appropriate sense of "number system"... (unless to my knowledge).