Numerical system that includes the limit targets such as $0^+$, $0^-$, $1^+$ etc

Question

I wonder what is the name of a mathematical system extending the real numbers that includes signed zero along with unsigned zero as well as other "limit targets", such as $1^+=1+0^+$, $5^-$ etc, so adding to each real number exactly two companions, the right companion and the left companion?

I have read the Wikipedia's article on arithmetic with signed zero, but it seems they equate positive zero with just zero, but I am asking for a system that has all three, $0$, $0^+$, $0^-$.

It seems that such system (if also extended with $\infty$, $\infty^+$ and $\infty^-$ or in alternative notation, $\tilde{\infty}$, $\infty$, $-\infty$) would be very consistent because it would include every possible limit target on the real line, but more simple than hyperreal numbers.

Answer 1

I think the sentiment behind this question is justified in the sense that it seems that there are "too many" infinitesimals in the system such as the hyperreals. There are simpler systems such as the dual numbers that look more like what you have in mind. However, it is not clear how to define the sine function on those number systems. A similar problem exists with the surreal number system. In the end, the abundance of infinitesimals in the hyperreal number system is necessitated by the need to ensure that the transfer principle holds.

Answer 2

This Wikipedia article on signed zero is about computer programming, not a mathematical theory.

It seems that what you are looking for is a formalization of the “determinate forms.”

I think yes, it is possible to do that, and that’s basically how we work with limits in standard Calculus (ie Calculus without hyperreals etc), just without formalizing it.

First, make “=” asymmetric by allowing the rhs to be simpler, so you can write $0^+=0$ but not $0=0^+$, so you are solving from left to right. There will be undefined terms such as all the 7 indeterminate forms plus others such as $\frac70$ where $0$ results either form a previous inconvenient simplification of $0^\pm$ or from the fact that $0$ is not attained from one specific direction. And you will have a bunch of determinate forms which will be your rules, such as $c^\infty=\infty$ for real $c>1$, $\infty+\infty=\infty$, $\frac c{0^+}=\infty$ for real $c>0$ etc.

So you could write $$ \frac{4^+-2^-}{3^+-3^-}=\frac{2^+}{0^+}=\frac{2}{0^+}=+\infty $$ as well as $$ \frac{4^+-2^-}{3^+-3^-}=\frac{4-2}{3-3}=\frac00=? $$

In any case, I see little benefit in including an “unsigned infinity” or $\infty^-$, since reasonable functions may approach either $+\infty$ or $-\infty$ when you take side limits (not both) and it is always from the left or right respectively.

Once you formalize this space, it will be very handy and that’s what people already do in practice. But it won’t by itself solve every limit problem.