Creation of minus zero demands 3 new neutral signs?

Question

Zero is usually known as neither positive nor negative, as it's right in the middle of the number line, and lies in-between $-1$ and $+1$.

However, in most programming languages, $-0$ and $+0$ are interpreted as two seperate numbers, which have the same logical value, but may result in different results when combined using mathematical operations. For instance, in a JS console, when typing in 1/-0 returns -Infinity while typing 1/0 returns +Infinity.

But what if we said negative zero and positive zero are different numbers in not only computer programming, but general mathematics too? In order for this to happen, we need to modify the real numbers a lot.

In this case, we would have to define $-0$ as a new, distinct integer lying between $-1$ and $+0$ and $+0$ as in between $-0$ and $+1$. In this case, our old, unsigned zero has vanished. Both $-0$ and $+0$ have discrete arithmetic rules, but now, there's a new problem. What is supposed to be between the integers $-0$ and $+0$, assuming there is a distance of $1$ between them?

To tackle this, I proposed establishing a sort of "neutral zone" in-between $0$ and $-0$. I originally thought numbers in this range would all be fractions, and would have some kind of neutral sign which is neither positive nor negative. For example, neutral $1/2$ would be right in the middle of -0 and +0.

But soon after, I realized that there would be an issue of directionality with such a third sign, as the third sign would go in neither the positive direction or the negative direction. So, I split that neutral sign in to three new signs, a positive-neutral sign (labeled $\pm$) which follows the positive direction, a negative-neutral sign (labeled $\mp$) which follows the negative direction, and the midpoint $N0.5$, i.e. the previously mentioned neutral half.

Using this system, number line would like so:

Now, the question I have is, does such a modified real number algebra cause any problems in arithmetic, or have any practical use?

Answer 1

Ok, firstly your system would wreak havoc on concepts of infinity.

What you are doing is creating a whole new (partial?) number line between your two "zeroes", and that opens up a new world of math, but we would have to scratch a lot of our other math.

For example, how would you define dividing by zero? What is the distance between the two zeroes? Is there even a way of writing out the concept of the old zero anymore?

Besides, your system would just be adding a few numbers to the number line in a realistic sense. It's like changing -0 to -1 and 0 to 1. o would still be +- 1/2, so to speak.

What this means is that it is impossible to come up with any system that (1) is logically consistent, (2) has two different zeroes, (3) satisfies the other axioms of real numbers, and (4) defines zero in the usual way that I described above. So I think that in order to proceed with this question, you have to ask yourself whether you want a different definition of zero. If not, you have to ask yourself which of the other axioms of real numbers you are willing to give up, and then try to build something consistent from there.

Answer 2

The usual definition of 0 in mathematics is that it is the "additive identity" for the real numbers. This means that it is a number that satisfies $$ x + 0 = x $$ for any $x \in \mathbb{R}$.

Now there is a very simple proof that if you define 0 this way, there can only be one unique number called 0. The proof is by contradiction. Suppose that you have two zeroes and call them $0_1$ and $0_2$. They both have to satisfy the definition above. Now consider the quantity $0_1 + 0_2$. Since $0_2$ is an additive identity, then $0_1 + 0_2 = 0_1$. Similarly, since $0_1$ is an additive identity, $0_1 + 0_2 = 0_2$. Therefore, $0_1 = 0_2$, so the zero must actually be unique after all.

What this means is that it is impossible to come up with any system that (1) is logically consistent, (2) has two different zeroes, (3) satisfies the other axioms of real numbers, and (4) defines zero in the usual way that I described above. So I think that in order to proceed with this question, you have to ask yourself whether you want a different definition of zero. If not, you have to ask yourself which of the other axioms of real numbers you are willing to give up, and then try to build something consistent from there.