Signs as part of a number coordinate

Question

Suppose we have a 1-dimensional space: a number line. We can name each element of it using its module and a "sign"(+/-).

Is there a system which have something like additional signs that are used to name numbers? In other words: Is there any property like signness which unite "+" and "-", and can we create new object with such property?

I am not talking of signs of operation like (multiplication $\times$) (division $\div$) etc. that are working on the same number line. They can not be used to name a number like "$\times4$"... as far as I get it

I came up with a system of n rays that are sent from one point, so we can name each ray with some symbol and use it to name numbers on it. Example:

1. we have 3 rays and 3 operations that are "pulling" a result to the corresponding $\infty$ : addition, upperSubtraction and lowerSubtraction.

upperMinus infinity
\
 \
  \________ plus infinity
  /
 /
/
lowerMinus infinity

2. we have upperMinus 4
3. we do plus 8 and get plus 4
4. we lowerMinus 8 and get lowerMinus 4
5. we upperMinus 8 and get upperMinus 4 again

Maybe my presupposition about ability to call number *n(multiply n) is wrong. We can create a number line with 1/identity as origin and * will be treated like plus sign or sign to the right of identity and / will be a minus sign or left sign. But there are still only two off them in one system!

I think that i might be confused by having concepts like:

• number "$-4$" per se and not an operation "-" with magnitude 4
• not thinking of number line having an intrinsic operation hence the "sign" used as a part of coordinate. Example: we have 2 number lines first is based on addition and it has intrinsic addition/negative addition(subtruction) and we can use sign/negate sign of intrinsic operation as part of coordinate (+ we can "create" a multiplication in this basis)

 |  |  |  |  |  |  |
-3 -2 -1  0  1  2  3 

second is using multiplication so we have $\div$ and $\times$ as intrinsic signs

 |  |  |  |  |  |  |
/4 /3 /2  1 *2 *3 *4 

so probably one can even call it not a number line but an ~operation line. I think that in lambda calculus the number(Church numeral) itself is an operation with some magnitude

Answer 1

The system you create is kind of neat, but your operations lack some of the basic properties we generally like to see in mathematical systems (particularly for operations called "$+$"):

1. No associativity, ie $(x + y) + z \neq x + (y + z)$ in your system. For example:

• $(2 + upperMinus 4) + lowerMinus 4 = lowerMinus 2$, but
• $2 + (upperMinus4 + lowerMinus4) = 0$.

2. No unique inverses. Since $2$ has two different things it can be added to that make $0$, this means that you can't have a concept of the negative of a number, which means you can't have a well-defined subtraction operation.

Without these, you're going to struggle to find any useful applications for this system, as you'll be unable to do most algebraic manipulations or use basic algebraic concepts like "the sum of three numbers".

A more standard approach to extending the real number line is to extend it to the complex number plane.

Answer 2

Let me take this back to a really fundamental level for a moment.

If we want to start a Group we need a set and an operation. What the set and operation are can be decided later, but for argument's sake I will use $S={0,1,2,3,4,5...}$

In order to successfully define an operation, the set $S$ must first be closed under this operation.

So I can define $a\circ b$ however I like so long as $a\circ b$ is a member of $S$ whenever $a$ and $b$ are members of $S$.

As was mentioned earlier, associativity is also a requirement, so we need $a\circ(b\circ c)=(a\circ b)\circ c$ for all $a,b,c$ in $S$.

Commutativity would be a bonus, and unless we want $S$ to be a semi-group we also require every element to have a unique inverse within $S$.

Not to mention a single identity element.

Now... what operations that we already know follow these requirements?

RBG colors can be considered a 3D number system or a six-digit code. If that is the case, I would argue that Complex numbers are similarly one-dimensional if you are willing to adjust your definition of dimension.