What I propose here doesn't have a direct relation to tetration, but since I don't really know where to propose a new extension of math that falls out of the horizon of mainstream math (without being ridiculed at the start), I thought this is the right forum for this proposal, given that I read many proposals of "experimental" extensions of current math here.

My idea is rooted in the question "How to extend math in the most fundamental way?", that is, not working inside known structures like R or C, yet also not introducing overly complex new structures.

I thought of a way to naturally extend the integers in the same way that the integers extend the natural numbers. As far as I know no one has written this down yet, maybe no one thought is was interesting or worth thinking about.

Really, it is very simple. But it can seem complicated or plain weird or "wrong" at first - just try to follow my explanation:

Subtraction is simply the opposite of addition. From the positive numbers we go towards zero instead away from it - until zero, where we go towards increasingly negative numbers.

We can do the same with respect to both positive and negative numbers, which creates a new operation (and a new algebraic sign). I call it "§".

By doing §-tion we go towards zero from both the negative and positive numbers - until zero, where we go towards the §-numbers.

Let's look at a few examples.

From positive numbers we go towards zero and beyond zero towards §-tive numbers:

+5§3=2

+1§1=0

+8§3=+5

+3§5=§2

...but also towards zero (and beyond zero towards §-tive numbers) from the negative numbers:

-5§3=-2

-1§1=0

-8§3=-5

-3§5=§2

From 0 or §-numbers we go towards increasingly §-tive numbers:

0§5=§5

§3§5=§8

Note that +1§1=-1§1=0.

I hope the examples make it clear what the operation does.

The §-numbers seem to make it possible to solve the equation -X+Y=X+Y by (for example) inserting X=3 and Y=§3 giving -3§3=3§3=0. This also means that §X-§X!=0 and instead §X-§X=§X, because otherwise you could derive -3=3 by subtracting Y on both sides above.

This seems to imply §1*-1=0, even though that may be counterintuitive.

I am not yet sure what properties of integers exactly are preserved and which not, and I am also not yet sure what §1*§1 is. It would be nice if §1*§1=-1 so that -1 has a squareroot, but I don't know yet whether this is consistent. It may even be possible that §1*§1 doesn't have a unique solution or can't have a solution within the defined numbers (which I think is unlikely). I also have no clue as to what the practical utility of these numbers are (if there is any).

I really encourage you to study these numbers, since I am unfortunately not remotely as good in figuring out mathematical properties as in inventing new principles , but I suspect they are quite interesting.

If you see any errors in my reasoning here, please correct me!

My idea is rooted in the question "How to extend math in the most fundamental way?", that is, not working inside known structures like R or C, yet also not introducing overly complex new structures.

I thought of a way to naturally extend the integers in the same way that the integers extend the natural numbers. As far as I know no one has written this down yet, maybe no one thought is was interesting or worth thinking about.

Really, it is very simple. But it can seem complicated or plain weird or "wrong" at first - just try to follow my explanation:

Subtraction is simply the opposite of addition. From the positive numbers we go towards zero instead away from it - until zero, where we go towards increasingly negative numbers.

We can do the same with respect to both positive and negative numbers, which creates a new operation (and a new algebraic sign). I call it "§".

By doing §-tion we go towards zero from both the negative and positive numbers - until zero, where we go towards the §-numbers.

Let's look at a few examples.

From positive numbers we go towards zero and beyond zero towards §-tive numbers:

+5§3=2

+1§1=0

+8§3=+5

+3§5=§2

...but also towards zero (and beyond zero towards §-tive numbers) from the negative numbers:

-5§3=-2

-1§1=0

-8§3=-5

-3§5=§2

From 0 or §-numbers we go towards increasingly §-tive numbers:

0§5=§5

§3§5=§8

Note that +1§1=-1§1=0.

I hope the examples make it clear what the operation does.

The §-numbers seem to make it possible to solve the equation -X+Y=X+Y by (for example) inserting X=3 and Y=§3 giving -3§3=3§3=0. This also means that §X-§X!=0 and instead §X-§X=§X, because otherwise you could derive -3=3 by subtracting Y on both sides above.

This seems to imply §1*-1=0, even though that may be counterintuitive.

I am not yet sure what properties of integers exactly are preserved and which not, and I am also not yet sure what §1*§1 is. It would be nice if §1*§1=-1 so that -1 has a squareroot, but I don't know yet whether this is consistent. It may even be possible that §1*§1 doesn't have a unique solution or can't have a solution within the defined numbers (which I think is unlikely). I also have no clue as to what the practical utility of these numbers are (if there is any).

I really encourage you to study these numbers, since I am unfortunately not remotely as good in figuring out mathematical properties as in inventing new principles , but I suspect they are quite interesting.

If you see any errors in my reasoning here, please correct me!