Some time ago I had the idea of extending the real numbers with a new direction/algebraic sign, similarly how negative numbers extend the positive numbers by adding a new sign. I call this sign §, and the numbers §-numbers. This numbers are, in a way, opposite to both positive and negative numbers, that is, they reduce negative numbers towards zero and positive numbers towards zero, and beyond that, they become numbers with the §-sign. Conversly, both positive and negative numbers reduce §-numbers towards zero.
For example -5§3=-2 5§3=2 3§3=0 3§4=§1 -3§3=0 -3§4=§1 §2§3=§5 §4-3=§1 §4-5=-1 -4§3=-1, etc...
Geometrically or quantitatively these numbers are not trivial to interpret, they can be interpreted as being between - +, or the three directions can be interpreted as going from zero into direction with an angle of 120° between them, but this doesn't properly illustrate that they are, in some way, opposites.
These numbers seem to have very interesting behaviour (even though I have just calculated some fairly simple functions). If we define §1*§1=-1, they can also provide a nice solution for the sqrt of -1, which can be §1 (and the sqrt(§1) could be -1 again). Compared to i they have the advantage of "mixing" with the non-imaginary numbers, so we don't have to introduce an extra seperate dimension. Additionally, they solve the equation X=-X with X=§1, if we define §1*-1=§1, which also seems to be a very astounishing property. These definitions continue the logic of the "new" sign "beating" the old sign in multiplication; as - "beats" + in multiplication, § beats both - and +. It also continues the logic of "[sign] squared goes towards +" (§1*§1 goes towards + in the sense that it gives -1, which gives +1 if squared again) or "[sign] to the power of four gives +".
Anway, my proposals seems so obvious that I hardly can believe that I am the first one to make it, so can I find informations about these kind of numbers anywhere? Have they been studied? If there really isn't any information or research about these numbers, I would stronlgy encourage mathematicians to study them! I really suspect they are quite a fundamental structure and they might provide important or even revolutionary solutions in math and science.