Is there any information/research about these extension of the real numbers?

Question

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.

Answer 1

Your hypotheses imply that $ 2 = 0\ $ so any such ring cannot be an extension of $\rm\:\mathbb R\:.\:$ Denote $\rm\ w = \%1\:.\:$ Then $\rm\ w = -w,\ \ {-}1 = w^2 = w(-w) = -w^2 = 1\ $ so $\rm\ 2 = 0\:.\:$ Further since $\rm\ w^2 = -1 = 1\ $ we infer that $\rm\ (w-1)\:(w+1) = 0\ $ so if $\rm\:w\ne \pm1\ $ then your ring has zero-divisors, so it cannot be a field.

You cannot possibly discover "new" extensions of $\:\mathbb R\:$ in this manner because the finite dimensional extension rings of $\:\mathbb R\:$ without nilpotent elements were classified long ago. Namely, Weierstrass (1884) and Dedekind (1885) showed that every finite dimensional commutative extension ring of $\mathbb R$ without nilpotents ($\rm\:x^n = 0 \ \Rightarrow\ x = 0\:$) is isomorphic as a ring to a direct sum of copies of $\rm\:\mathbb R\:$ and $\rm\:\mathbb C\:.\:$ Wedderburn and Artin proved a generalization that every finite-dimensional associative algebra without nilpotent elements over a field $\rm\:F\:$ is a finite direct sum of fields. For much further discussion see my post here and its links, which include detailed analyses of another proposal to develop a number system with multiple "signs".

Answer 2

Like Bruno said, you have to fully specify the algebraic properties of your numbers. Suppose, for instance, we say $(§a+b)-c=§a+(b-c)$; that is, $§$-numbers maintain the associativity of addition. But then $$-1 = (§1+1) - 1 = §1 + (1-1) = §1$$ Similarly, you can show $1=§1$. Therefore, $1=-1$, and hence all integers are equal! So that's no good. This means that we can no longer say that $a+(b+c)=(a+b)+c$ for any $a$, $b$, and $c$.

If interested in this kind of stuff, I would look at abstract algebra.