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

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.

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Definitions rarely provide solutions to anything. One still has to have an idea. There isn't any indication that what you've thought of can be justly called a "number", or that it amounts to anything more than crackpottery. Is your operation associative? Is it commutative? Does it mix well with ordinary addition? Is ordinary multiplication distributive over it? Why do you think it is interesting? It's easy to come up with a bunch of exotic binary operations on a set such as $\mathbf Z$. If you think you have found something interesting, the burden of proof is on you. –  Bruno Joyal Jan 12 '12 at 22:13
??? You haven't adressded my question. Why wouldn't you call it a number, it behaves just as usual numbers do, it just introduces a new alegbraic sign that is akin to - with respect to both - and +. It seems to be associative and of course it can't be commutative, since it works like minus. What is "mixing well" with addition supposed to mean? Does minus mix well with plus? § mixes the same way with plus. If you didn't have minus § would behave exactly as - does. Is it interesting to find a solution to X^2=-1? If it is, my numbers are interesting, since they provide one. –  Benny Jan 12 '12 at 22:37

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".

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OK, it seems you are right, but the equation above seems to work if -1*-1=+1 as with the integers. Then -w^2 is -(§1*§1)=-(-1)=1 –  Benny Jan 12 '12 at 23:34
@Benny In fact $\ (-1)(-1) = 1\$ in every ring, including above. This remains true even though your hypotheses imply that $\ -1 = 1\:.\:$ –  Bill Dubuque Jan 12 '12 at 23:56
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 $c$.