Splitting of the Number Line

Abstract

We introduce the novel concept of a split number line, in which the positive part is as usual and the negative part undergoes an either symmetric or asymmetric bifurcation at zero. An algrbra is made between the two kinds of negative components and positive compoments in the form of a stable triplex. The implications of this create a further abstraction on the concept of rings over number fields, such as to narrow the application of the Frobenius Theorem.

Triplexes and Concept

According to [Source: Ben] a stable simplex on a number field cannot be created, implications of Frobenius Theorem leading any real division algebra isomorphic to either \(\mathbb{R},\mathbb{C}\) or \(\mathbb{H}\) where, the set of quaternions \(\mathbb{H}\) are the only non-Abelian algebra. Exclusion of hypercomplex, and therefore non-associative alegbras as we stick to real division algebra only.

We form an abstraction on the signality of a number \( n \in N\), where \(N\) is a ring over real numbers.

Instead of the binary +,- number precursors indicating greater or less than zero we allow, +,-,| where + indicated greater than zero and - and | both indicate a number less than zero. There is a component form to these negative numbers, the “full” value is defined to be \[-n \to a\hat{e_{-}} + b\hat{e_{|}}\]

with a,b positive constants, and \(\hat{e_{-}}, \hat{e_{|}}\) basis elements that transform according to some algebraic rule along with the basis element \(\hat{e_{+}}\)