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Splitting of the Number Line

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.

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 \(\hat{e_{+}}\)

We take the concept of the traditional number line, (-ve ... +ve) from (\(-\infty\) to \(\infty\)). Keep the positive part from \([0,\infty)\) the same, but introduce a split to the negative number line to produce two negative “polarisations” “-” and “|”. These splittings are defined by two angles \(\theta\) and \(\varphi\) from the line of the “tranditional number line”. Thus, any negative number which would be \(-n\) on the traditional line will now have a mixed value of the polarisations given by \[-n \to -\frac{\big(ncos(\theta)\hat{e_{-}} + ncos(\varphi)\hat{e_{|}}\big)}{2cos(\theta)cos(\varphi)}\]

that is by definition. It is a requirement of the theory that when \(\theta=\varphi=0\), then, \(\hat{e_{-}}=\hat{e_{|}}=\hat{e_{+}}\) and equation [ABOVE] reduces back to \(-n=-n\).

Letting \(\theta=\frac{\pi}{3}=\varphi\). It is still true that \(-n\cdot-n=n^2\).

The components then transform according to the algebra with Cayley table \[\begin{array}{| c | c c c |} \hline \cdot & \hat{e_{+}} & \hat{e_{-}} & \hat{e_{|}} \\ \hline \hat{e_{+}} & \hat{e_{+}} & \hat{e_{-}} & \hat{e_{|}} \\ \hat{e_{-}} & \hat{e_{-}} & \hat{e_{+}} & \hat{e_{+}} \\ \hat{e_{|}} & \hat{e_{|}} & \hat{e_{+}} & \hat{e_{+}} \\ \hline \end{array}\]

It needs to be true that from equation ... \(-n \cdot -n =n^2\) then it must be true that\[n^2 = \frac{n^2}{4}\Bigg[ \frac{cos^2(\theta)+2cos(\theta)cos(\varphi)+cos^2(\varphi)}{cos^2(\theta)cos^2(\varphi)}\Bigg]\]

This has symmetric solutions of the form \(\theta=\varphi\), the equation reduces to: \[cos^2(\theta) = 1\] which are \(\theta=0,180\) which correspond to the normal number line, (both components fold in) and the mod of the number line \(|n \in \mathbb{R}|\) (the normal negative component folds onto the real part).

There exist non symmetric (Irwin-Worthy) solutions to the equation \[4cos^2(\theta)cos^2(\varphi) = cos^2(\theta)+2cos(\theta)cos(\varphi)+cos^2(\varphi)\\ 4(cos(\theta)cos(\varphi))^2=(cos(\theta)+cos(\varphi))^2\] These solutions are displayed in the graph below.

A more inventive test is the multiplication \((m-n)(m-n)=m^2-2mn+n^2\), this same expression can be displayed as \[(me_+ - \frac{ne_-}{2cos(\varphi)} -\frac{ne_|}{2cos(\theta)})(me_+ - \frac{ne_-}{2cos(\varphi)} -\frac{ne_|}{2cos(\theta)})\]

working through this leads to \[m^2e_+ +n^2\bigg(\frac{1}{4cos^2(\theta)}+\frac{1}{2cos(\theta)cos(\varphi)}+\frac{1}{4cos^2(\varphi)} \bigg)e_+ -mn\bigg( \frac{e_-}{cos(\varphi)}+\frac{e_|}{cos(\theta)}\bigg)\]

This imposes that the first bracket with the three fractions in is \(1\) and that the second bracket is \(2\), which is true from the original definition. THe first bracket is the same set of solutions shown in the graph below. This means there was no contradiction in the number system for any of the appropriate pairs of asymmetric angles, and problems involving bracketed multiplication appear to work in this system.

Norms: It would be useful to have a conjugate operation on the negative parts of numbers. Then one could define a norm in analogy to complex numbers \(|z|=\sqrt(z*z)\). This operation however requires a relationship between angles which narrows down the pairs of choosable asymmetric angles. \[|-n|=n \\ \bigg| -\frac{ne_-}{2cos(\varphi)} - \frac{ne_|}{2cos(\theta)} \bigg| = \\ \bigg(-\frac{ne_-}{2cos(\varphi)} - \frac{ne_|}{2cos(\theta)} \bigg)\bigg(-\frac{ne_-}{2cos(\varphi)} + \frac{ne_|}{2cos(\theta)} \bigg)= \\ n^2\bigg(\frac{1}{4cos^2(\theta)}-\frac{1}{4cos^2(\varphi)}\bigg)e_+\]

This highlights the condition that must be met for a | conjugation operator.

A suggested Cayley table by D.Worthy is \[\begin{array}{| c | c c c |} \hline \cdot & \hat{e_{+}} & \hat{e_{-}} & \hat{e_{|}} \\ \hline \hat{e_{+}} & \hat{e_{+}} & \hat{e_{-}} & \hat{e_{|}} \\ \hat{e_{-}} & \hat{e_{-}} & \hat{e_{|}} & \hat{e_{+}} \\ \hat{e_{|}} & \hat{e_{|}} & \hat{e_{+}} & \hat{e_{-}} \\ \hline \end{array}\]

There is a suggested set of angles in which \(\theta=\varphi=\frac{\pi}{3}\) for symmetric properties by R.Garner. The only symmetric (Garner) solutions appear to be 0 and 180 degree angles.

If one required the numbers to be held on the complex plane then the basis elements are \[\hat{e_{-}} = -\hat{e_{+}}\exp(i\theta) \\ \hat{e_{|}} = -\hat{e_{+}}\exp(-i\varphi) \\\]

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