Asymmetry in the Real Number Line and: A Proof that π + e is an Irrational NumberAuthor(s): Alexander C. Sarich
The following document is provided as a preliminary discussion regarding certain equivalence properties which arise between Mathematical Asymmetry and operations on Irrational numbers. The purpose for writing this preface is to offer the reader, if one so chooses to continue, certain advice and warnings about this reasoning. To which, I first must indulge that upon peer review, I have become acutely aware of these problems, but due to current invested interest and work regarding the field of Irrational numbers, it is my belief that there may exist pieces herein that are useful in a philosophically reductionist capacity. Of note, that forcefully this paper disregards the use of the Archimedean Property; where, without strong proof, asserts this to be a necessary conclusion since the Reals comprise a larger cardinality than the Rationals. Second, though the Rationals are assumed to be Dense on themselves, they are defined as not Dense on the Reals and as such is claimed that any arbitrary consecutive ordering of Rationals in a continuous interval on the Reals requires the Rationals to be equally spaced. Lastly, the application of these concepts to the problem of π+e implies contradiction if these assumptions are disputed. From the introduction onward, no statements have been revised to supplement these claims; instead, a more intensive and sophisticated undertaking has since begun in parallel, which ought to eventually supplement the weaknesses of this paper. As such, it is clear that the field of Irrational numbers, R/Q and techniques for simplifying analysis, is still a very open problem in Mathematics.