Authors: Jonathan W. Tooker
We give a geometric definition of real numbers. We demonstrate the existence of a broad class of real numbers which are not elements of any number field: those in the neighborhood of infinity. After considering the reals and the affinely extended reals, we prove that numbers in the neighborhood of infinity are ordinary real numbers. We show that real numbers in the neighborhood of infinity obey the Archimedes property of real numbers. As an application in complex analysis, we show that the Riemann zeta function has infinitely many non-trivial zeros off the critical line.
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