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dos Reis, Tiago S.; Anderson, James A.D.W. (2014)
Languages: English
Types: Unknown
A geometrical construction of the transcomplex numbers was given\ud elsewhere. Here we simplify the transcomplex plane and construct the\ud set of transcomplex numbers from the set of complex numbers. Thus\ud transcomplex numbers and their arithmetic arise as consequences of their\ud construction, not by an axiomatic development. This simplifes transcom-\ud plex arithmetic, compared to the previous treatment, but retains totality\ud so that every arithmetical operation can be applied to any transcomplex\ud number(s) such that the result is a transcomplex number. Our proof\ud establishes the consistency of transcomplex and transreal arithmetic and\ud establishes the expected containment relationships amongst transcomplex,\ud complex, transreal and real numbers. We discuss some of the advantages\ud the transarithmetics have over their partial counterparts.
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