Perspex Machine XIII: Construction of the Transcomplex Numbers from the Complex Numbers

New title: Construction of the Transcomplex Numbers From the Complex Numbers.

Transcomplex arithmetic is total. In particular this allows division by zero. Hence transcomplex arithmetic permits computations at mathematical singularities, thereby supporting novel computer applications and the design of novel computing machines. A geometrical construction of the transcomplex numbers was given elsewhere. Here we simplify the transcomplex plane and construct the set of transcomplex numbers from the set of complex numbers. Thus transcomplex numbers and their arithmetic arise as consequences of their construction, not by an axiomatic development. This simplies transcomplex arithmetic, compared to the previous treatment, but retains totality so that every arithmetical operation can be applied to any transcomplex number(s) such that the result is a transcomplex number.