Extending the reach of computation Original Research Article

The only well-defined mathematical model of the real number system based on the field axioms is the system of terminating decimals. This is reconstructed as the new real number system and built on the basic integers 0 and 1, the additive and multiplicative identities, respectively, and the addition and multiplication tables of elementary arithmetic. Then standard Cauchy sequences are introduced whose Cauchy limits well-define the nonterminating decimals. Cauchy convergence induces the Cauchy norm and the closure of the terminating decimals in the Cauchy norm is the space of nonterminating decimals and the continuum d∗d∗ that glues together the decimals into the continuum View the MathML sourceR∗ which is non-Archimedean and non-Hausdorff. The decimals form a countably infinite, hence, discrete subspace of View the MathML sourceR∗ that is both Archimedean and Hausdorff.