Analysis on a Nonarchimedean Extension of the Real Numbers

A field extension R of the real numbers is presented. It has similar algebraic properties as ; for example, all roots of positive numbers exist, and the structure C obtained by adjoining the imaginary unit is algebraically complete. The set can be totally ordered and contains infinitely small and infinitely large quantities. Under the topology induced by the ordering, the set becomes Cauchy complete; but different from , there is a second natural way of introducing a topology. It is shown that R is the smallest totally ordered algebraically complete extension of . Power series have identical convergence properties as in , and thus important transcendental functions exist and behave as in . Furthermore, there is a natural way to extend any other real function under preservation of its smoothness properties. In addition to these common functions, delta functions can be introduced directly. A calculus involving continuity, differentiability and integrability is developed. Central concepts like the intermediate value theorem, mean value theorem, and Taylor’s theorem with remainder hold under slightly stronger conditions. It is shown that, up to infinitely small errors, derivatives are differential quotients, i.e. slopes of infinitely small secants. While justifying the intuitive concept of derivatives of the fathers of analysis, it also offers a practical way of calculating exact derivatives numerically. The existence of infinitely small and large numbers allows an introduction of delta functions in a natural way, and the important theorems about delta functions can be shown using the calculus on R.