# Analysis on a Nonarchimedean Extension of the Real Numbers

### Abstract

A field extension R of the real numbers is presented. It has similar algebraic properties
as R; 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 R, there is a second
natural way of introducing a topology. It is shown that R is the smallest totally ordered
algebraically complete extension of R.
Power series have identical convergence properties as in R, and thus important
transcendental functions exist and behave as in R. 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.

M. Berz, *Lecture Notes, 1992 and 1995 Mathematics Summer Graduate Schools of the German National Merit Foundation, * **MSUCL-933**, Department of Physics and Astronomy, Michigan State University (1994)

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