Hyperreal Structures Arising from an Infinite Base Logarithm

This paper presents new concepts in the use of infinite and infinitesimal numbers in real analysis. The theory is based upon the hyperreal number system developed by Abraham Robinson in the 1960's in his invention of “nonstandard analysis”. The paper begins with a short exposition of the construction of the hyperreal number system and the fundamental results of nonstandard analysis which are used throughout the paper. The new theory which is built upon this foundation organizes the set of hyperreal numbers through structures which depend on an infinite base logarithm. Several new relations are introduced whose properties enable the simplification of calculations involving infinite and infinitesimal numbers. The paper explores two areas of application of these results to standard problems in elementary calculus. The first is to the evaluation of limits which assume certain indeterminate forms. The second is to the determination of convergence of infinite series. Both applications provide methods which greatly reduce the amount of computation necessary in many situations.