Calculus with infinitesimals

Abstract

Abraham Robinson discovered a rigorous approach to calculus with infinitesimals in 1960 and published it in [9]. This solved a 300 year old problem dating to Leibniz and Newton. Extending the ordered field of (Dedekind) real numbers to include infinitesimals is not difficult algebraically, but calculus depends on approximations with transcendental functions. Robinson used mathematical logic to show how to extend all real functions in a way that preserves their propertires in a precise sense. These properties can be used to develop calculus with infinitesimals. Infinitesimal numbers have always fit basic intuitive approximation when certain quantities arc small enough, but Leibniz, Euler, and many others could not make the approach free of contradiction. Section 1 of this article uses some intuitive approximations to derive a few fundamental results of analysis. We use approximate equality, x ≈ y, only in an intuitive sense that x; is sufficiently close to y. © 2007 Springer-Verlag Wien.