Analysis with Ultrasmall Numbers presents an intuitive treatment of mathematics using ultrasmall numbers. With this modern approach to infinitesimals, proofs become simpler and more focused on the combinatorial heart of arguments, unlike traditional treatments that use epsilon–delta methods. Students can fully prove fundamental results, such as the Extreme Value Theorem, from the axioms immediately, without needing to master notions of supremum or compactness.
The book is suitable for a calculus course at the undergraduate or high school level or for self-study with an emphasis on nonstandard methods. The first part of the text offers material for an elementary calculus course while the second part covers more advanced calculus topics.
The text provides straightforward definitions of basic concepts, enabling students to form good intuition and actually prove things by themselves. It does not require any additional "black boxes" once the initial axioms have been presented. The text also includes numerous exercises throughout and at the end of each chapter.
"This book presents an alternative approach to differential and integral calculus of functions of one variable. … For readers familiar with the classical methods of mathematical analysis, this book can provide an interesting alternative view."
—Zentralblatt MATH 1317
"What the book does exceptionally well is explain and develop the basic notions and machinery slowly, invitingly, methodically, and enjoyably. … Numerous solved exercises make the book highly efficient in learning the nonstandard technique it advocates but also for learning the elements of calculus … the book is a well-aimed stab at the heart of the teaching of analysis and presents a very interesting nonstandard approach. Any student intrigued by the subject of nonstandard analysis will find the book to be entertaining and well-written, and to present a coherent approach at a very elementary level."
—MAA Reviews, March 2015
Continuity and limits
Basic concepts revisited
Sequences and series
Topology of Real Numbers