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Elementary Calculus: An Infinitesimal Approach

H. Jerome Keisler
Dover Publications
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This book is an interesting attempt to bring non-standard analysis to the masses (or at least the masses of freshman calculus students). The present volume is a corrected reprint of the 1986 Second Edition published by Prindle, Weber & Schmidt, and is also available as a free download from the author’s web site; Dover is printing the book for those who prefer to work from a paper copy.

Abraham Robinson rehabilitated the concept of infinitesimal in 1960, from the unlikely direction of mathematical logic, by showing how to extend the real numbers to an ordered field of “hyperreal” numbers that include well-defined infinitesimal numbers and infinite numbers (the reciprocals of infinitesimals). The key point, and the one that required mathematical logic, is that every function on the reals has a unique “natural extension” to a function on the hyperreals, and this allows us to move back and forth between the two fields. This book takes an axiomatic rather than a constructive approach to Robinson’s hyperreal numbers. This has the advantage of being simple (there are only a few axioms) but the serious drawback that we never see any specific examples of infinitesimal numbers. The book includes an Epilog that sketches the construction of the hyperreals in more detail, and Keisler’s book Foundations of Infinitesimal Calculus (also available at the author’s web site in an updated version) is an instructor’s companion volume to the present book that develops the needed properties of the hyperreals in much more detail.

I think the big advantage of the infinitesimal approach presented here is in setting up geometric and physical problems. It makes it easier to account for everything without worrying about which parts are negligible, and from non-standard analysis we know that once we have that starting point there is a rigorous solution. On the other hand, a lot of details about infinitesimals are not explained adequately and there is a lot of handwaving in the proofs. For example, I thought the proof of the Intermediate Value Theorem on p. 162 was totally unbelievable (and the companion instructor’s book gives a completely different proof rather than attempting to rigorize this one). For another example, the study of sequences is dependent on the Extension Principle saying that sequences can be extended to subscripts that are infinite hyperintegers, but there’s no clue given as to what these extensions look like.

Apart from the use of infinitesimals, this is a very conventional calculus course, covering single-variable calculus, infinite series, partial derivatives, multiple integrals, and a little bit of vector calculus and differential equations. Infinitesimals are used throughout the book, but as more calculus infrastructure is built up they take a smaller role in the narrative. The book includes a large number of examples with the calculations worked out in detail, and a massive number of drill exercises, with answers to selected exercises given in the back of the book.

Bottom line: an innovative and relatively accessible approach to freshman calculus, but very different from conventional approaches and probably requiring a lot of preparatory work by the instructor. Not for the faint of heart.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

1. Real and Hyperreal Numbers
2. Differentiation
3. Continous Functions
4. Integration
5. Limits, Analytic Geometry, and Approximations
6. Applications of the Integral
7. Trigonometric Functions
8. Exponential and Logarithmic Functions
9. Infinite Series
10. Vectors
11. Partial Differentiation
12. Multiple Integrals
13. Vector Calculus
14. Differential Equations
Appendix: Tables
Answers to Selected Problems