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The Real Number System in an Algebraic Setting

J. B. Roberts
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

This is a construction of the real numbers, aimed at undergraduates and having no prerequisites. The “algebraic setting” of the title seems to mean that it deals with the arithmetic of numbers and not with real analysis or the topology of the real line. This is a Dover 2018 unaltered reprint of the 1962 W. H. Freeman publication.

The development is fairly standard. We essentially take the positive integers to be known, although we codify our knowledge in a set of nine axioms covering their additive, multiplicative, ordering, and well-ordering properties. We then progressively bootstrap to the positive rationals through equivalence classes, then to the positive reals through Cauchy sequences, then we finally introduce negative reals. The complex numbers are outlined in an appendix. There are lots of interesting exercises. The book starts out before the integers with a chapter on set theory, mappings, equivalence classes, and the like. I found this chapter to be something of a hodge-podge and a little confusing.

The standard book for constructing the real numbers (for those who want a whole book) is probably Landau’s Foundations of Analysis. The two books are not that different in approach, although I think the present book is much more accessible: it is lighter on formalism and has lots of examples. Landau follows his usual Satz-Beweis style that is heavy on formulas, light on narrative, and has no pictures; Roberts is much more chatty and has lots of pictures. The main technical difference are that Landau starts with the Peano postulates rather than Roberts’s more elaborate set of axioms, and that Landau uses Dedekind cuts to construct the reals while Roberts uses Cauchy sequences. Landau doesn’t deal with sets explicitly, but talks about pairs of numbers being equivalent and about sequences of numbers.

The author, J. B. Roberts, is better known as Joe Roberts, a long-time professor at Reed College (retired in 2014 after 62 years there). He is the author of the MAA Spectrum book Lure of the Integers and of the all-calligraphed Elementary Number Theory: A Problem Oriented Approach. The present book also takes a problem-oriented approach, in that there are a large number of challenging problems compared to a small number of theorems, but the main development is in the theorems and not the problems.

Surprisingly the book started as a set of notes for a freshman course at Reed in the 1950s, and even more surprisingly the author thought of it as a “math appreciation” course suitable for all students. The Preface states,

The course, while being quite detailed and technical, is of great cultural value to nonscience students. It seems that one cannot have any real understanding of what mathematics is about, what its methods are, and what is meant by mathematical creativity without having detailed experience in some technical aspect of mathematics.

An excellent recent book that has this same outlook is Burger & Starbird’s The Heart of Mathematics, although it deals with many areas of contemporary mathematics and not with the foundations of the reals.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.

Chapter I: Introduction
Chapter II: The Natural Numbers
Chapter III: The Positive Rational Numbers
Appendix to Chapter III: Cardinal Numbers
Chapter IV: Interlude, In Which the Way is Prepared
Chapter V: The Nonnegative Real Numbers
Chapter VI: The Real Numbers
Appendix A: Cardinal Numbers (continued)
Appendix B: The Complex Numbers
Appendix C: Peano Postulates
Appendix D: Turning an Error to Good Advantage