Which Numbers are Real?

Classroom Resource Materials

by Michael Henle

The set of real numbers is one of the fundamental concepts of mathematics. This book surveys alternative number systems: systems that generalise the real numbers yet stay close to the properties that make the reals central to mathematics. There are many alternative number systems, such as multidimensional numbers (complex numbers, quarternions), infinitely small and infinitely large numbers (hyperreal numbers) and numbers that represent positions in games (surreal numbers). Each system has a well-developed theory with applications in other areas of mathematics and science. They all feature in active areas of research and each has unique features that are explored in this book. Alternative number systems reveal the central role of the real numbers and motivate some exciting and eccentric areas of mathematics. What Numbers Are Real? will be an illuminating read for anyone with an interest in numbers, but specifically for advanced undergraduates, graduate students and teachers of university-level mathematics.

Genres: Mathematics

224 pages, Hardcover

First published January 1, 2012

Reviews

[Beau · Rating 4 out of 5]

This is a do-it-yourself little textbook concerning different number systems: the proofs of most theorems are left as exercises. The first couple of chapters provide an outline for constructing the real number system and proving that the axioms for a complete, linearly ordered field are categorical. Beyond these chapters, the book is a fun and readable introduction to the quaternions, the constructive reals, the hyperreals, and the surreals. The last couple of chapters introduce the typical mathematics student to some very elegant and beautiful material that they wouldn't ordinarily see in the classroom.

[Charles · Rating 5 out of 5]

Starting with the natural numbers and then adding zero, the negative integers, fractions, irrationals and finally including the square root of negative one is an excellent way to teach mathematical history and consistency. By adding additional operations and expanding the definitions, one recapitulates the history of commerce, science and engineering along with the history of mathematics.
Henle begins with the real numbers, their properties and their formal construction, makes the extension into the complex numbers and the quaternions and then moves to the more modern extensions of the constructive reals, the hyperreals and the surreals. Formal definitions of the construction of these extensions are given as well as explanations as to what properties exist and are shared between them. Specific representations and uses such as performing calculus with the hyperreals is briefly introduced. Exercises appear at the end of the sections but no solutions are included.
While this book could be used as a text in a course that covers numbers systems that could be considered extensions of the real numbers I do not see where such a course would be placed in the standard mathematical curriculum. Naturally it could be used for a special topics course, but that is a grab bag that can contain anything. The topic is an excellent one to jar the mathematical senses of upper level students so that they realize that few things are really settled in mathematics, new ways to look at old topics keeps the field dynamically unstable so that progress can continue to be made.

Published in Journal of Recreational Mathematics, reprinted with permission and this review appears on Amazon

[Pedro · Rating 5 out of 5]

Well written for a learning audience, by someone who clearly knows the different pitfalls and sources of confusion. Many exercises, clarifying the theory and learning the reader how to prove many theorems. Every chapter ends with references to the annotated bibliography.

[Akiva ꙮ]

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