This work is a delightfully concise treatment of number systems. The number systems constructed here include the real, complex, quaternion, hyperreal, and surreal. Although numerous papers and books have been published about each of these systems, this treatise provides an introduction to all of them. As it is a categorical axiom system that characterizes the reals, all other number systems are compared to the real numbers. Henle (Oberlin College) constructs the reals twice, using both Cantor's construction and Dedekind cuts. He uses each of these constructions of the reals to motivate the construction of alternative number systems. In particular, the construction of the hyperreals utilizes ideas from Cantor's construction of the reals, and Dedekind cuts provide the motivation in constructing the surreals. The chapter on the constructive reals provides the reader with the historical perspective necessary to appreciate alternative number systems. The author also presents the geometry and calculus of each of the number systems included in this text within the context of the appropriate system. --J.T. Zerger, CHOICE
This book is intended for math students who want to go beyond the standard textbook results about real numbers. The book starts with Part I, which contains the expected material: what is a field, what is an equivalence class, what is mathematical induction, Cantor's construction of real numbers as Cauchy sequences, Dedekind's construction of real numbers as Dedekind cuts, all the way to continuity derivative. Part II extends these results to complex numbers and quaternions. What is really unusual is Part III, which contains topics which are rarely covered in similar books for students: constructive (algorithmic) real numbers, extensions with infinitesimals in the style of non-standard analysis, and John H. Conway's surreal numbers (presented in their original game form). The exposition of all three topics is very pedagogical; most advanced students will be able to study it largely on their own, with minimal help from an instructor. My only suggestion to the instructor is that it may be helpful to explicitly mention the deep relation between surreal numbers and infinitesimals: namely, infinitesimals are a particular case of Conway's general scheme; see, e.g., [P. Ehrlich, Bull. Symbolic Logic 18 (2012), no. 1, 1-45; MR2798267].
This is an excellent book for advanced math students; it can also be used by professional mathematicians who want to learn the basics of constructive math, non-standard analysis, and surreal numbers. --V. Ya. Kreinovich, Mathematical Reviews
Book Description
This book explores alternative number systems that generalise the real numbers yet stay close to the properties that make the real numbers the fundamental quantities of mathematics. These exciting and sometimes eccentric number systems are active areas of research and will be of interest to students and teachers of mathematics.
About the Author
Michael Henle is Professor of Mathematics and Computer Science at Oberlin College and has had two visiting appointments, at Howard University and the Massachusetts Institute of Technology, as well as two semesters teaching in London in Oberlin's own program. He is the author of two books: A Combinatorial Introduction to Topology (W. H. Freeman and Co., 1978, reissued by Dover Publications, 1994) and Modern Geometries: The Analytic Approach (Prentice-Hall, 1996). He is currently editor of The College Mathematics Journal.
