Which Numbers Are Real? (Classroom Resource Materials)
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, CHOICEThis 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'ssurrealnumbers (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