Description
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.
An exploration of number systems that extend and generalise the real numbers, of interest to students, mathematics teachers and enthusiasts.
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.
Reviews
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
Book Information
ISBN 9780883857779
Author Michael Henle
Format Hardback
Page Count 229
Imprint Mathematical Association of America
Publisher Mathematical Association of America
Weight(grams) 430g
Dimensions(mm) 235mm * 157mm * 15mm
An exploration of number systems that extend and generalise the real numbers, of interest to students, mathematics teachers and enthusiasts.
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.
Reviews
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
Book Information
ISBN 9780883857779
Author Michael Henle
Format Hardback
Page Count 229
Imprint Mathematical Association of America
Publisher Mathematical Association of America
Weight(grams) 430g
Dimensions(mm) 235mm * 157mm * 15mm
