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Hardcover ISBN:  9780883857779 
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Hardcover ISBN:  9780883857779 
Product Code:  CLRM/42 
List Price:  $59.00 
MAA Member Price:  $44.25 
AMS Member Price:  $44.25 
eBook ISBN:  9781614441076 
Product Code:  CLRM/42.E 
List Price:  $55.00 
MAA Member Price:  $41.25 
AMS Member Price:  $41.25 
Hardcover ISBN:  9780883857779 
eBook ISBN:  9781614441076 
Product Code:  CLRM/42.B 
List Price:  $114.00 $86.50 
MAA Member Price:  $85.50 $64.88 
AMS Member Price:  $85.50 $64.88 

Book DetailsClassroom Resource MaterialsVolume: 42; 2012; 219 pp
Everyone knows the real numbers, those fundamental quantities that make possible all of mathematics from high school algebra and Euclidean geometry through the Calculus and beyond; and also serve as the basis for measurement in science, industry, and ordinary life. This book surveys alternative real number systems: systems that generalize and extend the real numbers yet stay close to these properties that make the reals central to mathematics. Alternative real numbers include many different kinds of numbers, for example multidimensional numbers (the complex numbers, the quaternions and others), infinitely small and infinitely large numbers (the hyperreal numbers and the surreal numbers), and numbers that represent positions in games (the surreal numbers). Each system has a welldeveloped theory, including applications to other areas of mathematics and science, such as physics, the theory of games, multidimensional geometry, and formal logic. They are all active areas of current mathematical research and each has unique features, in particular, characteristic methods of proof and implications for the philosophy of mathematics, both highlighted in this book. Alternative real number systems illuminate the central, unifying role of the real numbers and include some exciting and eccentric parts of mathematics. Which Numbers Are Real? Will be of interest to anyone with an interest in numbers, but specifically to upperlevel undergraduates, graduate students, and professional mathematicians, particularly college mathematics teachers.

Table of Contents

Chapters

Part I. The reals

1. Axioms for the Reals

2. Construction of the Reals

Part II. Multidimensional numbers

3. The Complex Numbers

4. The Quaternions

Part III. Alternative lines

5. The Constructive Reals

6. The Hyperreals

7. The Surreals


Additional Material

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


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 Book Details
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Everyone knows the real numbers, those fundamental quantities that make possible all of mathematics from high school algebra and Euclidean geometry through the Calculus and beyond; and also serve as the basis for measurement in science, industry, and ordinary life. This book surveys alternative real number systems: systems that generalize and extend the real numbers yet stay close to these properties that make the reals central to mathematics. Alternative real numbers include many different kinds of numbers, for example multidimensional numbers (the complex numbers, the quaternions and others), infinitely small and infinitely large numbers (the hyperreal numbers and the surreal numbers), and numbers that represent positions in games (the surreal numbers). Each system has a welldeveloped theory, including applications to other areas of mathematics and science, such as physics, the theory of games, multidimensional geometry, and formal logic. They are all active areas of current mathematical research and each has unique features, in particular, characteristic methods of proof and implications for the philosophy of mathematics, both highlighted in this book. Alternative real number systems illuminate the central, unifying role of the real numbers and include some exciting and eccentric parts of mathematics. Which Numbers Are Real? Will be of interest to anyone with an interest in numbers, but specifically to upperlevel undergraduates, graduate students, and professional mathematicians, particularly college mathematics teachers.

Chapters

Part I. The reals

1. Axioms for the Reals

2. Construction of the Reals

Part II. Multidimensional numbers

3. The Complex Numbers

4. The Quaternions

Part III. Alternative lines

5. The Constructive Reals

6. The Hyperreals

7. The Surreals

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