Which Numbers Are Real? (Classroom Resource Materials)

✍ Scribed by Michael Henle

• Publisher: The Mathematical Association of America
• Year: 2012
• Tongue: English
• Leaves: 230
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

cover
copyright page
title page
Introduction
Contents
I THE REALS
Axioms for the Reals
How to Build a Number System
The Field Axioms
The Order Axioms
The Completeness Axiom
Construction of the Reals
Cantor's Construction
Dedekind's Construction of the Reals
Uniqueness of the Reals
The Differential Calculus
A Final Word about the Reals
II MULTI- DIMENSIONAL NUMBERS
The Complex Numbers
Two-Dimensional Algebra and Geometry
The Polar Form of a Complex Number
Uniqueness of the Complex Numbers
Complex Calculus
A Final Word about the Complexes
The Quaternions
Four-Dimensional Algebra and Geometry
The Polar Form of a Quaternion
Complex Quaternions and the Quaternion Calculus
A Final Word about the Quaternions
III ALTERNATIVE LINES
The Constructive Reals
Constructivist Criticism of Classical Mathematics
The Constructivization of Mathematics
The Definition of the Constructive Reals
The Geometry of the Constructive Reals
Completeness of the Constructive Reals
The Constructive Calculus
A Final Word about the Constructive Reals
The Hyperreals
Formal Languages
A Language for the Hyperreals
Construction of the Hyperreals
The Transfer Principle
The Nature of the Hyperreal Line
The Hyperreal Calculus
Construction of an Ultrafilter
A Final Word about the Hyperreals
The Surreals
Combinatorial Games
The Preferential Ordering of Games
The Arithmetic of Games
The Surreal Numbers
The Nature of the Surreal Line
More Surreal Numbers
Analyzing Games with Numbers
A Final Word about the Surreals
Bibliography
Index
About the Author