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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 well-developed theory, including applications to other areas of mathematics and science, such as physics, the theory of games, multi-dimensional 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 upper-level undergraduates, graduate students, and professional mathematicians, particularly college mathematics teachers.
Introduction
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
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 4.2 The Polar Form of a Quaternion
Complex Quaternions and the Quaternion Calculus
A Final Word about the Quaternions
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 7
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