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henle m which numbers real 2012 rep

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Textbook in PDF format 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

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henle m which numbers real 2012 rep