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Which Numbers Are Real?
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.
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.
- GenresMathematics
230 pages, ebook
First published January 1, 2012
14 people want to read
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Displaying 1 - 4 of 4 reviews
January 27, 2019
This is a do-it-yourself little textbook concerning different number systems: the proofs of most theorems are left as exercises. The first couple of chapters provide an outline for constructing the real number system and proving that the axioms for a complete, linearly ordered field are categorical. Beyond these chapters, the book is a fun and readable introduction to the quaternions, the constructive reals, the hyperreals, and the surreals. The last couple of chapters introduce the typical mathematics student to some very elegant and beautiful material that they wouldn't ordinarily see in the classroom.
January 1, 2015
Starting with the natural numbers and then adding zero, the negative integers, fractions, irrationals and finally including the square root of negative one is an excellent way to teach mathematical history and consistency. By adding additional operations and expanding the definitions, one recapitulates the history of commerce, science and engineering along with the history of mathematics.
Henle begins with the real numbers, their properties and their formal construction, makes the extension into the complex numbers and the quaternions and then moves to the more modern extensions of the constructive reals, the hyperreals and the surreals. Formal definitions of the construction of these extensions are given as well as explanations as to what properties exist and are shared between them. Specific representations and uses such as performing calculus with the hyperreals is briefly introduced. Exercises appear at the end of the sections but no solutions are included.
While this book could be used as a text in a course that covers numbers systems that could be considered extensions of the real numbers I do not see where such a course would be placed in the standard mathematical curriculum. Naturally it could be used for a special topics course, but that is a grab bag that can contain anything. The topic is an excellent one to jar the mathematical senses of upper level students so that they realize that few things are really settled in mathematics, new ways to look at old topics keeps the field dynamically unstable so that progress can continue to be made.
Published in Journal of Recreational Mathematics, reprinted with permission and this review appears on Amazon
Henle begins with the real numbers, their properties and their formal construction, makes the extension into the complex numbers and the quaternions and then moves to the more modern extensions of the constructive reals, the hyperreals and the surreals. Formal definitions of the construction of these extensions are given as well as explanations as to what properties exist and are shared between them. Specific representations and uses such as performing calculus with the hyperreals is briefly introduced. Exercises appear at the end of the sections but no solutions are included.
While this book could be used as a text in a course that covers numbers systems that could be considered extensions of the real numbers I do not see where such a course would be placed in the standard mathematical curriculum. Naturally it could be used for a special topics course, but that is a grab bag that can contain anything. The topic is an excellent one to jar the mathematical senses of upper level students so that they realize that few things are really settled in mathematics, new ways to look at old topics keeps the field dynamically unstable so that progress can continue to be made.
Published in Journal of Recreational Mathematics, reprinted with permission and this review appears on Amazon
January 1, 2013
Well written for a learning audience, by someone who clearly knows the different pitfalls and sources of confusion. Many exercises, clarifying the theory and learning the reader how to prove many theorems. Every chapter ends with references to the annotated bibliography.
Displaying 1 - 4 of 4 reviews