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Which Numbers are Real?
by Michael Henle

This work is a delightfully concise treatment of number systems. J.T. Zerger, CHOICE

This is an excellent book for advanced math students; it can also be used by professional mathematicians who want to learn the basics of constructive math, non-standard analysis, and surreal numbers. V. Ya. Kreinovich, Mathematical Reviews

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. 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. 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.

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  • Michael Henle
  • 224 pp.
  • Hardback
  • Series: Classroom Resource Materials
  • ISBN: 9780883857779
  • 2012
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