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.

About the Author

Michael Henle is Professor of Mathematics and Computer Science at Oberlin College and has had two visiting appointments, at Howard University and the Massachusetts Institute of Technology, as well as two semesters teaching in London in Oberlin's own program. He is the author of two books: A Combinatorial Introduction to Topology (W. H. Freeman and Co., 1978, reissued by Dover Publications, 1994) and Modern Geometries: The Analytic Approach (Prentice-Hall, 1996). He is currently editor of The College Mathematics Journal.

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