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.
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