Which Numbers Are Real? THE SURREAL NUMBERS. THE HYPERREAL NUMBERS. THE CONSTRUCTIVE REALS. THE COMPLEX NUMBERS. THE QUATERNIONS

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.

Introduction; Part I. The Reals: 1. Axioms for the reals; 2. Construction of the reals; Part II. Multidimensional Numbers: 3. The complex numbers; 4. The quaternions; Part III. Alternative Lines: 5. The constructive reals; 6. The hyperreals; 7. The surreals; Bibliography; Index.

This book explores alternative number systems that generalise the real numbers yet stay close to the properties that make the real numbers the fundamental quantities of mathematics. These exciting and sometimes eccentric number systems are active areas of research and will be of interest to students and teachers of mathematics.