Axiomatic Arithmetic: An Introduction To Number Systems

The goal of this book is to construct and explore the properties of the most commonly encountered number systems, beginning with the Peano axioms for the natural numbers. The systems discussed include the natural numbers, integers, rational numbers, real numbers, and complex numbers. Along the way, the book covers a range of related topics such as the ring-theoretic properties of integers, Euclid's algorithm, the fundamental theorem of arithmetic, modular arithmetic, and cardinality. It also delves into decimal and continued fraction representations of the reals, the rational zeroes theorem, Liouville's approximation theorem, the extended real number system, and the fundamental theorem of algebra. The book contains 233 exercises with detailed solutions provided at the end, making it well-suited for self-study. These exercises also introduce additional number systems including the quaternions, octonions, and p-adic numbers. The intended audience is first- or second-year undergraduate students in Mathematics or the Applied Sciences who have completed a first course in Calculus. It may also be valuable to school teachers interested in the foundations of algebra and calculus, as well as students in teacher education programs. This book can serve as a supplementary text for introductory courses in real analysis or algebra. It also includes 57 figures to aid understanding throughout.

Amol Sasane received his PhD in Mathematics from the University of Groningen in the Netherlands in 2001. He also holds an undergraduate degree in Electrical Engineering from the Indian Institute of Technology, Mumbai, and a Masters degree in Theoretical Physics from Lund University, Sweden. He is currently a Professor of Mathematics at the London School of Economics, which he joined in 2004. His research interests lie in Applicable Analysis, and he is the author of two research monographs and six textbooks.