Number Systems: A Path into Rigorous Mathematics aims to introduce number systems to an undergraduate audience in a way that emphasises the importance of rigour, and with a focus on providing detailed but accessible explanations of theorems and their proofs. The book continually seeks to build upon students' intuitive ideas of how numbers and arithmetic work, and to guide them towards the means to embed this natural understanding into a more structured framework of understanding. The author’s motivation for writing this book is that most previous texts, which have complete coverage of the subject, have not provided the level of explanation needed for first-year students. On the other hand, those that do give good explanations tend to focus broadly on Foundations or Analysis and provide incomplete coverage of Number Systems. Features Approachable for first year undergraduates, but still of interest to more advanced students and postgraduates Does not merely present definitions, theorems and proofs, but also motivates them in terms of intuitive knowledge and discusses methods of proof Draws attention to connections with other areas of mathematics Plenty of exercises for students, both straightforward problems and more in-depth investigations Introduces many concepts that are required in more advanced topics in mathematics New to the second edition Complete solutions to all exercises, and hints for the in-depth investigations Extensive changes to chapters 4 and 5, including defining integral domains as distinct from commutative rings, a more complete discussion of irreducibles, primes and unique factorisation, and more topics in elementary number theory A completely revised chapter 8, giving a more coherent account of quadratic rings and their unique (or non-unique) factorisation properties A thorough correction of typos and errors across all chapters Updates to the bibliography