Stephen Abbott, Understanding Analysis (Second Edition), 2015. This lively introductory text exposes the student to the rewards of a rigorous study of functions of a real variable. In each chapter, informal discussions of questions that give analysis its inherent fascination are followed by precise, but not overly formal, developments of the techniques needed to make sense of them. By focusing on the unifying themes of approximation and the resolution of paradoxes that arise in the transition from the finite to the infinite, the text turns what could be a daunting cascade of definitions and theorems into a coherent and engaging progression of ideas. Acutely aware of the need for rigor, the student is much better prepared to understand what constitutes a proper mathematical proof and how to write one.
Miklós Bóna, Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory (Fourth Edition), 2016. The selected advanced topics are: Ramsey theory, pattern avoidance, the probabilistic method, partially ordered sets, the theory of designs, enumeration under group action, generating functions of labeled and unlabeled structures and algorithms and complexity. The book encourages students to learn more combinatorics, provides them with a not only useful but also enjoyable and engaging reading.
H. Davenport, The Higher Arithmetic: An Introduction to the Theory of Numbers (Eighth Edition), 2008. The Higher Arithmetic introduces the classic concepts and theorems of number theory in a way that does not require the reader to have an in-depth knowledge of the theory of numbers The theory of numbers is considered to be the purest branch of pure mathematics and is also one of the most highly active and engaging areas of mathematics today. Since earlier editions, additional material written by J. H. Davenport has been added, on topics such as Wiles’ proof of Fermat’s Last Theorem, computers & number theory, and primality testing.
Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics: A Foundation for Computer Science (Second Edition), 1994. Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. The subject matter is primarily an expansion of the Mathematical Preliminaries section in Knuth's classic Art of Computer Programming, but the style of presentation is more leisurely, and individual topics are covered more deeply. Major topics include: Sums, Recurrences, Integer functions, Elementary number theory, Binomial coefficients, Generating functions, Discrete probability, Asymptotic methods. Particularly enjoyable are the marginal graffiti contributed by students who have taken courses based on this material. The authors want to convey not only the importance of the techniques presented, but some of the fun in learning and using them.
Isaiah Lankham et al., Linear Algebra As an Introduction to Abstract Mathematics, 2016. This is an introductory textbook designed for undergraduate mathematics majors with an emphasis on abstraction and in particular, the concept of proofs in the setting of linear algebra. Typically such a student would have taken calculus, though the only prerequisite is suitable mathematical grounding. The purpose of this book is to bridge the gap between the more conceptual and computational oriented undergraduate classes to the more abstract oriented classes. The book begins with systems of linear equations and complex numbers, then relates these to the abstract notion of linear maps on finite-dimensional vector spaces, and covers diagonalization, eigenspaces, determinants, and the Spectral Theorem. Each chapter concludes with both proof-writing and computational exercises.
Ivan Niven et al., An Introduction to the Theory of Numbers (Fifth Edition), 1991. The Fifth Edition of one of the standard works on number theory, written by internationally-recognized mathematicians. Chapters are relatively self-contained for greater flexibility. New features include expanded treatment of the binomial theorem, techniques of numerical calculation and a section on public key cryptography. Contains an outstanding set of problems.
Sheldon Ross, A First Course in Probability (Tenth Edition), 2022. A First Course in Probability is an elementary introduction to the theory of probability for students in mathematics, statistics, engineering, and the sciences. Through clear and intuitive explanations, it presents not only the mathematics of probability theory, but also the many diverse possible applications of this subject through numerous examples. The 10th Edition includes many new and updated problems, exercises, and text material chosen both for interest level and for use in building student intuition about probability.
Joseph Silverman, A friendly introduction to number theory (Fourth Edition), 2013. A Friendly Introduction to Number Theory is designed to introduce students to the overall themes and methodology of mathematics through the detailed study of one particular facet-number theory. Starting with nothing more than basic high school algebra, students are gradually led to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing is appropriate for the undergraduate audience and includes many numerical examples, which are analyzed for patterns and used to make conjectures. Emphasis is on the methods used for proving theorems rather than on specific results.
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