The Art of Proof : Basic Training for Deeper Mathematics The Discrete Integers Natural Numbers and Induction Some Points of Logic Recursion Underlying Notions in Set Theory Equivalence Relations and Modular Arithmetic Arithmetic in Base Ten The Continuous Real Numbers Embedding Z in R Limits and Other Consequences of Completeness Rational and Irrational Numbers Decimal Expansions Cardinality Final Remarks Further Topics Continuity and Uniform Continuity Public Key Cryptography Complex Numbers Groups and Graphs Generating Functions Cardinal Number and Ordinal Number Remarks on Euclidean Geometry. The Art of Proof is designed for a one semester or two quarter course. A typical student will have studied calculus (perhaps also linear algebra) with reasonable success. With an artful mixture of chatty style and interesting examples the students previous intuitive knowledge is placed on solid intellectual ground. The topics covered include: integers induction algorithms real numbers rational numbers modular arithmetic limits and uncountable sets. Methods such as axiom theorem and proof are taught while discussing the mathematics rather than in abstract isolation. Some of the proofs are presented in detail while others (some with hints) may be assigned to the student or presented by the instructor. The authors recommend that the two parts of the book Discrete and Continuous be given equal attention. The book ends with short essays on further topics suitable for seminar style presentation by small teams of students either in class or in a mathematics club setting. These include: continuity cryptography groups complex numbers ordinal number and generating functions.