Description
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 student’s 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. 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.
Love this book. PROOFS 4 LIFE
I think the unique format of this textbook is fantastic. When I read it, I can almost hear the authors voice. The material is very dense so it’s definitely not meant for speed reading. The way the information is presented is straightforward — it doesn’t try to go over your head.
Get ready to learn mathematics from the ground up! Exciting stuff!
PERFECT!!!
It’s not a book for self-study, but I’ve found it to be just about perfect for a transition-to-higher-math course:
There are plenty of propositions & theorems interspersed through the text and left for the instructor to complete in class, or for the student to complete in assignments and/or for practice in lieu of problems at the end of sections or chapters. I’ve found that this makes homework assignments feel wonderfully relevant (and much more interesting than proving random theorems at the end of a section). In addition, it gives an understanding of the structure of mathematics and how it works: One can’t prove a theorem that uses something that has not yet been proved; translated, only theorems, propositions, corollaries, and lemmas appearing earlier in the text can be used to prove the current statement.
Unproven statements that are not assigned as homework can be proved by the instructor concurrently with presentation of the material. This gives the students an opportunity to see some proofs written by someone other than the text authors — every mathematician has a different style and approach and it’s important for students to know that.
Nice projects throughout the text give an opportunity for students to explore mathematics: discover the theorem and then prove it (as opposed to the standard prove or disprove, though there are projects of that form as well).
The reminders at the end of every chapter are a wonderful idea. Many students at this level have only experienced “reading a math book” as going back in the section to find an example similar to the problem on which he or she is currently working. That isn’t what it is, of course, and around midsemester when everything is getting busier for everyone, it’s a good idea for students to be reminded of it. (The other reviewer’s joke about these comments is simply juvenile and ignorant.)
My students loved this book. It’s readable, clear, and concise. And I very much enjoyed using it.