I do, however, know of plenty of definitions satisfying one or the other, which leads me to suspect that people rely one one definition for the one purpose and another definition for the other. I do not know of any definition of “rigour” satisfying both of these conditions. Since no one ever argues for this, it must apparently be evident already from the definition why this is so. It is always taken for granted as a virtual truism that a “rigorous” proof is better than a “non-rigorous” one. It must make it clear why being “rigorous” is desirable. Whether something is “rigorous” or not is supposed to be an objective, straightforward question, rather like telling whether a shirt is red or blue.Ģ. It must entail a clear-cut way of telling “rigorous” from “non-rigorous” methods. Judging by the way people use the term, the definition of “rigorous,” what ever it is, must, it seems, fulfil two key conditions:ġ. And I very much doubt that all those zealous advocates of “rigorous” mathematics do either. You may want to consider other kind of textbook to compensate for this.Some people would insist that the kinds of proofs I give in my calculus textbook are “not rigorous.” What does this even mean? They are not what exactly? What is the definition of “rigorous”? I do not know any credible definition of this term. If you do have to teach this topic, I strongly recommend you to take a look.Īs a final warning, I must tell you that (at least in my own experience) contest-geared textbooks tend to focus in quick development of problem-solving skills rather than rigorous mathematical exposition. Polynomials by Barbeau is a more leisurely treatment of the basic theory of polynomials (in case you're dissatisfied with any of the previous suggestions).Ĭomplex Numbers from A to Z by Andreescu and Andrica is a comprehensive exposition about complex numbers. Here you can find many challenging problems from areas usually excluded from high school contests (e.g. Putnam and Beyond by Gelca and Andreescu is a textbook focusing on undergraduate-level contests. The focus being on effective problem-solving, theory is really scant but perusing the algebra sections you may find interesting problems.ġ01 Algebra Problems from the Training of the USA IMO Team by Andreescu and Feng is a more specialized compendium. Problem-solving strategies by Engel is a famous compendium of problems. The Art and Craft of Problem-Solving teaches basic-level problem solving, including an algebra section. Topics in Algebra and Analysis: Preparing for the Mathematical Olympiad by Bulajich, Gómez and Valdez is what most closely resembles a comprehensive treatment among the books I know. Hopefully you'll find something useful in each of them. I give some references for Olympiad-style problem solving. It conveys the message that proofs and creative problem-solving are central to mathematics. It is directed at the most able students. It is a substitute for, rather than just a complement to, a regular school algebra textbook. I'd like to clarify that I'm not asking for something identical to these books, just something as close as possible to their spirit. I have in mind a student who can also easily read French, German or Hebrew if something better can be found in those languages.Įdit. $\frac$ to know what a line is.Īlso, previous questions have perhaps focused implicitly on material in English. To give you an idea, here are a sample of typical problems from the grade-8 book. Specifically, I am looking for something similar in spirit to a series of excellent Russian books by Vilenkin for students in so-called "mathematical schools" from grades 8 to 11, although I am only looking for the equivalent of the grade 8 and 9 books, which are at precalculus level. It should also have problems that range from exercises acquainting students with the basic algebraic manipulations on polynomials to much more difficult ones. It should include necessary theory (e.g., Bezout's remainder theorem on polynomials, proof of the fundamental theorem of arithmetic, Euclid's algorithm, a more honest discussion of real numbers than usual, proofs of the properties of rational exponents, etc., and a general attitude that all statements are to be proved, with few exceptions). Ideally, the book(s) should provide a comprehensive introduction to algebra at this level, starting from the most basic operations on polynomials. I am interested in the best materials available in English, French, German or Hebrew. I am looking for high school algebra/mathematics textbooks targeted at talented students, as preparation for fully rigorous calculus à la Spivak. Cross-posted to Math Educators Stack Exchange.
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