![]() ![]() While these books are good within their genre - and I would prefer them to the more commonly recommended books by Dolciani or Allendoerfer - I would emphasize that apart from the one issue with exponential functions, I feel these are inferior alternatives to Parsonson for a reader of high ability. For example, the following American precalculus books have chapters on this: Pre-Calculus Mathematics by Shanks et al., The Elementary Functions by Fleenor et al., Elementary Functions and Coordinate Geometry by Hu, Advanced Mathematics by Coxford and Payne. That is exponential and logarithmic functions. It should be accessible after about the first 14 chapters of Lang's Basic Mathematics.Īdded: There is one topic that is regarded as already known in Parsonson that might be worth looking at in another book at a higher level than Lang or Axler. It is certainly reasonable to read at least the first volume before starting calculus. The preface to the first volume says that it supposes the student is simultaneously studying calculus, but in practice I've found that calculus is rarely needed except in some of the more advanced probability chapters. They have hard problems, and can be considered something of a "one-stop shop" for the standard non-calculus subjects that are not always included in more elementary books: vector geometry, more advanced analytic trigonometry, combinatorics and probability, matrices and basic linear algebra, complex numbers and polynomials, partial fractions, conic sections and quadric surfaces. This means everything a candidate for Cambridge or Oxford would have been expected to know, except calculus. They were written to cover the entire A-level math curriculum - apart from calculus - in England in the 1970s. This series is aimed at bright high schoolers particularly interested in mathematics.įinally, I'd like to recommend the books Pure Mathematics I, II by Parsonson. 1, 3, 15, 19, 20, 34 in the Anneli Lax New Mathematical Library. ![]() It is impossible to be comprehensive on what good supplementary reading would be, but I would recommend reading these books of Gelfand's alongside the basic textbook: Algebra, The Method of Coordinates, Functions and Graphs (the second coming before the third).Īlso consider working through some of nos. I think this is very helpful if you intend to learn calculus from a rigorous book like Spivak or Apostol. If you have a genuine interest in mathematics, you will want to supplement your reading with various other books for these reasons: (1) to further explore topics in elementary math (2) to work on harder problems (3) to improve your ability to write proofs. This might be best reserved for a second pass through elementary geometry, if you want one.) (There is also the wonderful, but very hard Lessons in Geometry by Hadamard, the first volume of which now has an English translation. But there is little there that is not covered in Parsonson's books - see below.)įor geometry, please have a look at the answer here. A good book that carries trigonometry further roughly from the point where Lang leaves off would be Trigonometry by Nobbs. (Edit: You mentioned that Lang didn't go far enough in trigonometry. Trigonometry by Gelfand and Saul (This is in the same collection as the other books of Gelfand's that I mention below, but it is much closer to an ordinary textbook than they are.).Elementary Trigonometry by Durell (see here).In trigonometry, it would be reasonable but not strictly necessary to use a second source, such as: ![]() Algebra and Trigonometry by Sheldon Axler.Therefore, for a basic algebra textbook, I have only a couple of rather pedestrian recommendations to make, both authored by mathematicians. There are many excellent books in English that are meant to supplement, rather than replace, a basic algebra textbook. A better strategy is to use a decent, but not necessarily comprehensive, main textbook and rely on various kinds of supplementary reading to round out their knowledge. ![]() So as regards learning the basics, an English reader would do well not to dwell too much on trying to find a perfect textbook. The "Art of Problem Solving" textbooks have been mentioned, but from what I've seen of them, they also have significant shortcomings (though I think their problem books are much better). Then there are the late 19th and early 20th century British textbooks (and a few American ones that emulate them), but these have their own serious problems. As Dave Renfro alluded to in the comments, there are the American "New Math" textbooks of the 1960s, but the emphasis on logical formalism in them is not matched by interesting substantive mathematics. But let me first say that you ask a difficult one, because I've found there is unfortunately a dearth of well-written rigorous high school textbooks in English. $\text$ I will try to answer your question. ![]()
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