Rigorous and comprehensive textbooks on precalculus

Question

I am looking for comprehensive and rigorous textbooks on precalculus that provide proof for all the formulas and theorems they mention. You can suggest multiple books on different topics like trigonometry, algebra, and geometry.
I am aware that there are similar questions on this website but this question distinguishes from them mainly by the requirement of rigorousness.

Answer 1

$\text{Hello Bonsoir.}$ I will try to answer your question. 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. 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. 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. 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).

So as regards learning the basics, an English reader would do well not to dwell too much on trying to find a perfect 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. There are many excellent books in English that are meant to supplement, rather than replace, a basic algebra textbook.

Therefore, for a basic algebra textbook, I have only a couple of rather pedestrian recommendations to make, both authored by mathematicians. They also cover basic trigonometry.

• Basic Mathematics by Serge Lang
• Algebra and Trigonometry by Sheldon Axler

In trigonometry, it would be reasonable but not strictly necessary to use a second source, such as:

• Elementary Trigonometry by Durell (see here)
• 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.)

(Edit: You mentioned that Lang didn't go far enough in trigonometry. A good book that carries trigonometry further roughly from the point where Lang leaves off would be Trigonometry by Nobbs. But there is little there that is not covered in Parsonson's books - see below.)

For geometry, please have a look at the answer here. (There is also the wonderful, but very hard Lessons in Geometry by Hadamard, the first volume of which now has an English translation. This might be best reserved for a second pass through elementary geometry, if you want one.)

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. I think this is very helpful if you intend to learn calculus from a rigorous book like Spivak or Apostol.

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).

Also consider working through some of nos. 1, 3, 15, 19, 20, 34 in the Anneli Lax New Mathematical Library. This series is aimed at bright high schoolers particularly interested in mathematics.

Finally, I'd like to recommend the books Pure Mathematics I, II by Parsonson. They were written to cover the entire A-level math curriculum - apart from calculus - in England in the 1970s. This means everything a candidate for Cambridge or Oxford would have been expected to know, except calculus. 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. 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. It is certainly reasonable to read at least the first volume before starting calculus. It should be accessible after about the first 14 chapters of Lang's Basic Mathematics.

Added: 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. That is exponential and logarithmic functions. 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. 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.

There are also some algebra textbooks that treat algebra at a higher level than Parsonson (but without being about abstract algebra). These could supplement parts of the second volume of Parsonson: Higher Algebra by Ferrar and Introduction to Higher Algebra by Mostowski and Stark.