High school algebra textbooks for gifted students

Cross-posted to Math Educators Stack Exchange. (link)

I am looking for high school algebra/mathematics textbooks targeted at talented students, as preparation for fully rigorous calculus à la Spivak. I am interested in the best materials available in English, French, German or Hebrew.

Ideally, the book(s) should provide a comprehensive introduction to algebra at this level, starting from the most basic operations on polynomials. 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). It should also have problems that range from exercises acquainting students with the basic algebraic manipulations on polynomials to much more difficult ones.

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. To give you an idea, here are a sample of typical problems from the grade-8 book.

1. Perform the indicated operations. $\frac{3p^2mq}{2a^2 b^2} \cdot \frac{3abc}{8x^2 y^2} : \frac{9a^2 b^2 c^3}{28pxy}$

2. Prove that when $a \ne 0$, the polynomial $x^{2n} + a^{2n}$ is divisible neither by $x + a$ nor by $x - a$.

3. Prove that if $a + b + c = 0$, then $a^3 + b^3 + c^3 + 3(a + b)(a + c) (b + c) = 0$.

4. Prove that if $a > 1$, then $a^4 + 4$ is a composite number.

5. Prove that if $n$ is relatively prime to $6$, then $n^2 - 1$ is divisible by 24.

6. Simplify $\sqrt{36x^2}$.

7. Simplify $\sqrt{12 + \sqrt{63}}$.

8. Prove that the difference of the roots of the equation $5x^2 -2(5a + 3)x + 5a^2 + 6a + 1 = 0$ does not depend on $a$.

9. Solve the inequality $|x - 6| \leq |x^2 - 5x + 2|$.

And here are the chapter titles for the grade 8 and 9 books.

Grade 8: Fractions. Polynomials. Divisibility; prime and composite numbers. Real numbers. Quadratic equations; systems of nonlinear equations; resolution of inequalities.

Grade 9: Elements of set theory. Functions. Powers and roots. Equations and inequalities, and systems thereof. Sequences. Elements of trigonometry. Elements of combinatorics and probability theory.

Broadly similar questions have been asked elsewhere, however the suggestions made there are not satisfactory for my purposes.

1. The English translations of Gelfand's books are good; however they are not a sufficiently broad introduction to high school algebra, and do not have enough material on computational technique. They are more in the nature of supplements to an ordinary textbook.

2. Some 19th century books like Hall and Knight have been suggested. On conceptual material, these tend to be too old in language and outlook.

3. Basic Mathematics by Serge Lang seems more to dabble in various topics than to provide a thorough introduction to algebra.

4. I am not inclined towards books with a very strong "New Math" orientation (1971-1983 France, for example). I don't think a student should need to understand the group of affine transformations of $\mathbb{R}$ to know what a line is.

Also, previous questions have perhaps focused implicitly on material in English. I have in mind a student who can also easily read French, German or Hebrew if something better can be found in those languages.

Edit. 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. Fundamentally, this means: 1. It is a substitute for, rather than just a complement to, a regular school algebra textbook. 2. It is directed at the most able students. 3. It conveys the message that proofs and creative problem-solving are central to mathematics.

Solution 1:

Here is my second try. I give some references for Olympiad-style problem solving. Hopefully you'll find something useful in each of them.

• 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. Very student-friendly.

• The Art and Craft of Problem-Solving teaches basic-level problem solving, including an algebra section.

• Problem-solving strategies by Engel is a famous compendium of problems. The focus being on effective problem-solving, theory is really scant but perusing the algebra sections you may find interesting problems.

• 101 Algebra Problems from the Training of the USA IMO Team by Andreescu and Feng is a more specialized compendium.

• Putnam and Beyond by Gelca and Andreescu is a textbook focusing on undergraduate-level contests. Here you can find many challenging problems from areas usually excluded from high school contests (e.g. calculus and linear algebra).

• 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).

• Complex Numbers from A to Z by Andreescu and Andrica is a comprehensive exposition about complex numbers. If you do have to teach this topic, I strongly recommend you to take a look.

As 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. You may want to consider other kind of textbook to compensate for this.

Solution 2:

I would suggest you to take a look at the book Mathematical Thinking: Problem-Solving and Proofs by John P. D'Angelo and Douglas B. West. It covers a broad array of undergraduate-level topics in a self-contained way and starting from basic notions that are likely to be familiar to the kind of student you describe (mathematical logic, methods of proof, sets and functions). The authors present a blend of rigorous theoretical exposition with a practical problem-solving approach through lots of exercises. However a disadvantage is that the scope of some topics is rather limited (most likely due to constraints of space, but I think this is inevitable given the number of subjects covered). Hope you find it useful!