Customer Reviews

Algebra

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4 star:		(2)
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This text, which is intended to supplement a high school algebra course, is a concise and remarkably clear treatment of algebra that delves into topics not covered in the standard high school curriculum. The numerous exercises are well-chosen and often quite challenging.

The text begins with the laws of arithmetic and algebra. The authors then cover polynomials, the binomial expansion, rational expressions, arithmetic and geometric progressions, sums of terms in arithmetic and geometric progressions, polynomial equations and inequalities, roots and rational exponents, and inequalities relating the arithmetic, geometric, harmonic, and quadratic (root-mean-square) means. The book closes with an elegant proof of the Cauchy-Schwarz inequality.

Topics are chosen with higher mathematics in mind. In addition to gaining facility with algebraic manipulation, the reader will also gain insights that will help her or him in more advanced courses.

The exercises, which are numerous, often involve searching for patterns that will enable the reader to tackle the problem at hand. Many of the exercises are quite challenging because they require some ingenuity. Some of the exercises are followed by complete solutions. These are instructive to read because the authors present alternate solutions that offer additional insights into the problem.

I also highly recommend the other texts in the Gelfand School Outreach Program. They include The Method of Coordinates, Functions and Graphs (Dover Books on Mathematics), and Trigonometry. Also, to gain additional insights into the inequalities at the end of this text, the reader may wish to consult an Introduction to Inequalities (New Mathematical Library) by Edwin Beckenbach and Richard Bellman.

This book inspires even those with minimal interest in mathematics. If you are passionate about math, this is a must for you. The book is simply a refresher for high school algebra. It contains numerous gems that you could hardly find in a standard algebra text. If you are a teacher, you would have learned much to improve your teaching style and knows how to make your math classes more interesting...overall, a key source to keep on your bookshelf

This is a great book for people who are trying to learn or re-learn algebra. Explanations are very clear and there are many examples to work through.

Well, H. Wu on his page and N.F Taussig here have written quite good reviews, so I guess I can't really add anything new. Still, I feel the need to praise this book some more. Could it be used for a main text or should it be just a supplement? I don't know, but there is much more mathematics contained in these 149 pages than in any standard 500 page high school text on the market today. That's the unsurprising result of accomplished mathematicians writing a math book. Sure, some topics are missing. You won't find 3 or 4 chapters devoted to the several "different" ways to graph a line. There aren't fifty problems in a row that start with "suppose Sam rows upstream at 5 miles per hour and it takes her seven times as long as..." Unfortunately, there isn't even a treatment of complex numbers, the only omission that seems wrong.

You will find several interesting and serious topics that would be dangerous to bright students who insist they hate math, or rather what they've been told is math. Imagine their initial embarrassment when they find out that they can enjoy the subject! Maybe more importantly, imagine their relief when they realize that there IS a reason why we "FOIL", there IS a reason why negative times negative is positive, there IS a reason why we say a^(-1)=1/a, and it's not because "the teacher said so" or "that's just the rule" (ok, it is the rule, but now you'll see why). And there's no attempt to sneak anything by the reader. The authors are quick to acknowledge any gaps in their reasoning, and to assure the reader that in the future he or she will fill them.

It's this honesty and attention to rigor without being too formal or dry that give this book some extra charm. It moves smoothly from basic arithmetic (which everyone should still read if only to learn a different way of explaining it to a student/younger sibling/child) all the way to proofs, both algebraic and visual when possible, of some important inequalities. Cauchy's inductive proof, first for powers of two and then filling in the gaps, of the AM-GM inequality is here, as is the standard proof of Cauchy-Schwarz by the discriminant of a polynomial. Go to your local high school and look at its algebra book. I doubt that's in there.

I'll end with a few of the funny, sometimes weird, little remarks:

After illustrating the associative law using the example (sugar + coffee) + milk = sugar + (coffee + milk), the next problem is: "Problem 25. Try it."

"Please keep in mind that a monomial is a polynomial, so sometimes for a mathematician one is many."

"Probably you are discouraged by this solution because it seems impossible to invent it. The authors share your feeling."

One section begins: "62. How to confuse students on an exam: As usual, there are many evil ways to make use of knowledge."

An excellent source of exercises and examples to work out Algebra, at a school level. Gives just what school lessons usually lack of. Enjoyable.

This is a good, intelligent introduction. I assume most readers will know their algebra already and look for novelties departing from the norm in this book. There are very few of those; the list of novelties below is virtually complete.

The most innovative theoretical presentations are the following. Discussions of multiplication of negatives and negative exponents motivated not only by the standard "preserve the rules" argument but also an extrapolation argument based on continuation of sequences (§§ 14, 18). "Vieta's theorem" that the roots of a quadratic equation sums to the negative of the coefficient for x and multiply to the constant term, and some interesting problems involving this, such as a generalisation to qubics and a proof that there are infinitely many solutions to a^2-2b^2=1 (§ 52).

Some pretty rare sections have novel conceptual problems, such as the following. "Prove that a polynomial of degree not exceeding n is defined uniquely by its n+1 values" (§ 38). "The remainder of a polynomial P when divided by x^2-1 is ... ax+b ... How can you find a and b if you know the values of P when x=-1 and x=1 [the roots of x^2-1]?" (§ 37).

The are also a few interesting application-type problems, reproduced here. The difference in the sequence of squares is a geometric series increasing by 2---why? (§ 24). The partial sums of the powers of two {2^n} are always one less than the next term---why? (§ 36). Is x^2+x+41 always prime (Euler, § 38)? "Prove that the square has the maximum area among all rectangle having the same perimeter" (§ 59), and some other optimisation problems (§ 59, 68), the novelty being that these are treated as applications of graphing.

I bought this book for my daughter (10 years old) and we read it together. We went very slow and I supplement it with a work book. She likes it. I was impressed by the beauty of this book. It might be a little too slim for a textbook but every kid who wants to learn algebra should read it. More than teaching algebra it shows what math should be: simple and beautiful.

My daughter's math textbook is 5 pounds and I can't even stand looking at it. I understand that not every is enthusiastic about math and not everyone can feel the beauty of math. But you don't have to make math so ugly.

Learning math with a 5 lb textbook is simply terrifying but if your kid goes to public school you probably have no choice. Let you kid read a good book like this one, as early as possible, before he(she) grows a life time aversion to math.

I use this text to teach my 9-year old son algebra. The book is good, but beware that the scope is quite limited. I consider it a good text for elementary algebra. Roughly it corresponds to 7th - 9th level in countries like Hong Kong/Japan/Singapore.

Its a great book! You can't deny it! But, if you have more than some experience in algebra, you may want to consider something else. I would have liked it even more if had hard problems from Olympiads.