Advanced Mathematical Concepts

Rating:
Summary: Great Book
Review: I used this book for high school trig, and it its excellent. It is clear and concise, easy to understand. It includes real life applications, and has interesting chapter projects. I would reccomend this book immensely to any one taking trig.

Rating:
Summary: A helpful Textbook
Review: It is very important to have a good math book for the understanding of the concepts. This book has helped me greatly to comprehend the topics by presenting useful examples in each chapter, and it also did a good job pointing out main objectives/rules. If you need to do well in Pre-Calc start with this text!

Rating:
Summary: A behemoth of solecism and cynicism
Review: This book is not advanced, not mathematical, and certainly not about concepts. (And what a pretentious title!) I cannot recommend it to anyone with a drop of creativity or interest in mathematics.

The errors of content are few, but grievous. (The worst howler I've found so far is the treatment of logarithms of negative numbers: a sample problem uses Euler's formula to evaluate ln(-540) as i pi + ln(540), with no mention of multiple values or periodicity.) More distressing are the trespasses against mathematical style. There is a fussiness and fustiness throughout; eccentric and needless definitions abound. Whatever merits may be claimed for distinguishing between arcsin (infinitely many values) and Arcsin (range restricted to [-pi/2, pi/2]), there is no good reason to introduce Sin, the restriction of the ordinary sine function to the domain [-pi/2, pi/2]. What the authors call the "location principle" is known to the rest of the mathematical world as the intermediate value theorem. Calling the angle a plane vector makes with the positive x-axis its "amplitude" is potentially confusing in light of the other meanings of that term in trigonometry and complex analysis. And indexing the coefficient of x^j in a degree-n polynomial as a_{n-j} bespeaks a seriously damaged mind.

All this is unfortunate but possibly pardonable. The book's most damnable fault is the mathematical worldview it promulgates. Advanced Conceptual Mathematics is all about memorizing picky definitions and applying formulas proclaimed ex cathedra to exercises whose tedium is sometimes lightened by "real-life" examples. Rarely does an idea intrude; curiosity has no place. Argument is sacrificed to drill. No effort is made in the exposition to differentiate among the various parts of the mathematical enterprise: groundwork and grunt work, sensible conventions, genuine insight. In the interest of parceling the subject into lesson-sized chunks, the authors pass by opportunities for pointing out the interconnectedness that is among the greatest wonders of mathematics. (One particularly tragic example: graph theory comes just two chapters after combinatorics, yet no connection is made between binomial coefficients and the formula for the number of edges of a complete graph, which is simply announced with no discussion.)

This narrowness comes, above all, at the expense of interesting problems. The authors miss chances to review past work in any way more meaningful than the "mixed review" exercises that just rehash the same stultifying stuff. Here is another example of what goes wrong and how it could be better. In the section on natural logarithms, there is an exercise (p. 645, 38) titled "sociology," which states that "a sociologist has shown"-note the appeal to authority-that the spread of a rumor through a population can be modeled by a formula, which is then given (a logistic curve). Now imagine what might come next. Even if a calculus-based derivation of the equation is beyond the scope of the text, so much is accessible: a few plug-and-chug exercises to get a feel for the algebra; investigation of the meaning of the parameters; the initial- and final-time behavior; the shape of the graph; most important, why the model is reasonable for the situation. Alas, the problem in the book begins and ends with the plug-and-chugs. A student encountering this problem would be perfectly justified in concluding that mathematics is every bit as dull as it is commonly held to be.

Cynicism oozes from each of the book's thousand pages. Each section begins with an "application," as if the intellectual content of the math itself weren't sufficiently interesting to motivate and maintain students' interests-as, in fact, in this presentation, it isn't. "FYI" boxes scattered through the text offer, perhaps as a fillip against boredom, such instructive irrelevancies as "The earliest printed map is one of western China, dated A.D. 1115" and "NASCAR is the National Association for Stock Car Auto Racing." There are inserts on buying a car, balancing a budget, choosing a home computer. The book is chock full of profiles, presumably meant to be inspiring, of college students who use mathematics in their chosen fields, which range from music production to food technology. Naturally, the profile subjects are ethnically and racially diverse (only disabled Caucasians are overrepresented), and have all the correct virtues. ("I want to be a petroleum engineer who doesn't have a lax attitude about the environment," says Deborah Hempel, a senior at UT Austin.) Mary Ann Terrazas (senior, USC) cheerily predicts, "A career in computer science will allow me to get a job almost anywhere, anytime"-and indeed, a mercenary spirit pervades the text. I am reminded of the story of Euclid's student who complained about the uselessness of the mathematics he was studying. Euclid had his slave boy give the student a coin, "since he must make profit from what he learns." Our students deserve better.