It's been some time since I taught an introductory math course at the arithmetic/algebra/pre-calculus level, but I remember well the difficulty in trying to motivate students to learn (or re-learn) fractions and the quadratic formula and the rules of exponents. Thus, it was with some trepidation that I began reading this book. I expected the usual dry and unmotivated portrayal of the rules of algebra and geometry, followed by the typical collection of cookie-cutter exercises. Imagine, then, my surprise when I discovered the many delights that awaited me inside the pages of Achatz's wonderful book.

To begin with, this book is intended for a terminal applied math course at the high school, community college, or trade school level. In particular, it is not meant as a pre-calculus book and thus does not cover any of the abstract topics such as limits, infinite series, or even graphing that would be needed by future calculus students. (For an interesting discussion of the current debate over pre-calculus courses, see the recent book, A Fresh Start for Collegiate Mathematics: Rethinking the Courses below Calculus , ed. by Nancy Baxter Hastings.)

The book opens with a bit of set theory, a review of arithmetic, and a discussion of the number line; it's the first book I've seen since grammar school with a multiplication table for the integers. From there, the book covers fractions, percents, exponents, and basic algebra, and then continues with linear and quadratic equations and basic geometry. It finishes with three chapters on trigonometry, extensive appendices with conversion formulas, and solutions to selected exercises.

None of this seems particularly remarkable, but it is in the application of this material that the book truly shines. The author, Thomas Achatz, is a professional engineer who works for General Motors, and it is clear that this man has spent a lot of time in a machine shop. The exercises and examples refer to exotic (to me) objects like bushings, milling cutters, extrusion dies, vernier height gauges, taper shafts, and sine plates, and many of the diagrams and problems look like they were lifted directly from a blueprint or from a technical specification sheet.

A typical problem from the section on trigonometry (towards the end of the book) looks like this:

Some jig-bore machines (which drill holes into metal plates) can be programmed by giving (cartesian) coordinates for where to drill. Suppose you want to drill ten small holes, equally spaced around a circle of diameter 10 inches. The origin of the coordinate system is the lower-left corner of the square that circumscribes the 10-inch circle. What coordinates do you give to the machine?

What I find so delightful about problems like this is that it instantly silences any question of "when are we ever going to use this?" (assuming, of course one's students are planning on working in industry). It's also wonderful to know that people are using mathematics like this in factories and on shop floors.

Let's look at two more examples from the text. In the section on fractions, we find this nice exercise: "Determine the number of discs that can be blanked from a 24-foot roll of #18-gauge cold rolled steel if each blank is 1 1/8 inches in diameter and there is a 3/32-inch web between each disc." (One should imagine the appropriate illustration as well.) Later on, the chapter on percents contains an intriguing discussion on discounts. Apparently, some suppliers give a series of discounts depending on the size of the order, as given here: "widgets: list $1.65, less 15%, 10%, 6%". The first discount of 15% would apply for an order of, say, 6-10 widgets. For an order of 11 to 20 widgets, the discount would be 15% followed by an additional 10% off that new price, and likewise for ever larger orders one would get a final (third) discount of 6%. As one can quickly discover, these discounts are not additive, but it takes a moment with pencil and paper to discover what the total, true discount is.

Naturally, not every section in a math book can have such nice exercises and some chapters (such as the one on factoring polynomials) are like those in any standard textbook. Fortunately, these are balanced by lovely examples elsewhere.

Despite these appealing factors, the book is not perfect. The book is not typeset with TeX, which means that Greek letters like pi and rho don't mesh well with other letters and numbers, and while most of the diagrams are lovely, there are a few circles which are actually printed as 36-gons (this gets rather distracting after a while). At one point, the author states that the length of the side of a square is .7071 times the length of the diagonal (instead of the more familiar and accurate formula of the length of the diagonal divided by the square root of 2). Finally, there are a few odd choices the author makes, such as using Hero instead of Heron or introducing imaginary numbers only to quickly abandon them.

Overall, though, this is an excellent text for a technical mathematics course, and would serve as a valuable resource for anyone teaching a basic math course. The examples and diagrams alone make it worth the price, and at less than $30 at Amazon.com, it's a painless way to acquire a book full of wonderful exercises in basic math.

Gregory P. Dresden is Associate Professor of Mathematics at Washington & Lee University in Lexington, VA.