Read This!

The MAA Online book review column

Briefly Noted

June 2006

At this point, we are adding some 30 reviews a month to MAA Reviews. This column contains abbreviated versions of some of the reviews that have appeared there (fairly) recently.

This is an expanded version of a book first published in 2000, which, to my knowledge, is unique in its declared aim of providing a ‘robust bridge between high school and university mathematics’. Moreover, it is one of the relatively few (contemporary) mathematics books declaring an emphasis on ‘pure’ mathematics. Several questions therefore spring to mind: What should be included within the guise of ‘pure mathematics’? Is there a gap between high school and university mathematics? If so, what is needed to bridge it? When should this bridging take place?

With respect to the first of the above questions, the author’s view can be gleaned by the following description of the contents. Several chapters provide an introduction to number systems and analysis; five chapters cover the basic theory of the integers (including modular arithmetic); four chapters amount to an introduction to discrete mathematics (counting, sets, functions, permutations); and three chapters deal with Functions, relations and countability.

As for the second question, the gap between high school and university mathematics is assumed to arise from what is supposed to be the handle-turning algorithmic approach that students are said to bring to their undergraduate studies, and the book aims to correct this imbalance by providing a ‘fascinating introduction to the culture of mathematics’. (Why, one wonders, should any mention of applied mathematics be excluded from this corrective process?)

However, experience of teaching undergraduates tells us that there isn’t a single definable gap in students’ mathematical knowledge, but a myriad of deficiencies — with wide variation from one student to another. Some such deficiencies may be addressed via much of the content covered in this book, but its scope is (inevitably) too narrow for it to cater for all needs. Whilst different tutors may form their own lists of gaps in student background knowledge, Liebeck has the tacit two-fold aim of ‘gap-filling’ and ‘attitude bridging’.

For me, where this book succeeds is in its provision of insights into topics that students are likely to encounter as maths majors. Chapters 11 to 15 lay good foundations for later incursions into number theory. Then there is an all-pervading emphasis on methods of proof, which are discussed in a wide variety of contexts. Another of the book’s strengths lies in the author’s accessible style of presentation and the provision, at the end of each chapter, of interesting sets of exercises. It also commences a formalisation of many set-theoretic topics and makes some incursion into the world of discrete mathematics.

Overall, I would recommend this publication as a possible source of directed supplementary reading for first year maths majors. Also, for those undergraduates who wish to change to mathematics, and who have not done maths since high school, this book could form part of a necessary bridging course.

[P. N. Ruane; posted to MAA Reviews 05/25/2006]

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.

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.

Overall, 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; posted to MAA Reviews 06/03/2006]

In the early 1960s, Alexandre Grothendieck and a large group of collaborators set out to remake the foundations of algebraic geometry. With Jean Dieudonné, he started writing the Éléments de Géométrie Algébrique, later known as EGA, in which they planned to completely recast the basics of the subject. Simultaneously, Grothendieck started an algebraic geometry seminar at the Institut des Hautes Études Scientifiques (IHES) in which he and others could develop the new ideas further. The published reports from the seminar made up the several "SGA" volumes. The stated goal was to eventually incorporate all or most of SGA into EGA.

The plan was extremely ambitious, and, as is well known, it was never actually completed. Four (of ten originally planned) parts of EGA were eventually published (only one in book format), and seven parts of SGA (several of which had multiple volumes).

SGA was originally published and distributed by IHES, with this first volume, containing material presented at the seminar in 1960 and 1961, appearing in 1963. In the late 1960s, Grothendieck resigned from IHES and prohibited it from having anything further to do with the work. As a result, the SGA volumes were reprinted (with only minor updates) in Springer's Lecture Notes in Mathematics series. At that point, they were still photo-reproductions from typed notes, not very easy to read or handle.

The new edition of SGA 1 is elegantly done in TeX, with a few added notes by Michel Raynaud updating the material when necessary. (Large chunks of the material in SGA 1 were superseded by EGA IV; this had already been noted in the 1970 reprint and is not belabored in the new version.) The page numbers from the original edition are reproduced in the margins, a nice touch that historians will value. The book still retains much of its character as seminar notes; for example, there are no notes from the seventh seminar, and this is simply noted in the table of contents as "VII: n'existe pas."

SGA 1 includes the 1970 introduction to the whole series, outlining Grothendieck's ambitious plans, listing all of the SGA volumes, and explaining his break with IHES. The main topic of the seminar is the theory of étale coverings and the related notion of the (algebraic) fundamental group. This material has, of course, been treated elsewhere since, but this remains a useful reference. Many thanks are due to the SMF for bringing these volumes back; let's hope they stay the course and get through the whole series! Even more, let's hope they do EGA also!

[Fernando Q. Gouvêa; posted to MAA Reviews 06/09/2006]

Publication Data

Technical Shop Mathematics, 3rd edition, by Thomas Achatz. Industrial Press, 2006. Hardcover, 561 pages, $44.95. ISBN 0-8311-3086-5.

Revêtements Étales et Groupe Fondamental (SGA 1), ed. by A. Grothendieck. Société Mathématique de France, 2003. Distributed by the American Mathematical Society. Hardcover, 325 pages, $40.00. ISBN 2856291414.

Peter Ruane is retired from university teaching and now dabbles in as many creative diversions as possible.

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

Fernando Q. Gouvêa is looking forward to retiring his old copies of the SGA volumes.

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Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouv&ecric;a, Dept. of Mathematics, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.