Read This!

The MAA Online book review column

Briefly Noted

December 2000

However, this is not a book for the "calculus-phobic." There are no exercises like "list the first five terms of the sequence," and very few of the problems would be suitable for a 50-minute midterm exam. Most of the problems assume that the reader is skillful in writing proofs, and they are meant to emphasize concepts. The statements are straightforward, without gradual questions designed to break the problems into easier parts or give hints as to which approach is better.

Inside, the student will find all the classical results that one should learn when studying analysis — for example, one problem asks us to show that if a_n tends to plus or minus infinity, then (1+1/a_n)^a_n tends to e, and another asks us to show that the sum of the reciprocals of the squares is equal to pi squared over six. The solutions, even when labeled "elementary" by the authors, are not short and easy (nor could they be), on the contrary, they require a solid background in calculus, algebra, and trigonometry. "Obvious" steps are usually skipped from the proofs.

Bottom line: if you love mathematics and are really serious about understanding analysis, this book is a must. Note that a second volume is currently being translated.[Ioana Mihaila]

If you enjoy reading popular mathematics books, you will probably enjoy Calvin C. Clawson's Mathematical Mysteries. The author has an engaging style of writing, and his enthusiasm for the subject shines through. Topics discussed include prime numbers (of course), number sequences (eg. The Fibonacci sequence), the golden ratio, the proof that square root of 2 is irrational, primes and secret codes, Ramanujan, Goldbach's Conjecture, the Riemann Zeta function, and Godel's incompleteness theorem. There's a good bit of history included throughout, and there is even a chapter that discusses "Numbers and the Occult." The topics do become weightier by the end of the book, and at that point it may be only mathematicians who are still reading. On the whole, most readers of MAA Online will find most of the topics familiar.

I enjoyed the book on a couple of different levels. I am teaching a liberal arts mathematics course this semester, and Clawson's book treats many of the topics we are discussing in that class, from a slightly different perspective. As a community college professor, his communication skills are well honed. I assigned his chapter on secret codes to a student who is doing a research project. It is the most readable account of RSA codes I have seen. From a mathematician's perspective, I also enjoyed his treatment of the Riemann Zeta function and his sketch of the proof of Godel's theorem. Not being a number theorist or a logician myself, he included enough details to remind me why these topics are interesting.

Of course the intended audience is not professional mathematicians. Clawson does not prove every result he discusses (or even hint at a proof in many cases). In the Ramanujan chapters in particular, he only justifies a small fraction of the equations he presents. This I found a little bit unsatisfying. Clearly, he does this to keep non-mathematicians from getting frustrated. With this caveat in mind, those readers interested in more details should know how to find them.

In summary, I found Clawson's book to be a very enjoyable read, a good presentation of familiar and some less familiar topics. This book would serve as an excellent resource for one who is teaching an elementary course that deals with number theory.[Joel Foisy]

Recently, many mathematics professors have become interested in what they can do to prepare students for mathematics-related jobs outside academia. A crucial problem for many of us is that, having never worked in industry, we have only the vaguest idea of what is involved. This book, which collects the reports from a workshop held in 1998, can help give some specific content to the concept of "industrial mathematics." The workshop was held at the Center for Research in Scientific Computation in North Carolina State University. Graduate students were brought together with "industrial scientists" who presented them with challenging real-world problems. The six reports resulting from this interaction are collected here. If you'd like to know more about what mathematicians can and do work on "in the real world," this book is definitely worth a look.[Fernando Q. Gouvęa]

Publication Data

Problems in Mathematical Analysis I, by W. J. Kaczor and M. T. Nowak. American Mathematical Society, 2000. Softcover, 380 pp., $39.00 ($31.00 to AMS members). ISBN 0-8218-2050-8.

Mathematical Mysteries: The Beauty and Magic of Numbers, by Calvin C. Clawson. Perseus Books, 2000. Softcover, 314 pages, $17.00. ISBN 0-7382-0259-2.

Industrial Mathematics: The 1998 CRSC Workshop, ed. by Pierre A. Gremaud, Zhilin Li, Ralph C. Smith, Hien T. Tran. SIAM, 2000. Softcover, 70 pp., $22.00 ($17.60 to SIAM members). ISBN 0-89871-467-2.

Ioana Mihaila (mihaila@ccucs.coastal.edu) is assistant professor of mathematics at Coastal Carolina University, SC. Her research area is analysis and she has a special interest in student math contests at all grade levels.

Joel Foisy (foisyjs@potsdam.edu) is an assistant professor of mathematics at SUNY Potsdam in Potsdam, NY.

Fernando Q. Gouvęa (fqgouvea@colby.edu) is the editor of FOCUS and MAA Online. He teaches both "History of Mathematics" and "Number Theory", among others, at Colby College. He is a number theorist whose main research focus is on p-adic modular forms and Galois representations.

Back Issues:

• April 1999: Reprints of "classics", books on applied mathematics.
• May 1999: Books on math and physics, both popular and technical.
• July 1999: Various kinds of geometry, and classics in applied mathematics.
• August 1999: What's happening, Dirichlet, and Cryptonomicon.
• October 1999: Graph theory sources, John von Neumann, and Noncommutative Geometry.
• November 1999: Number theory in nine chapters, mathematics at work, and Jacques Hadamard.
• February 2000: Fallacies, Flaws, and Flimflam, measuring the universe, and primary sources at USMA.
• March 2000: Cardano and Manifold: Time.
• May 2000: Grassmann, Smarandache, and a really big prime.
• June 2000: A problem book, a history book, a collection of biographies, and a historical survey of reciprocity laws.
• August 2000: acrostics, field theory, categories, and partial differential equations.
• September 2000: excursions, history, and some high-level expository writing.
• November 2000: a festival of math videos and the Fourier-analytic proof of quadratic reciprocity.

Go to... The main Read This! page The index of all reviews The MAA Online front page

Find out... About the Mathematical Association of America Why you should be a member of the MAA How you can help support the activities of the MAA

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 Gouvˆa, Dept. of Math&CS, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.