The William Lowell Putnam Mathematical
Competition, 1995-2000 follows the now familiar structure of its two
predecessors, with a few notable exceptions. The most important of these
are the inclusion of a significant bibliography and of a section of hints,
both of which make the book a better teaching resource for problem-solving
seminars. Students will also benefit from a nicely brief and
focused section "Advice to the student reader" in the introduction. The
well-balanced and structured introduction, written in a friendly style, also
includes a section on "The Putnam Competition over the years." It goes on
to discuss scoring and basic notation. All of this should be helpful and
anxiety-reducing for students picking up the book. The solutions are well presented and many include helpful diagrams, very informative remarks and references, as well as cross-references to other Putnam problems. One gets a feeling that the authors took care to present the Putnam Competition as a mathematical endeavor firmly connected to the mainstream of mathematics, rather than an esoteric event for the lovers of mathematical puzzles.
The book concludes with a chapter presenting the winners (both individual and team) of the 1985-2000 competitions, as well as one on Putnam trivia, followed by a lively article by Bruce Reznick entitled "Some Thoughts on Writing for the Putnam," which gives readers a glimpse into the process of generating problems for the competition.
I cannot end before mentioning the typography. I appreciate the choice of
the font, spacing and paper, all of which make the book more pleasant to
behold than its predecessors. The use of bold font in the index to indicate
references with the detailed explanation of the entry is a characteristic
example of the smart choices made by the creators of the book. All in all,
I consider this tome to be a must have resource for college libraries as
well as for individual readers interested in ingenuity and problem solving
in mathematics. [Leo Livshits]
Leonardo of Pisa's Liber Abaci (Book of
Calculation) is one of the most important books in the history of European
mathematics. Leonardo learned his mathematics from the Islamic mathematical
tradition (some have even argued that he should be considered part of that
tradition), but he wrote his books in Latin. As a result, he became one of
the most important mediators of that tradition to European readers.
The Liber Abaci focused on "calculation." From our point of view, most of the contents are either arithmetic or algebra. It starts off with an extensive and detailed discussion of the use of "the nine Indian figures" to represent numbers and to perform arithmetical operations with them. The "nine figures" are, of course, 1, 2, 3, 4, 5, 6, 7, 8, and 9, with 0 being treated as a separate sign. The book then goes on to deal with a large number of problems that we would describe as algebraic. Most of these reduce to linear equations (this includes systems of linear equations and also linear diophantine equations). The final chapter discusses quadratic equations. All of this is done, of course, without any sort of algebraic symbolism.
As Heinz Lueneberg has pointed out in a recent issue of FOCUS, this is an amazing book, and it is somewhat frustrating that we had to wait for the 800th anniversary of its original publication to see an English translation. But here it is at last, the Liber Abaci in English. It is now possible for mathematicians who are interested in history to read it, and for students to read portions of it in their history courses.
One final comment: attentive readers will note that the name "Fibonacci"
does not occur in this brief review. That is because Leonardo never used
that name: it is a nickname given to him by a 19th century historian of
mathematics. I'm enough of a purist to want to refer to Leonardo as he
referred to himself: Leonardo of Pisa, of the Bonacci family. He was a
great mathematician, and here is an accessible and readable edition of his
most famous book. Don't miss the opportunity to get a copy. [Fernando
Q. Gouvêa]
This is an interesting
idea. Conference proceedings sometimes contain a heterogeneous mix of
articles that make it hard to justify purchasing a copy. At best, one will
be interested in one or two papers. Surveys in Number Theory solves
that problem by being selective. Instead of containing the full proceedings
of the Millennial Conference on Number Theory (these were published
separately in three large hardcover volumes called
Number Theory for
the Millennium), this book collects only
those papers that might be of interest to a wider range of readers. For the
most part, these are papers that survey recent work in various parts of
number theory; most number theorists will find something to like here. And
since the resulting book is shorter and in paperback, its price is very
reasonable.
There are several very nice things in the book. Some of the topics covered are the Riemann Hypothesis, normal numbers, automorphic forms, Iwasawa theory, diophantine approximation, and Waring's Problem. Bjorn Poonen's article on how to compute rational points on curves is particularly good. There is an interesting essay by Robert A. Rankin on "G. H. Hardy As I Knew Him" that is likely to be of interest to historians of mathematics.
Given the size of the Millennial Conference, I think the idea of going for a two-tiered approach to publishing the conference proceedings was a good one. Large research libraries will very likely want to get the full three-volume set. Everyone else will probably be happy with this shorter version.[Fernando Q. Gouvêa]
, by Kiran S. Kedlaya, Bjorn Poonen and Ravi Vakil. Mathematical Association of America, 2002. Hardcover, 354 pp., $44.95 ($35.95 to MAA members). ISBN 0-88385-807-X.
Fibonacci's Liber Abaci, by L. E. Sigler. Springer-Verlag, 2002. Hardcover, 636 pp., $99.00. ISBN 0-387-95419-8.
Surveys in Number Theory: Papers from The Millennial Conference on Number Theory, ed. by M. A. Bennett, B. C. Berndt, N. Boston, H. G. Diamond, A. J. Hildebrand, and W. Philipp. A K Peters, 2003. Paperback, 363 pp., $30.00. ISBN 1-56881-162-4.
Leo Livshits (llivshi@colby.edu) is associate professor of mathematics at Colby College, and has directed many problem seminars and problem competitions.
Fernando Q. Gouvêa (fqgouvea@colby.edu) is the author of several books, including, most recently, Math through the Ages, written in collaboration with William Berlinghoff.
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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 Math&CS, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.
MAA Online is edited by Fernando Q. Gouvêa (fqgouvea@colby.edu). Last modified: Thu Mar 13 12:47:08 -0500 2003