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100 Years of Math Milestones: The Pi Mu Epsilon Centennial Collection

Stephan Ramon Garcia and Steven J. Miller
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Number of Pages: 
Problem Book
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Hunacek
, on
The year 2013 marked the 100th anniversary of the creation of Pi Mu Epsilon, the national mathematics honor society, and it is in honor of that occasion that this very nice book was written. The book’s format is interesting. Each chapter corresponds to one of the years from 1913 to 2012 with a discussion of one or more mathematical developments that took place in that year. This may be a theorem, an event in the life of a famous mathematician, or the creation of a mathematical entity such as the American Institute or Mathematics or the National Museum of Mathematics in New York.
For each year, there is generally some introductory background discussion, followed by a problem related to that topic, followed by a discussion of that problem (which sometimes, but not always, includes a solution). It is the problems that really motivated the creation of the text: co-author Miller is the editor of the problem department of the Pi Mu Epsilon Journal and wrote a series of articles on “centennial problems” of that journal; these articles morphed into this book.
The chapter for 1918, for example, constitutes a discussion of Georg Cantor and his work, including proofs of some of his theorems. This is followed by a problem (is there an uncountable collection of subsets of the natural numbers that is totally ordered by set inclusion?), which in turn is then given a short and elegant solution. (Unfortunately, a reader of this chapter may wonder what connection this material has to the year 1918; that’s the year Cantor died, but I don’t think this fact is made clear in the chapter.) 
The chapter format described above is not completely rigid. For example, the topic for the year 1984 is, literally, Nineteen Eighty-Four — the book by Orwell. The connection to mathematics arises from the famous “equation” 2 + 2 = 5 that appears in the book, and in the statement of the protagonist, Winston Smith, that “Freedom is the freedom to say that two plus two make four. If that is granted, all else follows.” The problem for this year focuses on the number 4 and is a variation on the famous “four fours puzzle: given four fours and the unlimited use of a finite set of mathematical operations, which natural numbers are constructible?” There are amusing references to Star Trek and to Douglas Adams. Then, after all this, there is a “bonus topic” — a brief discussion of the Bieberbach Conjecture, a solution to which was announced that year. 
Each chapter ends with a bibliography, usually containing at least a couple of entries, and sometimes many more. These bibliographies also demonstrate that at least one of the authors has a sly sense of humor. On page 207, for example, we find the statement that the square root of 2 “has been called the Rome of mathematics, since all roads lead to it.” This is followed by the reference “2, page 207”. And, of course, reference 2 turns out to be this book. 
Reading this book was a lot of fun. The problems are interesting. But even if you don’t invest much time in thinking about them, the introductory discussions by themselves are clear, well-written, informative and entertaining. Most professional mathematicians will already be familiar with a lot of the topics covered, but I imagine that even professionals will find something in these pages that is new to them. I certainly did. I also got a kick out of picking a year at random, trying to guess what topic would be covered in that year, and then checking the book to see if I was correct. (I often was not, but at least I got my birthday year correct: Arrow’s Impossibility Theorem.) I plan on keeping this book close at hand during the upcoming academic year; dipping into it strikes me as a splendid way to spend time while waiting to see if any students show up for office hours. I also suspect that there are other jokes hidden in the book that await discovery.


Mark Hunacek ( teaches mathematics at Iowa State University. 

See the publisher's web page.