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The Number Pi

American Mathematical Society
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There's an old story about a shoe store owner who meets a mathematician and complains that he doesn't know what size shoes to buy.

"No problem," says the mathematician. "There's a simple formula for that"; and he shows him the Gaussian normal distribution. The shoe seller stares for some time at the equation and asks, "What is that funny symbol?"

"That's the Greek letter pi," the mathematician responds.

"What is pi?"

"That's the ratio between the circumference and the diameter of a circle."

Upon hearing this, the shoe seller cries out, "What does a circle have to do with shoes?!"

The number π has fascinated people and has inspired mathematical activity for almost three millennia. To paraphrase Christopher Marlowe, this is the transcendent constant that launched a thousand geometrical, algebraic, and analytic excursions. In recent years, there has been a spate of excellent books about this constant, among them Pi Unleashed, by Jörg Arndt and Christoph Haenel, Pi and the AGM, by J. Borwein and P. Borwein, and Pi: A Source Book (Second edition), edited by L. Berggren, J. Borwein, and P. Borwein. In 1998, the film Pi (co-authored and directed by Darren Aronofsky) described a frenzied search for number-theoretic patterns (in the stock market and in the Torah, for example) by various groups of people. (The 2002 prize-winning novel The Life of Pi is not pertinent to our discussion.)

To emphasize that it is "dealing with a theme which cuts across the mathematics courses classically taught in the first four years of college," the book under review traces significant mathematical developments concerning pi in five chapters, arranged to correspond roughly to the four undergraduate years and the first year of graduate study. The fourth chapter ("Squaring the Circle") includes some Galois theory, and the fifth chapter explores the link between pi and elliptic integrals in some detail. There is a sixth chapter which contains solutions to the ninety-five exercises spread throughout the book. There is also a 105-item Bibliography.

Beginning with Archimedes and ending with the brothers Borwein and the brothers Chudnovsky, a stellar cast of characters have their (sometimes unexpected) connections to pi explained, generally with detailed proofs: Gauss, Bernoulli, Euler, Buffon, Viete, Wallis, Stirling, Machin, Euler, Lambert, Euler, Ramanujan,... Geometric and number-theoretic aspects of pi are treated thoroughly. For more extensive historical remarks and anecdotes, a reader would be better advised to consult Pi: A Source Book (Second edition), A History of Pi, or The Joy of π.

The overwhelming flavor of the book is analytic, with an abundance of integrals, infinite series, and infinite products. For example, there is a nice overview of the long and complex proof of Ramanujan's 1914 result

 \[\frac{1}{\pi} = \frac{\sqrt{8}}{9801}\sum_{n=0}^\infty \frac{(4n)!}{n^4}\frac{[1103+26390n]}{396^{4n}}\]

by the brothers Borwein. The last formula in the book is another amazing series due to Ramanujan,

\[\frac{1}{\pi} = \sum_{n=0}^\infty \binom{2n}{n}^3\frac{42n+5}{2^{12n+4}}\] 

which enables one to compute the digits of pi in base 2 from the nth to the 2nth without having to calculate the first n digits beforehand.

The book is a translation from the French, and the translator is apparently British ("our" replacing "or" as a word ending, for example). Although the translation is generally smooth, there are a few awkward spots. On p. 34, for example, the word "ensemble" (used twice in the same paragraph) should have been translated as "set". On p. 45, the term "points of suspension" seems an awkward term for "ellipsis" or just plain "dots". Section 3.5.4 discusses the "Euler indicator function," an unfamiliar (mistranslated?) term for Euler's phi-function or totient function. There are other places where the reader is brought up short, not knowing if the stylistic infelicities are due to the authors or to the translator. There is an obvious typo in Exercise 2.2, where an infinite product has its general term missing. Instead of a reference to a French edition of The Enjoyment of Mathematics by Rademacher and Toeplitz, the authors should have pointed to one of several English editions.

Nitpicking aside, The Number π is a marvelous, rich book, suitable for a large (mathematically trained) audience. For instance, with a good group of senior mathematics majors and some work by the instructor, the book could be used for a capstone course, a course which would provide a sense of history and continuity to the undergraduate experience. This is a valuable addition to the ever-expanding shelf of books on mathematical constants, even though it may leave shoe store owners cold.

Henry Ricardo ( is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book, A Modern Introduction to Differential Equations, was published by Houghton Mifflin in January, 2002; and he is currently writing a linear algebra text.

Date Received: 
Sunday, June 6, 2004
Include In BLL Rating: 
Pierre Eymard and Jean-Pierre Lafon
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Henry Ricardo
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Friday, January 13, 2012