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Pi: A Source Book

Lennart Berggren, Jonathan Borwein, and Peter Borwein, editors
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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This book should be on every mathematician's coffee-table!

Pi: A Source Book is a compendium of seventy articles on π (and related things, all cool). These are followed by four appendices, and "A Pamphlet on Pi." The latter is itself filled with arcana: a discussion of the normality question, the hugely fascinating business of Bailey-Borwein-Plouffe formulas which allow for the isolated calculation of individual binary digits of π (and what about other bases, such as 10?). The "Pamphlet" even sports Kaplansky's "A Song about Pi," which, according to Kaplansky's musicological data is a song of non-trivial complexity (only one in five songs, but fully half of the songs in Woody Allen's movies, possess such an exotic refrain-structure...).

The seventy articles comprising the source-book proper range from historical articles and classics by such players as Wallis, Huyghens, Newton, and Euler, to the articles on irrationality and transcendence (with e thrown in for good measure) by Lambert, Hermite, Lindemann, Weierstrass, and Hilbert. There are also the remarkable 17-line proof of π's irrationality given in 1946 (published in 1947) by Ivan Niven, Ramanujan's magical formulas, Mahler on approximation and Baker on linear forms in logarithms, and stuff by Van der Poorten on Apéry and the irrationality of the zeta function at 3. The Chudnovsky brothers weigh in with a discussion of approximations and complex multiplication à la Ramanujan.

Qua numerical analysis there is also a smorgasbord available, from Archimedes' first true algorithm for determining π, through (e.g.) Shanks' 1853 hand-calculation (!) of the first 607 decimal places of π, Brent's 1976 article on multi-precision arithmetic, and various things Borwein (et al), to Kananda's 1988 determination of the first 200 million digits of π in 1988. (Kanada's beyond a trillion now, I am told).

Indeed, Pi: A Source Book is truly an amazing book, irresistible in its own way, and filled with gems. And once it's on your coffee-table, feel free to do more than just browse: it's pretty well-suited for more in-depth study — how could it be otherwise, given the high density of classic stuff in these 800 pages? The authors provide a quintette of points of entry to the major themes represented in the book (caveat: they are arranged chronologically). For example, someone interested in questions of irrationality and transcendence should start with Niven's 1947 coup, go next to Van der Poorten on Apéry's work, and then to Hilbert on π and e. And on from there, of course.

Obviously the book is highly recommended.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles.

The table of contents is not available.