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Proofs from THE BOOK

M. Aigner and G. M. Ziegler
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Donald L. Vestal
, on

This collection of proofs is based on Paul Erdős’ idea of “The Book,” in which God has collected the most elegant and aesthetically pleasing proofs for all of the theorems in mathematics. Upon seeing a particularly beautiful proof, Erdős would remark, “That’s a proof from the 4th edition of The Book.” Okay, he didn’t say it that way, exactly, but this book is the fourth edition of Aigner and Ziegler’s attempt to find proofs that Erdős would find appealing. The first edition was published in 1998.

I never read any of the previous editions, but this one is a great collection of remarkable results with really nice proofs. The authors have done an excellent job choosing topics and proofs that Erdős would have appreciated. While the material requires some mathematical maturity, the proofs are largely accessible to readers with an undergraduate-level mathematics background. The five areas specifically covered are Number Theory, Geometry, Analysis, Combinatorics, and Graph Theory. If you have an appreciation for any of these areas, you’ll almost certainly find something here to enjoy.

A couple of highlights that stood out to me: If we have a family {f} of pairwise distinct analytic functions such that for each complex number z, the set of values {f(z)} is at most countable, does it then follow that the family is itself at most countable? Erdős proved that the answer depends on the continuum hypothesis. Another nice result that I had not seen before: Sperner’s Lemma, which is stated (and proved) here.

I love the fact that the chapters are relatively short and self-contained. I was surprised that the chapter on “Completing Latin Squares” includes no mention of Sudoku. Nonetheless, this is a very nice book.

Donald L. Vestal is Associate Professor of Mathematics at South Dakota State University. His interests include number theory, combinatorics, spending time with his family, and working on his hot sauce collection. He can be reached at Donald.Vestal(AT)