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Exploring Continued Fractions: From the Integer to Solar Eclipses

Andrew J. Simoson
Publication Date: 
Number of Pages: 
Dolciani Mathematical Expositions
[Reviewed by
Allen Stenger
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This is a collection of loosely-linked articles on a variety of real-life subjects, in most of which we seek to understand a ratio or a periodic behavior. Examples include different tuning systems for musical instruments, the 17- year life cycles of cicadas, the Antikythera mechanism, and a number repeating astronomical phenomena. The recurring idea that links the chapters is the problem of devising simple rational approximations to observational data. That is where continued fractions come in: they provide the best simple approximations to given real numbers (“best” in a sense that is made precise by Hurwitz’s Theorem, although that is not covered in this book). 
Despite the title, the book is primarily about approximations and not about continued fractions; continued fractions are the usual tool. Chapter IX covers most of the simple mathematical facts about continued fractions and convergents.
The book is very eclectic and has the feel of a popular math book, although the prerequisites are too high for that: some calculus, vectors, matrices, and an enormous number of calculations. There’s quite a lot of number theory included, and the author proposes the book as an undergraduate number theory text. 
The book is organized into pairs: one “strand” followed by one chapter. The strand introduces a real-life problem; the chapter generally does not solve the problem but takes off in another direction inspired by the mathematics of the strand. For example, Strand V is about just intonation and tempering in musical tuning, which brings up harmonic numbers, and then Chapter V deals with the harmonic series and several of it applications, such as Nathan Fine’s “jeep in the desert” problem.
The mathematical topics touched on in the book, besides continued fractions, include Diophantine analysis, Farey series, the Stern–Brocot tree, the harmonic series and Euler’s constant, Gaussian integers, Pell’s equation, derangements, Catalan numbers, Cantor’s diagonal construction, periodic functions, and much more. The book includes several alternative definitions of continued fractions (that is, how to pick the quotients) along with justifications.
There are a large number of problems. Like the text, the problems are usually written in real-world terms and most of the work is figuring out what mathematical problem to solve. There are hints for some of the problems in the back of the book.
Bottom line: a very wide-ranging and idiosyncratic work, that seems more like a text on applications of elementary number theory than on number theory itself.


Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.

See the publisher's web page.