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Nice Numbers

John Barnes
Publication Date: 
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[Reviewed by
Michael Caulfield
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In 2010 John Barnes published Gems of Geometry. The book was based on a series of lectures he had given on a variety of geometry topics that he finds fascinating. After fielding requests to do the same for numbers, Barnes agreed. He gave lectures on numbers and their properties at both Reading and Oxford. In 2016 those lectures were published under the title Nice Numbers.

The lectures comprise the ten chapters of the book. Some of the topics are very basic, and historically interesting, such as Measures, Time, and Notations. Other chapters discuss properties of primes and the RSA algorithm. In all cases Barnes presumes little to no background knowledge. He provides all of the necessary historical or mathematical information that allows the reader to understand his explanations. Thus, the book is suitable for any reader who can do high school level mathematics and has the interest and curiosity to follow Barnes on his wide-ranging explorations. Each chapter ends with suggestions for further reading; most chapters have some exercises for the reader to attempt, although no answers are provided. The more challenging exercises are marked with an asterisk.

The reader will encounter terms that are particular to British English. These include British monetary units or terms of measurement, such as “firkin,” “kilderkin” or “hogshead.” We also encounter bell-ringing terms, such as the “Plain Bob Minim.” And terms used to describe the gaits of animals, as in “pronk”, a term used to describe a jump in which the animal leaps from all 4 feet at once. But fear not: the terms are defined so that the unfamiliar reader can still follow the discussion. Some of these perhaps unfamiliar terms apply specifically to mathematics. He uses the term “vulgar fractions” to describe what also might be called a common fraction. And a “surd” can be defined as a non-integral root such as the square root of 2.

Barnes sprinkles topics of interest throughout the book. Several of these are not specifically number-theoretical. For example, he discusses in Chapter 5 how the length of a day varies throughout the year. In Chapter 2 he helps the reader interpret a positive result on a medical test, explaining how often it is that the true result is actually negative — and why we should not be surprised by that. Other topics have a historical flavor. In Chapter 3, after a segment on the unit fractions favored by the Egyptians, we see why that way of partitioning a fraction made good sense in that society. In Chapter 9 he goes in to great detail in discussing the problem of resolving the ratios within a musical octave. Then we find that in the nineteenth century, a 53-note octave scale was devised to solve the ratio problem. A harmonium using this scale was actually constructed!

Other facts are of interest from a strictly mathematical point of view. He describes the divisibility tests we learned as youngsters in terms of congruences, showing how in this light they are all similar. He compares the prime factorizations of amicable number pairs and their sums, noting that the sums tend to have much smaller largest primes in their factorizations than the numbers in the pairs do. And he shows that complex primes look much different than integer primes.

Nine appendices add to the lecture material, including such topics as Pascal’s Triangle, the Chinese Remainder Theorem, and Rubik’s Cube.

The writing style is intended to mimic the lecturing style upon which the book is based. Thus he often throws in a witty comment, as a speaker might make to an audience. This sometimes comes across as a bit haphazard, as it would be if a speaker suddenly had a thought and made a more-or-less unrelated remark.

There are times when we might quibble with Barnes’s approach. For example, when discussing the proper factors of an integer, he first refers simply to the factors, then the proper factors, then to the factors again, but in all cases he means the proper factors. And when discussing the ratios for the musical scale, he supposes that middle C is 240 Hz for the convenience of the calculations to come, even though this is incorrect (as he acknowledges in the book). On the other hand, his inclusion of photos of bottles with old measurement markings, or of old tape measures, and graphs such as the graphs used to illustrate bell-ringing patterns, add a great deal of value and help the reader to visualize what is happening.

If you are already well-versed in the problem of resolving the octave, and of the difficulties in matching our day to the actual revolution of the earth, and of the basis of the RSA algorithm, then you can skip Nice Numbers. But if not, you are likely to find yourself learning more about an unfamiliar topic or two. And that’s a valuable way to spend some leisure time.

Michael Caulfield is a Professor of Mathematics at Gannon University in Erie, Pennsylvania. His writing is focused on mathematical topics of historical interest. His career in mathematics began when he was a member of an “Outstanding” team in the COMAP Mathematical Contest in Modeling.

See the table of contents in the publisher's webpage.