Single Digits: In Praise of Small Numbers

Author of Single Digits Marc Chamberland provides his take on the significance of the single digits excluding zero (zero’s significance is so great it can fill a book all by itself). And though the single digits are well know to us all, the author found tricks that may be unfamiliar even to the most mathematically adept of readers.

The introduction to Single Digits show Chamberland understands the concerns of his audience, “Some of the topics, such as the Pizza Theorem, require little mathematical background and are understandable by a curious 12-year-old; other sections require modest amount of technical math, while a few selections, such as the section on E8, allude to sufficiently advanced material that it should not be read with small children present.”

Chamberland’s writing style is conversational with occasional flights of fancy as this sentence demonstrates, “Proving that a suspected number is irrational is often a difficult feat, but showing that it is transcendental requires more muscle than a four-by-four truck.”

There are numerous topics within each digit’s chapter; each topic is short, no more than a few pages with diagrams and illustrations. A few topics will indeed require the reader to have an advanced background in mathematics, i.e. some calculus, but there is no problem so difficult that it includes a $1 million dollar prize for its solution. And many of the topics are so well written, this reviewer would not be surprised if they find their way into the next appearance of The Best Writing in Mathematics.

Now for the digits themselves.

What can one say about the number one? The first digit is the starting point for topics in uniqueness as in, “The One.” From uniqueness the author moves on to origami, Fibonacci numbers, the Golden Ratio (as a representative of a value that consists only of symbols and the number 1), and prime numbers. There’s more to one than one may presume, there are knots, prime knots, and the cardinality of infinite sets; there are Gibreath’s conjecture, Benford’s Law, inverse problems, and inverse methods (inverse methods are methods that make medical imaging possible).

For the second digit Chamberland addresses algorithms and proofs involving pairs. There are Jordan Curves and parity arguments, bisecting rectangles, and symmetry. There is the twin prime conjecture, and the Ham Sandwich Theorem. There are formulae to calculate Pi that revolve around the number two. There are squaring numbers, Morse code, fractals, and many, many topics that are not covered in other popular mathematics texts that this reviewer has reviewed to date.

For the third digit Chamberland uses the three-body problem as a lead in to problems in astronomy, gravitation, and physics, chaos, surfaces, and proofs. As a bonus he mentions a favorite unmanned spacecraft, SOHO. The chapter on the third digit also includes a study of fractions including Egyptian fractions, which as a topic doesn’t quite adhere to the author’s self-imposed rule—there is no explicit connection made between fractions and the number three, but never mind.

The general theme identified, there’s no need to count past three. There are common mathematical themes across all the digits, including pure mathematics, proofs, and computer proofs versus logical argument. There are few but not many biographies of number theorists, Fermat among them. And called out for special attention are Noam Elkies and Don Zagier who both, in the 1980s, solved one of Euler’s conjectures made over 200 years ago. The odd thing is, not only did these modern mathematicians solve the same conjecture independently, they did so days apart.

Though Single Digits is a new take on a well-traveled popular mathematics subject, the author sometimes has to reach, though overall he does succeed in making the topic interesting, part of that is his commentary on real-world utility of mathematics. Single Digits would make a valuable companion text to mathematics courses provided the author added more problems or exercises for the reader to solve. From a different perspective, if authors of mathematics textbooks could distill Marc Chamberland’s ability to connect mathematics to the real world and add that to their own texts, they would write more interesting and consequently more valuable textbooks.