Single Digits: In Praise of Small Numbers

by Marc Chamberland

The remarkable properties of the numbers one through nine

In Single Digits , Marc Chamberland takes readers on a fascinating exploration of small numbers, from one to nine, looking at their history, applications, and connections to various areas of mathematics, including number theory, geometry, chaos theory, numerical analysis, and mathematical physics. For instance, why do eight perfect card shuffles leave a standard deck of cards unchanged? And, are there really "six degrees of separation" between all pairs of people? Chamberland explores these questions and covers vast numerical territory, such as illustrating the ways that the number three connects to chaos theory, the number of guards needed to protect an art gallery, problematic election results and so much more. The book's short sections can be read independently and digested in bite-sized chunks―especially good for learning about the Ham Sandwich Theorem and the Pizza Theorem. Appealing to high school and college students, professional mathematicians, and those mesmerized by patterns, this book shows that single digits offer a plethora of possibilities that readers can count on.

Genres: Mathematics, Science

240 pages, Hardcover

First published June 1, 2015

Reviews

[John B. · Rating 4 out of 5]

Over the years I have picked up a handful of popular math books that attempt to elevate the reader's love for and knowledge of math. The books Innumeracy: Mathematical Illiteracy and Its Consequences and The Mathematical Tourist: New & Updated Snapshots of Modern Mathematics immediately come to mind. My frustration with these books, and the popular math genre in general, is that authors go to great lengths to shield the reader from exposure to math beyond the level of high school algebra and geometry--if even that. Yes, math can be explained just using words, but a hand-holding explanation that is afraid of showing math notation does nothing more than give the reader a glimpse of the real beauty that is to be found. Marc Chamberland has written a book that shows the math and walks the reader through some very interesting and challenging concepts. The author is not afraid of showing integral signs, summation formulas, and diving into alternate dimensions.

I simply loved the presentation of the Gamma function in Chapter 1. Not only does the author attempt to explain how it works and what it means, he wants to show the reader the math behind it. The reader is assumed to be interested in math and competent in basic operations. The text is clear, and if one is willing to reread--and scribble on the back of an envelope, the concepts and topics are a pleasure to discover.

Not all the topics interest me in the same way. Mathematics is a whole world of interesting topics, and while the book attempts to give equal time and clarity to each topic the author has selected, I find that some days some topics and explanations are less exciting than others. Formulas for pi have always interested me more than prime numbers, so the twin prime conjecture didn't get as much of my attention as Viete's formula. Prime numbers have been in the news lately (most recently the distribution of prime numbers occurring after primes ending in 1 have shown hints that they are not uniformly distributed) so the sections that deal with prime number conjectures do catch my attention.

My son and I bring this book along for discussion when we have dinner out at restaurants. While we are waiting for our meal to arrive we read and discuss a topic, sometimes scribbling on a paper napkin. We find the gems in this book do pique our curiosity and, taken in small doses, stimulates some great discussion.

[Mark Buchignani · Rating 4 out of 5]

Single Digits is not a cultural survey of the small numbers it sings the praises of. One is the Loneliest Number, two’s company, three’s a crowd, Seven Brides for Seven Brothers, Eight is Enough – these you will not find here.

Instead, as the preface explains: “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 amounts of technical math, while a few sections, such as the section on E8, allude to such sufficiently advanced material that it should not be read with small children present.”

This is a volume of math for math's sake. If you can appreciate seemingly magical or coincidental results from what might as well be thin air, then this is your book. And the mathematics itself is not too complex – but the equations are not for novices (as the preface notes). You will have to work some to follow along, or just skip the proofs and enjoy the wizardry – or stand in the center of a four-way intersection, roll a tetrahedral die, and walk north for 1, east for 2, south for 3, or west for 4. Then repeat. And repeat. And repeat. Sooner or later you will return to your original spot. But not necessarily so for three dimensions instead of two. Fun stuff!

[Jody Ellis · Rating 3 out of 5]

This book has been excellently written for any diehard mathematician. For the average Joe who isn't a math guru though it does drag a bit and tends to become less interesting as you go.

I've given it a good star rating because it was only my personal lack of ability to follow and retain the info that meant I decided to put it down permanently. It will stay on my shelf but I think in the end my son will be the one eating up those pages!!

NOTE - I was given this book as part of the Goodreads giveaway, in exchange for a fair review.

[Maurizio Codogno · Rating 5 out of 5]

Non sono rari i libri che parlano delle proprietà dei numeri piccoli. Così su due piedi mi vengono in mente quelli di Hodges, Odifreddi e Stewart. In questo caso, però, il livello matematico è decisamente più alto. Non è trascendentale, intendiamoci: una persona curiosa riesce a seguire i risultati che vengono presentati. Sicuramente, però, si parla spesso di temi a livello superiore rispetto alle nozioni imparate alle superiori e poi più o meno faticosamente conservate. Questo a mio parere è un importante vantaggio: si può avere finalmente un'idea di quante cose ci siano in matematica oltre a quelle che si è costretti a imparare a scuola. Le spiegazioni sono molto chiare, e le curiosità, molte delle quali mi erano sconosciute, mostrano come la matematica riservi sempre sorprese. In definitiva, un testo altamente consigliato, e che sarebbe bello venisse tradotto in italiano.

[Peter Herrmann · Rating 5 out of 5]

Impressive array of info. I have an MS in Math (from 46 years ago) so already knew some of the material - but much (most?) was new to me. I'll quibble with a few of the author's entries being too concise .. or rather, they omitted some key clarifying details. E.G., in the section on Heegner numbers (Chapter Nine), the discussion re class number of a determinant went over my head. Perhaps it's simple, but a few more words would have helped. Granted, I can research this further elsewhere (if I wanted). There were, elsewhere in the book, more than a few instances of sloppy phraseology or insufficient diagram labeling that left me at a loss. E.G., 'duals' in Chapter Two: Figure 2.13. I had to re-read the descriptive paragraph multiple times before I got it - perhaps it's simple and I am dense, but a bit more labeling would have helped (at least me). Exercise problems would make this book really great (there were actually a few ... but more could be added in perhaps some future edition).

[William Crosby]

I used to think that I was good at and liked math. After going through this book, I have decided that I am not.

As a non-math student, I did not find the explanations accessible. Mostly I found incomprehensible formulas, unknown symbols, and unfamiliar terms.

Discusses various properties and math factoids of the numbers 1-9. Provides many examples showing how useful math is. Occasionally throws in jokes and puns.

[Paul Smith · Rating 2 out of 5]

I majored in math and while I've been out of college for almost 20 years, this book assumed a lot more advanced that I had knowledge of even back then when my skills were still fresh. If you're just a number geeks, this probably isn't the book for you. Still had some interesting stuff, but there were too many times I just flipped the pages because I had no idea what was going on.

[Apostolos Georgakis · Rating 4 out of 5]

One needs good prior knowledge in number theory in order not to get lost here.

[Moniek · Rating 2 out of 5]

You do need quite some math knowledge to understand, contrary to the introduction that states you could get by with high school math.
Additionally, each chapter corresponds to a number, but it more seems like a random set of math stories, some undoubtedly interesting for the a little more advanced in maths, but usually it was hard to find out why that particular story was connected to that number.
For me, this made a disappointing read.

[Mew Clawfur · Rating 3 out of 5]

I decided to try this out but ultimately, I think I'm the wrong audience. This is definitely a book for someone who enjoys math and numbers and that's just not me. I respect the work put into this though, so I'll give it three stars.

[Stephanie · Rating 2 out of 5]

I horribly misunderstood the summary when I picked up this book t the library - it appeared to be about what makes numbers cool (history of, invention, factoid collections, etc) but was really about the "cool math" related to single digit numbers. The parts where the author was speaking English (not math) were quite witty and enjoyable but as nearly all of the presented math was way beyond my educational level either from rusty use or total lack of knowledge. Really, I could only skim the book. This is not a general audiences book. However, if you really like math, this could be your new bible.

[Charles M. · Rating 3 out of 5]

A mathematician's delight of every conceivable theory, etc. involving the first nine numbers.

[hpboy13 · Rating 4 out of 5]

This book definitely had a lot of interesting nuggets that I enjoyed reading! That said, I wish the author had gone a little more in-depth explaining some of the concepts. I don't require 5-page-long proofs, but just saying "So-and-so finally solved this problem in 1859. Next section!" doesn't really cut it. I definitely had to keep my phone at hand to google as I read.

The book also seemed to lose a bit of steam towards the end; the chapters on 8 and 9 seemed like afterthoughts, and were only a few pages long. So this is a fun read for math geeks, but not quite the be-all-end-all book on cool stuff with small numbers.