Why are numbers interesting? What is it that makes mathematicians (and math enthusiasts) so intrigued and passionate about the natural numbers? The hidden treasures inside the surprisingly rich structure of the integers has been studied by number theorists for two millennia and going strong… So, what is it that people have analyzed for so long? After all, the natural numbers are so simple to define, so intuitive to the human mind… How is it possible that their ultimate structure is so difficult to understand? If you have ever wondered about these questions, rejoice! Here is the book you should read.
A new edition of What Makes Numbers Interesting (the actual title, From Zero to Infinity , as the author points out in the introduction, was a publisher choice; I will stick with the author's choice here) has been recently published, in commemoration of the 50th anniversary of the first edition of the book (back in 1955). This is actually the fifth edition of the book and it incorporates interesting updates, and new chapters which were not present in previous editions.
Constance Reid, the author of this little book (about 180 pages), shares with the reader her refreshing view on the natural numbers and, especially, she shares her contagious enthusiasm for the theory of numbers. Each chapter of What Makes Numbers Interesting talks about a different number: from 0 to 9, and the two new chapters on e and aleph-nought. However, each chapter uses the number as an excuse to talk about a large number of very interesting topics. For example, the chapter about the number 2 concentrates on binary digits and binary arithmetic, while the chapter on the number 6 opens with a discussion on perfect numbers. Actually, I am not entirely sure whether the numbers are the excuse to talk about these interesting topics or the topics are the excuse to talk about these truly interesting numbers. Anyhow, the author manages to cover an impressive array of topics, even including some proofs (or ideas thereof). But somehow, even though some of these topics are certainly not easy to grasp (for example, she talks about the Lucas test for primality, congruences, Wilson's theorem or quadratic reciprocity), the author manages to keep the text at a level that any interested reader should be able to follow regardless of their background, albeit some work. Some advice: if you are going to read the book buy a notepad and several pencils and play with the numbers as you go along. Each chapter ends with some "challenges" or problems which should keep the reader busy for a while.
Some passages are more challenging than others and some chapters will need to be re-read a couple of times if the reader has never seen the material before. Perhaps, the book goes too far a couple of times (the section on Waring's problem may confuse some readers) but overall, the level is just challenging enough to keep the reader alert and entertained throughout the whole book. I am sure the reader will enjoy every bit of it. I sure did. Popular science authors usually say that they hope that everyone, from the layman to the expert, learns something from their books. I sure did!
Álvaro Lozano-Robledo is H. C. Wang Assistant Professor at Cornell University.