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The Story of Numbers

Asok Kumar Mallik
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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Unlike, say, Leo Corry’s similiarly titled A Brief History of Numbers, this book is not intended as a historical account of how various kinds of numbers came to be discovered and accepted. Although it does have some historical content, this book is primarily intended as a collection of vignettes and anecdotes about various kinds of numbers. Its style is informal, and prerequisites are minimal; a good understanding of high school mathematics will take a reader through most of the book, though on some occasions limits, infinite series, derivatives and integrals are mentioned. Indeed, it says something about the general tone of this book that the library at Iowa State University shelves it in its “leisure” collection, rather than with the mathematics books.

In keeping with the elementary nature of this book, mathematical subtleties are generally ignored, and proofs are often omitted; if you want to see, for example, a development of the real numbers by Dedekind cuts, you’ll need to look elsewhere. But there are a lot of stories told here, as the author starts with the integers and works his way up through real and complex numbers, and then to some more esoteric kinds of numbers, including hyperreal numbers, quaternions, and p-adic numbers.

After an introductory chapter, there is a fairly long chapter on the integers. The material in this chapter ranges from amusing little arithmetic facts such as

\(\begin{align*}1 + 2 &= 3,\\4 + 5 + 6 &= 7 + 8,\\9 + 10 + 11 + 12 &= 13 + 14 + 15,\\16 + 17 + 18 + 19 + 20 &= 21 + 22 + 23 + 24,\end{align*}\)

and so on, to discussions of various kinds of prime numbers (Fermat, Mersenne, Double Mersenne, Sophie Germaine, Wilson, Twin and many others) to composite numbers (e.g., highly composite numbers, perfect numbers) to number sequences (Fibonacci, Padovan, etc.), and more besides. There are discussions of various kinds of very large numbers, as well as discussions of individual numbers (such as 4 or 7) with amusing properties (in the case of 4, the author notes that every positive integer can be written as the sum of 4 squares; in the case of 7, the author discusses the unsettled question of whether there is any \(n>7\) with the property that \(n! + 1\) is a perfect square).

The next chapter is on the real numbers. The author begins by discussing irrational and transcendental numbers and stating a few facts about them. He then looks at continued fractions and also sequences of numbers obtained by iteration, thereby allowing him to talk briefly about (among other things) chaos theory and various sequences converging to \(\sqrt2\). The chapter ends with two sections, the first talking about various specific rational numbers (including congruent numbers) and the second talking about various specific irrational ones (\(e\) and \(\pi\) are of course discussed, but so are a number of other ones as well).

Following this, there is a relatively short chapter on complex numbers. Beginning with a brief account of their history, the author touches on such topics as the geometric representation of complex numbers, Euler’s formula, the fundamental theorem of algebra, the curiosities of complex exponentiation, and Gaussian integers. There is also a fairly lengthy discussion of the Riemann Hypothesis (pitched at a somewhat higher level than much of the rest of the book). I am still wondering whether the author’s sentence “Riemann’s zeta function occupies a prime place in mathematics” is a deliberate play on words or a happy accident.

Finally, in a chapter that, like the discussion of the Riemann Hypothesis, may in large part not be as readily accessible to the target audience of high school students and lay people as is the rest of the book, the author discusses such “unusual” numbers as hyperreal numbers, dual numbers, quaternions and p-adic numbers.

The level of mathematical detail varies with the topic being discussed. As noted earlier, in many — actually, most — cases, results are just stated as fact, without proof. However, in those cases where a simple argument to show something can be given, the author gives it: he proves, for example, that there are infinitely many primes, that \(\sqrt2\) is irrational, and that there exists two irrational numbers \(\alpha\) and \(\beta\) with the property that \(\alpha^\beta\) is rational. (This last result, in case you haven’t seen it before, is particularly easy and elegant: consider \(\sqrt2^{\sqrt2}\). If this number is rational, take \(\alpha=\beta=\sqrt2\). If it is not rational, take \(\alpha\) to be this number, and \(\beta\) to be \(\sqrt2\). Of course, using more advanced results, one can show that \(\sqrt2^{\sqrt2}\) is in fact irrational, but the beauty of the proof above is that it does not depend on knowledge of this fact.)

A reader who is tantalized by some of the stories mentioned in this book — for example, the brief reference to legislation in the state of Indiana that purported to establish by law a value of \(\pi\) — can search online for more detail, or consult the reasonably good bibliography provided by the author. The books listed in the bibliography are generally of the “popular”, rather than technical, kind, so should be accessible to any reader of this book. (The author, by the way, says that the Indiana legislation establishes two different values of \(\pi\), but opinions vary; the legislation is so poorly written that some people have discerned many more values inherent in it. See, for example, Singmaster’s article “The Legal Values of Pi,” in the June 1985 Mathematical Intelligencer, in which he concludes that there are six different values inherent in the legislation itself, and three other values in earlier writing by the author of that bill.)

Though the author has done a good job in assembling a large collection of facts and interesting stories, nobody can think of everything, and there are, I think, some examples of missed opportunities in the text. More specifically:

  • Constructible numbers don’t appear. The set of these numbers is a field that is strictly between the rational numbers and the algebraic numbers, and plays a central role in the solution of some of the famous construction problems of antiquity. These numbers can be described, and their role discussed, without recourse to a great deal of mathematical background.
  • Cardinal and ordinal numbers are also not discussed. These form two interesting and unusual sets of numbers and there are certainly all sorts of interesting stories connected to them.
  • Another benefit of discussing cardinal numbers is that it can then be stated that the set of algebraic numbers is countable. This, together with the fact that the set of real numbers is not, allows the reader to see a beautiful example of how mathematicians can prove something exists without giving a single specific example of that thing.
  • Although Gaussian integers are mentioned, the author stops short of discussing other quadratic extensions of the integers. Doing so would allow a relatively easy discussion of how uniqueness of factorization into primes is not always the case in number systems.
  • The role of complex numbers in elementary Euclidean geometry could have been discussed in somewhat more detail. (Numerous examples are given in Complex Numbers from A to … Z, by Andreescu and Andrica.)
  • Although octonions are mentioned in the book, they are not discussed in any detail. Particularly in view of the fact that some physicists think that they may be the key to a unified field theory (see, e.g., The Geometry of the Octonions by Dray and Manoque), it might have been useful to spend a few pages on them.

This is not a textbook: there are no exercises, and there are certainly very few courses being offered which match up with the topic coverage. The book is simply intended to — and does — convey to high school students or other interested laypeople some new and interesting things about numbers. It’s also not a bad book for faculty members to have on their shelves; though most professionals won’t learn a great deal of new things from this book (although I picked up some interesting facts that I was not previously aware of), they may find the book a useful collection of interesting information with which to spice up a lecture.

Mark Hunacek ( teaches mathematics at Iowa State University.

  • Introduction
  • Integers
  • Real Numbers
  • Imaginary and Complex Numbers
  • Special Numbers
  • Appendix A:
    • Solution of Equation (1.2)
    • Brahmagupta's Equation and Its Solution
    • Solution to Equation (2.26)
  • Appendix B:
    • Sum of Integral Powers of Natural Numbers
  • Appendix C:
    • Origin of Curious Patterns (Section 3.7.6)