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Number: From Ahmes to Cantor

Midhat Gazalé
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Marvin Schaefer
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It is the best of books, it is the worst of books, it soars to abstract perfection, it bogs down into tedious detail, it shows creative originality, it reads like a student book report, its focus is sharp, its focus is missing, it inspires the curious reader, it depresses the cautious proof-reader, ... — in short, the book is so like an educational pot pourri that some of its highest authority insists on its being received, for good or for evil, in the superlative degree of comparison only. Its scope is all the canvas of the history of numbers in mathematics. But its treatment sometimes leaves one wondering what the Dickens it is about.

I expect that most of us owe our involvement in mathematics to reading popularizations in our youth. Certainly, many of us were turned on by the writings of Martin Gardner, George Gamow, Isaac Asimov, Lancelot Hogben, W. W. Rouse Ball , H. E. Dudeney and much of what is found in James R. Newman's World of Mathematics. These authors offered recreational wonders and challenges to the investigative mind: the more one toyed, the more one dug, the more wonders one found!

So it appears to have been for Midhat Gazalé, who largely credits his latest offering, Number: from Ahmes to Cantor, to inspiration he received from Martin Gardner's many Scientific American columns and books, and to Tobias Dantzig's Number: the Language of Science. Receiving additional inspiration from Donald Knuth, John Conway and Richard Guy, Eli Maor, and George Ifrah, M. Gazalé found himself pursuing "answers [to fundamental questions about number that] were not being provided within the confines of an engineering curriculum. How did the decimal notation come about? Are the decimal and binary systems the only legitimate ones? Are there nonuniform base number systems? Exactly what are the irrationals? Why is their representation not periodic? What is a transcendental number? What is a real number? What is the famous continuum? What is meant by infinity?"

From his reading, Gazalé, an engineer and former President of AT&T France, was inspired to ask such questions. Tobias Dantzig's fourth edition (a 1953 rewrite of the 1930 original) goes into wonderful and inspiring detail on most of these topics. So what distinguishes Gazalé's new book?

Gazalé tries to begin at the beginning, presenting an abbreviated prehistory of mathematics from Cro-Magnon through the Egyptians, Mayas, and Greeks. He starts with one-to-one correspondences and their abstraction to counting symbols, treating Egyptian unitary fractions and Babylonian positional notation along the way. As he investigates the Babylonian base-60 notation, he tells the story of zero and introduces the strangely mixed Mayan base-20/base-18 notation. From this, he generalizes to notations in which every position to the left of the "decimal" point represents a different base. He then extends to introduce the representation of fractions and irrational numbers in different bases. Along the way, he picks up the Euclidean algorithm, the GCD, Farey and Fibonacci sequences, and continued fractions. To this point, the treatment is light and highly motivated.

His third chapter is on divisibility and number systems, and represents a considerable increase in the level of difficulty over all other chapters in the book. Gazalé designates it as an optional chapter, and into it he crams a tremendous amount of elementary number theory: Pascal's divisibility test, the rationale for casting out nines, the Little Fermat and Euler Theorems (along with a generalization and a short appendix on Carmichael's Variation), pseudoprimes, primitive roots, cyclic numbers (an n-digit base m number that, when multiplied by any integer from 1 to n, results in a number whose digits are a cyclic rotation of the original digits), and more.

The remaining chapters build up to the rational numbers and the reals, which are initially examined through the Dedekind Cut and ultimately by Gazalé's pièce de résistance which he calls the cleavage. Gazalé defines the cleavage of a real number t as the set of all points (a, b) in the Cartesian plane such that a and b are integers, a is not zero, and at is in the interval [b,b+1). For each a = 1, 2, 3, .... one can find such a b, so one can give the cleavage by listing the values of b in order. It turns out that one can uniquely define a real number by listing the lattice points of its cleavage. So, omitting the abscissas, as examples we have the first 40 terms of:

Cleavage(1/7) = {0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, ...}

Cleavage(Square root of 2) = (1, 2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 24, 25, 26, 28, 29, 31, 32, 33, 35, 36, 38, 39, 41, 42, 43, 46, 48, 49, 50, 52, 53, 55, 56, ...}

Cleavage(Golden Ratio) = {1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, 30, 32, 33, 35, 37, 38, 40, 42, 43, 45, 46, 48, 50, 51, 53, 55, 56, 58, 59, 61, 63, 64, ...}

Cleavage(pi) = {3, 6, 9, 12, 15, 18, 21, 25, 28, 31, 34, 37, 40, 43, 47, 50, 53, 56, 59, 62, 65, 69, 72, 75, 78, 81, 84, 87, 91, 94, 97, 100, 103, 106, 109, 113, 116, 119, 122, 125...}

Though he does not use the term, Gazalé's cleavage is essentially the spectrum of a real number. Gazalé shows that for if t is rational, a line drawn from the origin with slope t passes through an infinite set of lattice points (ka, kb) with t = b/a. He also argues that for all irrational t, the line having slope t never intersects any lattice point but that the cleavage points (a, b) determine a converging sequence of rational approximations {a/b} which approaches t from below and which is bounded above by the converging sequence {(b+1)/a)}.

The book also discusses (to varying degrees of depth) complex numbers, Gaussian primes, Fermat's Last Theorem, algebraic numbers, the Cantor set, convergence of infinite series, and more. Gazalé introduces some unfamiliar mathematics like Midy's theorem (if the period of a repeating decimal for a/p has an even number of digits, the sum of the two halves is a string of 9s, where p is prime and a/p is a reduced fraction).

So what is there not to like about this book?

The book is terribly uneven in its coverage, its degree of maturity (in terms of expected background and rigor), and its accuracy. Some theorems are asserted, some are proved, some are "proved" through computational example. Gazalé appears to be very honest about citing where he got his inspiration, and to some extent his book appears to be an extended book report on those sources (I don't think he's had much mathematical training in number theory, but he has a wonderful imagination and a good deal of stick-to-itiveness). The text contains many notational quirks, typographical errors and misstatements both serious and folklorique. For example, the reader needs to get used to the unconventional choice of residues m (mod d), to the use of argument to mean integer part, etc. In a few places, the comma is used where English-speakers use the decimal point, and a decimal point is used between two or more integers where a raised dot is expected for a multiplication. On page 110, the properties of congruences are seriously misstated and could misguide a novice reader.

I found dozens of typographical errors including superscripts that should be subscripts, missing digits in examples of algorithms, points not present in geometric figures, etc. There is a circular proof of the Pythagorean Theorem (it uses trigonometric identities derived from sin2x + cos2x = 1). In an example derivation on page 134, the representation of a number 58608 to the mixed base (...7, 3, 7, 3, 7, 3.) is erroneously shown as (2 0 2 6 0 6 0) instead of (2 0 2 0 6 0 6 0), a typo that is bound to confound a young reader. The book lacks a bibliography, but many references are identified in footnotes. In reading, I was inspired by the book to check a few unsubstantiated statements in history of mathematics resources; these were usually correct. As I bogged down in chapter 3, I decided to read the frequently-cited book by Dantzig and found it to be a delight. One cannot doubt the gravity of Albert Einstein's review of it, "This is beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands."

Gazalé introduces Fermat's Last Theorem with a quotation in French(!) of Fermat's famous "I have found a remarkable proof of this proposition, but it will not fit in the margin." This didn't sit quite right, and when Gazalé wrote on page 173 that "Fermat himself, in his correspondence with Mersenne, had claimed in a marginal note, perhaps in earnest, that he had found a simple and elegant proof" I shot off to search my library for substantiation. Of course, Fermat (1601-1665) communicated with Mersenne (1588-1648), and it was in this correspondence that Fermat communicated his ill-fated assertion that the "Fermat numbers" Fn = 22n + 1 were all primes. However, I could not find anything anywhere in the literature even to hint at a private communication of FLT to substantiate Gazalé's claim. Indeed, the historical literature confirms the story we all grew up with: that after Fermat's death, his son Clément-Samuel first revealed FLT in his 1670 publication of Fermat's annotated edition of Diophantus' Arithmeticorum libri sex, et de numeris multangulis.

The publisher's blurbs identify this to be a book "the general reader can read ... like a coffee-table book" that "will fascinate all those who have an interest in the world of numbers, ..., who enjoy mathematical recreations and puzzles, and for those who delight in numeracy." But the text is often sophisticated by US educational standards, and probably is best read by those who have been exposed to at least some properties of infinite series, set theory, and abstract algebra — i.e., those who are ready for a first course in analytic number theory. Perhaps this book, which lacks both systematic development and rigor, would be best as a source for topics to complement such a course.

Whatever it is that makes mathematics mathematics, the rigorous elements of discipline: observation, [consistency] analysis, proof and generalization, appear to be essential to its spirit. Much of this — and far more — can be found in Gazalé's book. Yet there is a certain je ne sais quoi missing from the book.

This is not to say that this book is not worth your attention, nor to say that M. Gazalé is not good at his craft. For it is an intriguing book that covers a lot of ground and it is about numbers, about the history of mathematics, about mathematical processes, about mathematics. But do yourself a favor first — read a copy of Dantzig, a book that embodies mathematics!

Marvin Schaefer ( is a computer security expert and was chief scientist at the National Computer Security Center at the NSA, and at Arca Systems. He has been a member of the MAA for 39 years and now operates an antiquarian book store called Books With a Past.

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