The irrationals. A story of the numbers you can't count on (Q2898748)

Havil's new book tells the story of the theory of irrational numbers from Pythagoras and Euclid to Cantor and Dedekind. Quite recently, \textit{F. Toenniessen} [Das Geheimnis der transzendenten Zahlen. Spektrum Sachbuch. Heidelberg: Spektrum Akademischer Verlag (2010; Zbl 1183.00004)] has published an excellent book covering roughly the same grounds, yet none of the two replaces the other.NEWLINENEWLINEHavil's book begins with the contributions of Pythagoras, Thales, Theodorus, Theaetetus, Eudoxus and Euclid (in particular his Book X of the Elements). The next chapter already brings us a lot closer to the present by displaying Abu Kamil's solution of the system of equations \(x+y+z=10\), \(xz=y^2\), \(x^2+y^2 = z^2\), Fibonacci's proof that the real root of the equation \(x^3 + 2x^2 + 10x = 20\) is neither rational nor an irrational number of the type studied in Euclid's Book X, or van Roomen's challenge of solving a polynomial equation of degree 45. The third Chapter presents a detailed discussion of Euler's expansion of \(e\) as a continued fraction and Lambert's related work on the irrationality of \(\pi\). Chapter 4 gives the proofs of the irrationality of \(\pi\) based on the work of Hermite and Niven, and Chapter 5 discusses Apéry's proof of the irrationality of \(\zeta(3)\). In Chapter 7, transcendentals show up, accompanied by contributions of Liouville, Hermite and Lindemann. The remaining chapters deal with e.g. the Markov spectrum, normality, Cauchy sequences and Dedekind cuts. The four appendices contain material on the spiral of Theodorus, the rational parametrization of the unit circle, basic properties of continued fractions, and a story on the tomb of Roger Apéry.NEWLINENEWLINEThe minor complaints begin with typesetting the book's title as ``\( \pi\)He IRRATI\(\Phi\)NALS'', which reminds me of innumerous failed attempts at displaying some knowledge of Russian by writing the letter R backwards.NEWLINENEWLINEThe chapter on Pythagorean mathematics mentions the famous ``crisis'' [p. 14] after the discovery of incommensurability without the warning that supporting evidence for it is basically nonexistant. On p. 29, Havil starts telling the story of Theodorus and Theaetetus, but begins with ``the suicide of Socrates'', which is one of the strangest euphemisms for capital punishment that I have seen. Among the references for the problematic stopping at or before \(\sqrt{17}\) I have missed Szabo's book [\textit{A. Szabo}, The beginnings of Greek mathematics. Budapest: Akademiai Kiado. Dordrecht-Holland: D. Reidel Publishing Company (1978; Zbl 0383.01001)], where it is claimed that this question misses the point.NEWLINENEWLINEOn p. 97, a short passage from Euler's beautiful article on the continued fraction expansion of \(\tanh x\) in connection with Riccati's differential equation is quoted, without a word of warning that the terms ``irrational'' and ``transcendental'' were not used yet in their modern meaning. On p. 182 Havil even claims that \textit{L. Euler}'s ``irrational'' in a quote from his [Introductio in Analysin infinitorum. New York etc.: Springer-Verlag (1990; Zbl 0688.01015)] means ``algebraic'', which is wrong; in the part not quoted Euler explains that he means numbers commensurable in squares, i.e., square roots of nonsquare positive integers.NEWLINENEWLINEOn p. 204, Havil ``quotes'' Kronecker as having said that irrational numbers do not exist; those who still think that there is some truth in this can do worse than look at page 4 of his lectures on number theory, where \(\frac{\pi}4\) is defined by the Leibniz series, or just browse through Kronecker's collected works. The statement often but falsely attributed to Poincaré according to which set theory is a ``malady that would one day be cured'' occurs on the same page. See the review [\textit{J. W. Dauben}, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 21, 223--241 (2005; Zbl 1090.01008)] for references on both quotations.NEWLINENEWLINEPutting these quibbles aside, the book contains several gems that have little to do with its topic; among those I found particularly delightful was Descartes' method for finding the tangents to a parabola by inscribing a circle that touches it [p. 82]. My overall impression is that ``The Irrationals'' is a fantastic book that deserves a wide audience and should be compulsory reading for every undergraduate or beginning graduate student.