# The Irrationals: A Story of the Numbers You Can't Count On

Julian Havil
Pages: 280
https://www.jstor.org/stable/j.ctt7smdw

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. Acknowledgments
(pp. ix-xii)
4. Introduction
(pp. 1-8)

Irrational numbers have been acknowledged for about 2,500 years, yet properly understood for only the past 150 of them. This book is a guided tour of some of the important ideas, people and places associated with this long-term struggle.

The chronology must start around 450 b.c.e. and the geography in Greece, for it was then and there that the foundation stones of pure mathematics were laid, with one of them destined for highly premature collapse. And the first character to be identified must be Pythagoras of Samos, the mystic about whom very little is known with certainty, but in whom...

5. CHAPTER ONE Greek Beginnings
(pp. 9-51)

The birth of irrational numbers took place in the cradle of European mathematics: the Greece of several centuries b.c.e. For this conviction and for much more we must place reliance on a few fragments of contemporary papyrus, some complete but much later manuscripts, and the scholarship of many specialists who, even between themselves, sometimes disagree in fundamental ways. Of great importance is the following passage:

Thales,¹ who had travelled to Egypt, was the first to introduce this science [geometry] into Greece. He made many discoveries himself and taught the principles for many others to his successors, attacking some problems in...

6. CHAPTER TWO The Route to Germany
(pp. 52-91)

In Chapter 1 we considered the efforts of the ancients as they struggled to come to terms with incommensurability. Here we will move through time and place at some speed as we continue the story from where it was left, to move with it through ancient India and Arabia, medieval Europe and finally nineteenth-century Germany.

The ancient Greeks had not embraced irrationality, they had avoided it as much as possible and used geometry to best cope with it, and the succession of their civilization by that of the Romans did nothing to advance matters. What did the Romans do for...

7. CHAPTER THREE Two New Irrationals
(pp. 92-108)

We commented in the Introduction that “first proofs are often mirror-shy” and this chapter is devoted to two of them; neither can be placed “fairest of them all” and it is to great eighteenth-century (mathematical) kings that we look, not a wicked queen, as we gaze closely at them. In the end, it was not to be that π’s mysterious nature was first to be understood, since this occurred fully thirty years after the irrationality of e was established – a number not born in antiquity but in the eighteenth century itself. We have seen Wallis struggle with the nature...

8. CHAPTER FOUR Irrationals, Old and New
(pp. 109-136)

If the purpose of the previous chapter was simple, that of the present one is assuredly not so. We will go in search of irrational numbers: all manner of them. In doing so we will begin with more recent approaches which establish the irrationality of e and π, move on to more general methods which establish much more, approach new types of irrationals and finish with transcendentals.

It is strange that a mathematician of Euler’s stature had not detected that the irrationality of e was an inevitable consequence of its canonical representation as the infinite series, well known to him:...

9. CHAPTER FIVE A Very Special Irrational
(pp. 137-153)

Mathematical constants are either anonymous or famous, with fame a reflection of the constant’s importance. And we can distinguish between those numbers for which fame is an intrinsic quality and those which have had it thrust upon them: π and e compared with$\sqrt 2$, for example. None could doubt the star quality of the first two numbers but we have seen that it is through Pythagorean tradition that$\sqrt 2$holds its distinguished place in the mathematical firmament: this chapter is concerned with another number whose celebrity is incidental. While we agree with the sentiment that adding two numbers that...

10. CHAPTER SIX From the Rational to the Transcendental
(pp. 154-181)

An essential part of Apéry’s proof in the previous chapter is the use of the expression$|\zeta (3) - a_n /b_n |\~\1/b_n^2$, the left-hand side of which contains one of a sequence of rational approximations to ζ(3) and the right is a measure of the accuracy of each of those approximations. We needed only the estimate above to secure the contradiction required by his proof but in fact he had contrived a sequence of rational approximationsan/bnthe accuracy of which is better than this and, indeed, startlingly good, as table 6.1 indicates.

How well can an irrational be approximated by a rational? If...

11. CHAPTER SEVEN Transcendentals
(pp. 182-210)

In the last chapter we located the country of the transcendentals without identifying any of its inhabitants: here we will discover all manner of transcendental numbers from the lowly no-names to the important such as π and e.

The concept of transcendence appears to date back to Leibniz since, in 1682, he discussed sinxnot being an algebraic function ofx(that is, it cannot be written as a finite composition of powers, multiples, sums and roots ofx) and in 1704 he stated that

the number$\alpha=2^{1/a}, \ a= \sqrt 2$is intercendental

without, however, defining the term.

And recall the comment...

12. CHAPTER EIGHT Continued Fractions Revisited
(pp. 211-224)

In chapter 3 we considered the application of continued fractions in the eighteenth century, as they were used to grind out the first proofs of the irrationality of e and π. Their role in the study of irrational numbers is far greater though, with a number of the results of previous chapters implicitly dependent on them; here we make some of that dependence explicit.

We shall continue to restrict our interest to simple continued fractions and so to finite or infinite expressions of the form

$$\alpha = a_0 + \frac{1}{{a_1 + \frac{1}{{a_2 + \frac{1}{{a_3 + ...}}}}}}$$

and we will adopt the notation that thenth convergent of the continued fraction...

13. CHAPTER NINE The Question and Problem of Randomness
(pp. 225-234)

We will move from the question, howirrationalis an irrational number, to another: howrandomis the decimal expansion of an irrational number? Our measure of irrationality allows comparisons to be made, but what measure is there of randomness? In the above quotation, the genius von Neumann provides a hint of the complexities involved when we attempt to harness randomness and we shall expose some of the difficulties in what follows. First, we shall once again eliminate the rationals.

This is achieved by the combination of three results:

If the decimal expansion of a number is finite it is...

14. CHAPTER TEN One Question, Three Answers
(pp. 235-251)

We met Mary Cartwright in chapter 4, challenging prospective undergraduates to prove that π² is irrational. Ted Titchmarsh was her contemporary at Oxford and was later to hold the Savilian Chair of Geometry some 280 years after John Wallis, and then for a ‘mere’ 32 years. Our opening quotation from him may attract questions about the physical nature of$\sqrt { - 1}$but assuredly the misconception that$\sqrt 2$is an obvious concept while$\sqrt { - 1}$a profound one had long been widespread. Yet, as we have seen in chapter 2, without irrational numbers analytic geometry and the idea of limit, with its consequent...

15. CHAPTER ELEVEN Does Irrationality Matter?
(pp. 252-271)

The opening quotation is taken from Newcomb’s 1882 bookLogarithmic and Other Mathematical Tables: With Examples of Their Use and Hints on the Art of Computation. Microscopes have become a great deal more powerful than those available to him and so have telescopes and in consequence we would need to increase that number of decimal places considerably for his statement now to hold true. Yet true it would then be and, moreover, he declared that ‘for most practical applications’ five-figure accuracy is quite sufficient. For the man whose shared logarithmic tables used at the Nautical Almanac Office alerted him to...

16. APPENDIX A The Spiral of Theodorus
(pp. 272-277)
17. APPENDIX B Rational Parameterizations of the Circle
(pp. 278-280)
18. APPENDIX C Two Properties of Continued Fractions
(pp. 281-285)
19. APPENDIX D Finding the Tomb of Roger Apéry
(pp. 286-288)
20. APPENDIX E Equivalence Relations
(pp. 289-293)
21. APPENDIX F The Mean Value Theorem
(pp. 294-294)
22. Index
(pp. 295-298)