From its lively introduction straight through to a rousing finish this is a book which can be browsed for its collection of interesting facts or studied carefully by anyone with an interest in numbers and their history. Think Pythagoras discovered that the square root of 2 is irrational? Think again! The symbol itself did not appear until 1525 and what Pythagoras actually showed was that the side of a square is incommensurable with its diagonal.
How then to define the irrationals? Here are two classic attempts:
- A number which cannot be expressed as the ratio of two integers.
- A number whose decimal expansion is neither finite nor recurring.
But these are both stated in the negative — is there a positive characterization? Here’s the first one Havel offers:
The set of all real number having different distances from all rational numbers.
The set of real numbers for which the following holds:
After this fascinating introduction, Havel provides a lively retelling of the classical Greek studies of incommensurables and their fascination with regular polygons and the ratios of their sides — among which, of course, is the Golden Ratio, sometimes deemed the most irrational of irrationals. This is followed by a discussion of the treatment of surds and of the solution by radicals of polynomials. Among the gems to be found is this wonder from Bhaskara II:
For what purpose? In his own words: “This method has not been explained at length by previous writers. I do it for the instruction of the dull.” (page 55).
What motivated such studies — beyond the existence of dullards? Among other things the search for the length of the Great Year — the time (if it exists) “when the sun, moon and five planets … have returned to the same positions relative to one another.” (Cicero as quoted on page 65). Starting with Plato, lots of folk believed that the occurrence of this event would signal either the end of the world or some lesser calamity. If the periods of the seven objects are commensurable then the Great Year would be finite. Many people spent way too much time trying to prove they could not be. The results, while inconclusive, created some very interesting studies of surds. Perhaps the last such searcher after the Great Year was the astronomer Johannes Kepler, whose formula relating periods to radii along with the incommensurability of the radii under his system implied an infinite Great Year — a fact of which he was quite proud!
In spite of the advantage of the existence of irrationals as a source of celestial security, there were those who questioned whether they existed at all:
..when we seek to subject them to numeration we find that they flee away perpetually, so that not one of them can be apprehended… so an irrational number is not a true number, but lies hidden in a kind of cloud of infinity.
(Michael Stifel (1487–1567) as quoted on page 75)
What then of π? Here is Stifel yet again: “…the circumference of a circle receives no number, neither rational nor irrational.” (page 75). We can’t blame him for his frustration as the irrationality of π was proved after that of its transcendental sibling e. Chapter three begins that journey with a discussion of continued fractions. Included are several of the wonders we owe to Euler who “calculated without effort, just as men breather, as eagles sustain themselves in the air.” (Arago, as quoted on page 97). Here’s the one Euler used to establish that e is irrational.
The presence of an infinite arithmetic sequence in the partial quotients proves that (e+1)/(e–1) is irrational and hence that e is as well.
In chapter four we begin the process of demonstrating that the irrationals vastly outnumber the rational numbers. In the words of the philosopher Willard Quine
“The irrationals exist in such variety, indeed, that no notation whatever is capable of providing a separate name for each of them.” (quoted on page 109)
The chapter begins with Fourier’s proof (1815) of the irrationality of e which uses the power series representation of e. Among other interesting things we learn that if 0 < r ≠ 1 is rational, then ln(r) is irrational. Havel then discusses the use of the rational root theorem as a means of adding to our list of irrationals.
We also meet the Conway Constant, which arises from his “look and say” sequences. Start with a positive integer, say 23. The next term describes the first: it has one 2 and one 3 and hence the second term is 1213. The third term, using the same convention is then 211213 and so forth. Conway classifies these sequences in a very interesting way and derives λ = 1.303577269034296 as the asymptotic rate of growth of these sequences, which, while irrational, is algebraic. being the root of a polynomial of degree 71.
That leads us to chapters 6 and 7 and the discussion of transcendental numbers — the vast array of irrationals which are not solutions of polynomials over the rational numbers. Havel begins by developing yet another definition of an irrational number:
A real number x is irrational if and only if there are infinitely many rational numbers p/q such that |x – p/q| < 1/q2. (page 165)
We are then treated to Hermite’s (1873) proof that er is transcendental for every non-zero rational number r. Hermite’s method was modified by Lindemann in 1882 to establish that π is also transcendental. These two, linked by the amazing eiπ + 1 = 0, remain elusive. To this date it is not known whether π + e or πe are irrational, let alone transcendental. Here’s another question to ponder: does e occur anywhere in π? That is, is there a point after which the expansion of e is 31415…?
In Chapter 8 we return to the study of continued fractions and learn that there is hierarchy of irrationality with the golden ratio as the most irrational of all (despite its being algebraic). To rank irrationals we compute the partial fraction decomposition and look at the sequence of rational approximations it generates. A result of Hurwitz provides an upper limit on the error of each such approximation. We rank irrationals based on the ratio of the actual error to the Hurwitz upper bound. In the case of the golden ratio this ratio approaches 1 in the limit! It’s a neat idea and well worth a look.
In Chapter 9 Havel provides a nice discussion of the formal development of the real numbers by Cauchy, Weierstrass, and Dedekind along with a brief discussion of the Peano axioms for arithmetic.
Havel concludes with an answer to the perpetual classroom question: What’s this good for? In particular, Havel provides lots of situations in which the irrational top three (√2, e, π) are unavoidable. Among these are the period of a pendulum, the normal distribution in probability, and the probability that two randomly chosen integers are coprime.
This is a wonderful book which should appeal to a broad audience. Its level of difficulty ranges nicely from ideas accessible to high school students to some very deep mathematics. Highly recommended!
Richard Wilders is Professor of Mathematics and Marie and Bernice Gantzert Professor in the Liberal Arts and Sciences at North Central College in Naperville, IL. His primary interests are in the history of mathematics and science.