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

Julian Havil does it again. Like his previous book *Gamma: Exploring Euler's Constant* (2009) in which a key role was played by Euler's constant 0.57721..., this epic story follows the trace of the irrationals throughout the centuries. His pleasant narrative style, larded with, citations, anecdotes and historical facts keeps your attention going, following the attacks of mathematicians fighting like knights trying to unveil the secrets from an ever metamorphosing irrational dragon.

However, it is not just a superficial story leading you through the history of mathematics from the Greek till the present, it is also a serious book about mathematics with proofs and derivations. Some of them are easy at an undergraduate level, but others are at a much more advanced level.

Starting with the Greek where Pythagoras' theorem bears the seed of the first irrational √2, which was called incommensurable in those days. Square roots and surds kept playing an important role while algebra and calculus were being developed. It was however by the introduction of continued fractions that a new tool came available to stir the world of irrationals. Lambert proves the irrationality of $\pi$ (in 1761) (although a modern proof was only published in 1947 by Ivan Niven) and also Euler's number e enters the scene (proved to be irrational by Fourier in 1815). More intrinsic studies of the irrationals were undertaken. What combinations of irrationals are irrational? Niels Abel in 1929 and Evariste Galois in 1930 prove that there are algebraic numbers not expressible in radicals. One of the remarkable achievements of the 20th century is Apéry's proof of the irrationality of ζ(3) and how well it can be approximated. Enter transcendental numbers (Liouville's number $\sum_{r=1}^\infty 10^{−r!}$ seems to be the first and Hermite proved in 1863 that $e^r$ is transcendental). In fact a whole hierarchy of irrationality emerges (starting a search for the "most irrational number"). Then Cantor, Dedekind and others take the scene and come to an axiomatic definition of the reals.

Most results that can be proved in a reasonable number of lines are included. Sometimes it is a reconstruction of the historical proof, sometimes it is a more modern one. As the reader advances in the book, the proofs become on the average more advanced as well. Some items are included as an appendix (e.g. the spiral of Theodorus, equivalence relations, continued fractions, mean value theorem, etc.)

As a conclusion it is a book that can be warmly recommended to any mathematician or any reader who is generally interested in mathematics. One should be prepared to read some of the proofs. Skipping all the proofs would do injustice to the concept, leaving just a skinny skeleton, but skipping some of the most advanced ones is acceptable. The style, the well documented historical context and quotations mixed with references to modern situations make it a wonderful read.

**Submitted by Adhemar Bultheel |

**20 / Nov / 2012