Julian Havil

Ostensibly a popularization, but similar to Havil's earlier books _Gamma_ and _Nonplussed!_ in that any reader who wants to follow any substantial fraction of it in detail had better be conversant with a lot of college-level math. For me, the going first got especially rough in the chapter (#6) on rational approximations of irrationals (which is important later in identifying those irrationals, such as e and pi (and "almost all" others!), that are transcendental). There Havil himself admits, "We are wallowing in deep mathematical water, where statements are barely believable and proofs barely understandable." (p 176) Still, the book can give interested amateurs an impression of how the pros delve into the complexities of the real-number continuum.… (meer)

The constant γ (called the Euler, or the Euler-Mascheroni) constant plays a significant role in Number Theory. Being, like π or e, one of the ubiquitous mathematical constants, it is, still today, remarkably less well known than its famous counterparts: this lack of knowledge is ilustrated by the fact that no one knows if γ is either a rational or a irrational! This nice popular science book tells the story of γ (if one may say so...) starting with John Napier's celebrated work on logarithms, then going on to discuss the harmonic series (starting with the celebrated proof of its divergence by Nicholas Oresme, c.a. 1350), and the Zeta function, the Gamma function, and the definition of γ. It the proceeds with a digression about some properties of γ, unexpected relations of the harmonic series and the logarithm function to problems in other areas (such as the optimal choice problem, and Benford's law), and concluding with two chapters about the distribution of primes and the work of Riemann (including his famous hypothesis.) Overall, this is a very interesting book that offers a relaxed exploration of a number of important mathematical issues in an enjoyable style.… (meer)