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Gamma: Exploring Euler's Constant
Among the many constants that appear in mathematics, π, e, and i are the most familiar. Following closely behind is y, or gamma, a constant that arises in many mathematical areas yet maintains a profound sense of mystery.
In a tantalizing blend of history and mathematics, Julian Havil takes the reader on a journey through logarithms and the harmonic series, the two defining elements of gamma, toward the first account of gamma's place in mathematics.
Introduced by the Swiss mathematician Leonhard Euler (1707-1783), who figures prominently in this book, gamma is defined as the limit of the sum of 1 + 1/2 + 1/3 + . . . Up to 1/n, minus the natural logarithm of n--the numerical value being 0.5772156. . . . But unlike its more celebrated colleagues π and e, the exact nature of gamma remains a mystery--we don't even know if gamma can be expressed as a fraction.
Among the numerous topics that arise during this historical odyssey into fundamental mathematical ideas are the Prime Number Theorem and the most important open problem in mathematics today--the Riemann Hypothesis (though no proof of either is offered!).
Sure to be popular with not only students and instructors but all math aficionados, Gamma takes us through countries, centuries, lives, and works, unfolding along the way the stories of some remarkable mathematics from some remarkable mathematicians.
-- "Notices of the American Mathematical Society"
In a tantalizing blend of history and mathematics, Julian Havil takes the reader on a journey through logarithms and the harmonic series, the two defining elements of gamma, toward the first account of gamma's place in mathematics.
Introduced by the Swiss mathematician Leonhard Euler (1707-1783), who figures prominently in this book, gamma is defined as the limit of the sum of 1 + 1/2 + 1/3 + . . . Up to 1/n, minus the natural logarithm of n--the numerical value being 0.5772156. . . . But unlike its more celebrated colleagues π and e, the exact nature of gamma remains a mystery--we don't even know if gamma can be expressed as a fraction.
Among the numerous topics that arise during this historical odyssey into fundamental mathematical ideas are the Prime Number Theorem and the most important open problem in mathematics today--the Riemann Hypothesis (though no proof of either is offered!).
Sure to be popular with not only students and instructors but all math aficionados, Gamma takes us through countries, centuries, lives, and works, unfolding along the way the stories of some remarkable mathematics from some remarkable mathematicians.
-- "Notices of the American Mathematical Society"
266 pages, Hardcover
First published March 17, 2003
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Displaying 1 - 20 of 20 reviews
March 28, 2013
Still a favourite of mine, this is perhaps the best 'popular' maths book I've yet come across. I feel the existence of such accounts is something of a niche in mathematics, since most popular books on subjects like physics tend to be largely descriptive and deliberately avoid actual results and derivations for fear of becoming inaccessible. As with Havil's text here, and others like Dunham's, Maor's, Nahin's, et al. however, in mathematics comparable popular treatments will give the lay reader a flavour of actual mathematics unless explicitly descriptive, or as is often the case, biographical or event-centred.
This is not to say you won't find motivation here- there's plenty of that, with backgrounds, anecdotes and general historical exposition providing context for the significance of results and bridging the material. But the eponymous Euler-Mascheroni constant, though it seems at first an odd, and to some an off-putting choice as the subject, is itself a fine linking element for all its mysteries (and perhaps exactly because of them). Gamma's journey takes us to surprising places, beginning reasonably enough with harmonic series and proceeding to more general concepts like zeta and Gamma functions, Euler-Maclaurin summation, the prime number theorem and the Riemann hypothesis (for a very natural and attention-grabbing invitation to analytic number theory). However, while this path is steadily traversed, Havil takes frequent free-flowing excursions and introduces curiosities like Benford's law, integrals of peculiar forms and even a sorting algorithm at one point (quicksort if memory serves). Some very useful appendices follow.
Clearly elaborated proofs here are offered at important points or where reasonable generality can be had, but Havil is careful to mark caveats. While the breadth of the subjects touched upon makes following Gamma slightly more demanding in comparison to similar books, there's encouragement to be had here and plenty to follow even if you don't keep in perfect step with the derivations.
This is not to say you won't find motivation here- there's plenty of that, with backgrounds, anecdotes and general historical exposition providing context for the significance of results and bridging the material. But the eponymous Euler-Mascheroni constant, though it seems at first an odd, and to some an off-putting choice as the subject, is itself a fine linking element for all its mysteries (and perhaps exactly because of them). Gamma's journey takes us to surprising places, beginning reasonably enough with harmonic series and proceeding to more general concepts like zeta and Gamma functions, Euler-Maclaurin summation, the prime number theorem and the Riemann hypothesis (for a very natural and attention-grabbing invitation to analytic number theory). However, while this path is steadily traversed, Havil takes frequent free-flowing excursions and introduces curiosities like Benford's law, integrals of peculiar forms and even a sorting algorithm at one point (quicksort if memory serves). Some very useful appendices follow.
Clearly elaborated proofs here are offered at important points or where reasonable generality can be had, but Havil is careful to mark caveats. While the breadth of the subjects touched upon makes following Gamma slightly more demanding in comparison to similar books, there's encouragement to be had here and plenty to follow even if you don't keep in perfect step with the derivations.
June 14, 2020
The constant gamma is the positive real number defined by the limit as n goes to infinity of the difference between the partial n-sum of the series of 1/j and the natural logarithm on n. It is called the Euler, or the Euler-Mascheroni, constant and plays a significant role in Number Theory. Being, like "pi" 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 illustrated by the fact that no one knows if gamma is either a rational or an irrational number! This nice popular science book tells the story of gamma (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 gamma. It then proceeds with a digression about some properties of gamma, 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.
March 16, 2022
I genuinely enjoy reading the books in the special numbers series published by Princeton University Press. This book is now another one of my favourites (I've also read e and i)! Since my high school days (around grade 11 -12) I have been fascinated by the Gamma function, its properties, and related numbers. This continued to interest me into my university years. In all those years my exposure have come from Schaum's outlines and physics textbooks. Never knew a "popular" math book on this subject ever existed!
Pros
- Includes mathematical history
- Includes calculus and infinite series (not a popular math book for the layman)
Cons
- Many integral calculations were not carried out step-by-step (need to be examined carefully by the reader)
Pros
- Includes mathematical history
- Includes calculus and infinite series (not a popular math book for the layman)
Cons
- Many integral calculations were not carried out step-by-step (need to be examined carefully by the reader)
December 9, 2015
With the excuse of the gamma Euler constant,this book gives a very kind introduction to the gamma or generaliced factorial function ,to the z function and a brief introduction to de difficult subject of the analytic number theory.The book is full of striking results,and in a appendix gives a notions of complex variable,all understable with the mathematical maturity and backgrund of high school mathematics
June 15, 2019
We are introduced to Zeta function introduced by Euler in 1734, also known as Basel Problem. We then proceed to gamma constant and Gamma function, explored by Euler as well. In the second part of the book we move to Prime numbers and Euler's contribution to discipline of mathematics. No wonder Euler is considered as one of the best mathematicians ever.
July 2, 2013
heavier than I expected, but I am loving it.
July 2, 2019
I enjoyed the book, having studied and never really (at least not conciously) encountered the constant gamma. The author starts off with the history of the logarithm, the harmonic series and reaches after that the discovery of gamma. From here on, the journey continues to prime number and the Riemann assumption. The content is very technical which the author warns about right in the introduction, why I would only recommend to people having studied mathematics or at least something similar very quantitative.
May 13, 2025
absolutely accomplishes its aim, to interest readers further into math showing just a tiny bit of all the beauty in the results. It’s an exhilarating read with a mix of history and a mathematical rigour not found in other “pop” math books.
November 12, 2017
This book is a 5 on content and quality. I gave it a 4 because I didn't like it that much.
April 7, 2018
I read this on vacation earlier this year. Strikes a good balance between popular appeal and technical detail. Lots of fun.
February 14, 2023
Among the many constants that appear in mathematics, π, e, and i are the most familiar. Following closely behind is y, or gamma, a constant that arises in many mathematical areas yet maintains a profound sense of mystery. In a tantalizing blend of history and mathematics, Julian Havil takes the reader on a journey through logarithms and the harmonic series, the two defining elements of gamma, toward the first account of gamma's place in mathematics. Introduced by the Swiss mathematician Leonhard Euler (1707-1783), who figures prominently in this book, gamma is defined as the limit of the sum of 1 + 1/2 + 1/3 + . . . Up to 1/n, minus the natural logarithm of n--the numerical value being 0.5772156. . . . But unlike its more celebrated colleagues π and e, the exact nature of gamma remains a mystery--we don't even know if gamma can be expressed as a fraction. Among the numerous topics that arise during this historical odyssey into fundamental mathematical ideas are the Prime Number Theorem and the most important open problem in mathematics today--the Riemann Hypothesis (though no proof of either is offered!). Sure to be popular with not only students and instructors but all math aficionados, Gamma takes us through countries, centuries, lives, and works, unfolding along the way the stories of some remarkable mathematics from some remarkable mathematicians.
March 3, 2015
At the time this book came out, I was going through parts of Whittaker's famous "Modern Analysis". This book was written when mathematicians still "did" mathematics. They got their hands dirty. They didn't immediately seek full generalization. While mathematics will always possess what looks to the layman like hieroglyphic notation, the notation employed in Whittaker's book reminds us that it is to be an aid to thought, not an impediment or a pyrotechnic. This is less the case today, where mathematics is far more professionalized than then; which is why I was so delighted by Gamma. Gamma captures the spirit of the great age of 18th and 19th century mathematics: with its feet firmly planted on the ground yet discovering things of great beauty. It is a pleasure to follow Havil's exposition of Euler's thought processes as he uncovers Harmonic series, the Zeta function and so on. In fact, I recommend this to anyone who wants to get a quick feel for these--now very complicated--subjects. To my surprise, there was a nice discussion of Benford's Law. Besides the nice exposition, there's lots of work for, you the reader, to do! But you can do it without having to remember a memory stack of technical terms and unintiutive notations that mar so many modern expositions. Havil is a math teacher who loves math. That means he has wrestled, as all teachers have done, with those pesky concepts that often aren't as "obvious" or "elementary" as is assumed in professional texts.
August 7, 2008
I've read this book twice now. It's quite good.
Some tidbits:
p.32-33 Kempner series = Harmonic series with those terms whose denominators having a fixed digit dropped (e.g. only sum reciprocals of numbers without a '7' in them). This is bounded by geometric series, and therefore converges.
p.39 Fun proof of sum of reciprocal squares = pi^2 / 6, appropriate for calc class, once you know Taylor series for sin.
p. 44-45 \int_0^1 1/(x^x) dx = \sum_n=0^{\infy} 1/(n^n) using Taylor series for e^x and integration by parts. Totally sweet.
p. 133-134 'Worm on a Band' A point starts at one end of a line with length 1m. The point travels 1 cm/min. At the end of every minute, the line instantly increases length by 1m. By considering percents, one shows that the worm actually reaches the end of the band. At approximately e^(100-\gamma) minutes. Seriously?
p. 159 Benford's Law about leading digits not being uniformly distributed.
Some tidbits:
p.32-33 Kempner series = Harmonic series with those terms whose denominators having a fixed digit dropped (e.g. only sum reciprocals of numbers without a '7' in them). This is bounded by geometric series, and therefore converges.
p.39 Fun proof of sum of reciprocal squares = pi^2 / 6, appropriate for calc class, once you know Taylor series for sin.
p. 44-45 \int_0^1 1/(x^x) dx = \sum_n=0^{\infy} 1/(n^n) using Taylor series for e^x and integration by parts. Totally sweet.
p. 133-134 'Worm on a Band' A point starts at one end of a line with length 1m. The point travels 1 cm/min. At the end of every minute, the line instantly increases length by 1m. By considering percents, one shows that the worm actually reaches the end of the band. At approximately e^(100-\gamma) minutes. Seriously?
p. 159 Benford's Law about leading digits not being uniformly distributed.
August 18, 2017
A rare in-depth look at the gamma constant
I was hoping to find a book that went further into the rabbit hole of the gamma constant than most books. It pops up in many places, but why? Here, Havil goes further than I expected, and is easy to follow. He also remains entertaining while doing so.
And it is always interesting to end up with my favorite subject, the Riemann Hypothesis. My friends sometimes tease me for reading books on math and the Riemann Hypothesis (boring!), but they noticed my enthusiasm for this one.
I was hoping to find a book that went further into the rabbit hole of the gamma constant than most books. It pops up in many places, but why? Here, Havil goes further than I expected, and is easy to follow. He also remains entertaining while doing so.
And it is always interesting to end up with my favorite subject, the Riemann Hypothesis. My friends sometimes tease me for reading books on math and the Riemann Hypothesis (boring!), but they noticed my enthusiasm for this one.
July 8, 2013
Not gonna lie: I admit I skimmed over most of the mathematical detail in favor of the historical context I was really looking for. But I got enough out of it to have at least an interesting conversation or two on the Bolzano–Weierstrass theorem and Benford's Law (that's a fun one!) and some other odds and ends. Not really a general-audience publication, really, though.
Currently reading
June 20, 2011so far so good although i always feel like an idiot when i get caught up on even the "simple" proofs. and this happens a lot.
March 14, 2013
A look at Euler's constant from many directions. Requires a knowledge of calculus for full enjoyment
September 2, 2013
I found this book quite difficult, but it was a very engaging read none the less and exposed me to some areas of mathematics that I have not seen before, and am now eager to explore!
May 5, 2025
THe maths really was a bit complex for a light read, interesting, but not as good as others like it.
December 18, 2016
An unexpectedly wonderful romp through the wonderful world of number theory. If you've every been enthralled by numbers - even for a moment - this book will bring back that joy.
Displaying 1 - 20 of 20 reviews