Customer Reviews

"e": The Story of a Number (Princeton Science Library)

5 star:		(42)
4 star:		(17)
3 star:		(4)
2 star:		(3)
1 star:		(3)

 

 

This is the second book by Eli Maor that I have read and reviewed in as many months (the previous book was "To Infinity and beyond"). As I was reading this latest book I thought several times that the title was wrong. I think a more appropriate title might be "A popular introduction to calculus" or "The road to calculus." Then, again, he does more than just calculus, too. So I'm not sure what to call it. It's more than just about e, and it's more than just about calculus. It's all that, with a lot of other interesting tidbits tied in as well. While Eli does spend quite a bit of time discussing e, this book goes well beyond a simple linear history of a number that's fundamental to modern mathematics.
Eli begins his story with John Napier and the invention/use of logarithms as tools for calculation. I found this introduction interesting because it reminded me how valuable calculation tools were, in the days before electronic calculators. I even found myself rummaging through my desk for that long-forgotten slide rule and remembering with a degree of nostalgia the many hours spent working through problems in mathematics and physics during my high school years, and how I'd pride myself on being able to carry the a full three significant digits through a complex sting of calculations.
It seems as though the initial chapters of Maor's book deal more with the history of e than does the middle of the book. Somewhere around page 40 Maor moves away from mathematical history aimed squarely at natural logarithms and focuses more on what is (I suspect) his true love: calculus. This is one of the best introductions to calculus I've seen, primarily because Maor did such a nice job of bring together all the historical footnotes.
Coincidentally, as I was reading Mayor's book my wife was taking a class for teachers, aimed at educators who teach calculus in the middle and high schools. She found the book immensely helpful in both dealing with the actual mathematics in her class as well as providing insight into ways of introducing concepts relating to higher-level mathematics to young students. She introduced Mayor's book to other students in her class, as well as the professor (who had read it already, of course), all of whom enjoyed it immensely.
In terms of the history that he covers, I thought the discussion relating to Newton and Leibniz was the most interesting. My own coursework in Physics used Newton's dot notation, while my courses in mathematics adopted Leibniz's differential notation. Reading Maor's book provided a bit more insight into the historical quirks that led to the notation in common use today.
Especially interesting was his discussion about Newton's approach to the calculus. I think that if students had to use the notation and approach first used by Newton, calculus might still be relegated largely to the college curriculum. I really had no idea, before reading Maor's book, how convoluted Newton's approach was in comparison to that used by Leibniz. Newton is often portrayed (rightfully so) as a genius, and Mayor's description of Newton's calculus left me marveling that Newton managed to work through it as he did, given the (relatively) more difficult approach he took.
The end of Maor's book uses the calculus to illustrate several examples showing how e appears in various mathematical and physical problems. There are examples using aerodynamic drag, music, spirals, hanging chains and the cycloid. No discussion of e would be complete without a nice explanation of the function that is its own derivative, which Maor tells with characteristic clarity.
Frequently while reading Maor's book I found myself wishing I'd had this introduction before taking several of the classes I took during my school years. His treatment of the complex plane, for example, is as clear as his introduction to basic ideas in calculus. Looking back on my first class in complex variables, I recall the fog that surrounded my initial introduction to conformal mapping. Maor, though, makes it easy. With the skill of a master educator, he manages to explain the concept with such ease that you learn the essential ideas almost before you realize where he is taking you. Though most texts of this sort would not tread on a subject as foreboding to the general public as the Cauchy-Riemann equations, Maor explains the basic concepts as clearly and almost as effortlessly as he does conformal mapping. Ordinarily I wouldn't think it's possible to explain Cauchy-Riemann in a book that's intended for the general public with an interest in mathematics, but that's what Maor does, and he does it well.
In short, this nice little book manages to cover a lot of mathematical territory with the skill that only a master educator can muster. It is definitely a whole lot more than just the story of the second-most famous number in mathematics.

To those of you who are not familiar with Maor, let me point out that he is a mathematician (as opposed to a lot of the other people who write popular math books) with an immense knowledge of math history and also an excellent writer. Some reviewers have compared this book to books like "An Imaginary Tale" by Paul J. Nahin and "History of Pi" by Petr Beckmann. This is totally missing the point. Both of those books are written by non-mathematicians, and contain error that will annoy mathematicians. Maor on the other hand is a superb scholar. I've read all his four books quite carefull, and I've not found any errors.

This book will give you a great understanding of what calculus is all about.

The beginning and the end of Maor's story are compelling. He spells out exactly what John Napier put in his original "logarithmic" tables--it turns out that these were logs to the base 1/e, shifted by a factor of 10,000,000, even though their creator wouldn't have put it that way. I was, however, disappointed that no actual *example* is given of a calculation that was made possible by these unusual original tables. Maor tells us how excited Kepler and others were by the possibilities, and hints that computations involving sines were especially aided, but there's not a single example of how the pre-Briggs (log to base 10) logarithm was ever used. (And let me point out that this is not an obvious matter; after extensive googling I have only been able to locate very artificial examples of what Napier's very incomplete tables were good for.)

Still, the opening chapters on the "pre-history" of e (before the invention of calculus) are one of the strongest parts of this book. Where Maor gets bogged down is in the long digression telling of the invention of calculus and the bitter priority dispute. In my opinion, there's a solid block of dead weight beginning from the first page of Chapter 8, and Maor doesn't get his steam back until the latter part of Chapter 11 (when we meet the truly "mirabilis" logarithmic spiral).

Some of the sidebars are excellent--e.g. the math behind terminal velocity, which makes parachuting possible ("The Parachutist") and the Weber-Fechner law, which claims to give a mathematical model of human response to affective stimuli ("Can Perceptions Be Quantified?").

As in his "Trigonometric Delights," Maor excels in presenting the world of complex analysis that was opened up by Leonhard Euler in the 18th century. Some background is really required to enjoy all this, but, if you have it, you are treated to the Cauchy-Riemann equations, and to excellent discussions (partly relegated to the appendices) of the equiangularity of the logarithmic spiral (proved with an elegant conformal mapping) and the full range of geometric and analytic analogies between the circular functions (sine, cosine, etc.) and their hyperbolic counterparts (cosh, sinh, etc.).

All in all, I recommend this title, but do skim over the tedious exposition of calculus & the priority dispute.

I am a bit bothered by all those 5-star reviews and feel obligated to tell any potential readers of this book the other side of the story. (Please also check out another review of this book by guttes on January 21, 2001.)

First of all, this book lacks a focus. It jumps back and forth with things related or even unrelated to the number e (it spends more than a chapter on the discovery of calculus, and on and off on the topic of pi). While it is more than verbose on something not (totally) relevant, it simply does not have some topics that you wanna know more about -- for instance, who/why there is such a notation 'e' for the number, or what are the latest hot topics about the number. It has no consistency in presenting its mathematical formulation either, in the first half of the book it assumes readers with minimal calculus background, but then in the second half of the book it assumes readers with background on complex and multi-variable analysis. In the very last chapter, it touches on the topic of transcendence, but again, fails to deliver anything substantial and signifies an hollow story for such an interesting and promising topic.

In a nutshell, the problems of the whole book are the choice of materials presented and the organization of the materials for the topic. It may be one of the very few books out there with a title related to the number 'e', but it surely is not a book telling you a story of this number.

For those who also finds this book disappointing, there are indeed way better "popular" math books out there. I would recommend books written by William Dunham, such as "Journey Through Genius" (for some easy reading) or "The Calculus Gallery" (for those with more calculus backgroud).

Galileo wrote that philosophy is written in the grand book of the universe, in a language of characters, circles, triangles, and other figures. Somewhere in this grand offering came the number e, which is the limit of the expression (1+1/n)^n, as n approaches infinity. It is a curiosity number, one that bridges Napier's original logarithms (which are to the base 1/e) and the origins of calculus. It was discovered at a time of exploding international trade, which is based on compound interest, whose formula you will recognize in the definition of e. It is the base of natural logarithms, a non-terminating, non-repeating decimal. e cannot be the solution to a quadratic equation that has integer coefficients.

This is a splendid book about a number as strange and useful as pi. Well written, this book can be handled by bright high school students and college students who have an interest not in solving math problems (the way we usually teach math), but in the history of math and this curious number. I read it for general interest and was very pleased with the entire book.

Of the "great" constants in mathematics such as 0,1,i,and pi, e is the black sheep of the family. To the non-mathemetician, e rarely is thought of nor is its numerical value well-known. (I am not spoiling anything by saying that e approximately equals 2.718281828).

Maor does a good job in showing how important this number actually is and how modern math would not exist without it. e pops up in physics, astronomy and finance, as Maor demonstrates. This book gives a history of e, which has only been known for a few hundred years (as opposed to its cousin, pi, which has been around for thousands).

This is a good, educational and entertaining book, admittedly a bit of a challenge for the non-mathematician, even though he limits his more complicated proofs to the appendicies. My only real gripe with the book is the use of footnotes, which can sometimes be confused with exponents. In many books, this might not be a problem, but with this topic, a different approach is needed. Still, this is probably as much a publisher's problem as Maor's, and takes little away from an otherwise good book.

A definitely recommended read, pure mathematical beauty - this book shows how the mathematicians worked - it shows how Napier evolved the logarithms, how leibnitz and newton approached calculus, and how the number e itself was discovered... its so simple to read, and it also shows exactly how the firsts proved their theorems... unlike textbooks we have read proofs in. And a very quick and simple read too - where one is exposed to the beauty of mathematics when one sees how 'Pi' came to have its myriad infinite series expansions..

A chapter here is devoted to the Bernoulli family - which is where I want to draw your attention. This is perhaps the most influential family in mathematics, where three generations of the family have made staggering contributions to physics, mathematics, astronomy, finance, etc etc. And the family is as insane as can be - The brothers Jacob and Johann were the first to sit and understand Leibnitz' calculus, and Johann was a staunch supporter in the fued with Newton, on Leibnitz' side...(btw - Newton destroyed Leibnitz professionally, after Leibnitz' death... and Leibnitz funeral was attended by his secretary only... none else). So soon Jacob and Johann fight like mad about who made certain proofs - and Johann leaves the house, promising to return only after Jacob's death. Which he does. Later Johann and his son, who were working together starting arguing on who's name should come first on the documents they published,... so they soon quarelled and began to work alone... and then later, when Daniel starting publishing more than his illustrious father, Johann expelled him from the house with a warning to never return...In the above book, Eli Maor also compares the Bernoulli achievements in maths, with the other family that had a 150 years of three generations of contributions to its field - which was also a contemporary - viz. the Bach's and music. Awesome book - clean and nice, and leaves you admiring the poetry of maths... The book closes on the marvellous work of Euler who shows 'e power i, pi = -1', and thus links the three most vexatious numbers in maths, e, pi and i (sqrt -1), in a most wonderfully simple equation. What sheer elegance, and what beauty indeed.

First, this book is poorly organized. The flow from one chapter to the next makes little or no sense. Rather than come up with a coherent narrative thread to tie together a story about the number "e", Maor rambles from one anecdote to another, often touching on subjects that have little relevance to the number "e." He has written this more like a mathematical proof than a book with references to topics that only later are discussed breaking up any narrative flow.

Second, Maor has little sensitivity to the relative mathematical sophistication of his reader. At times, he seems to assume only a rudimentary understanding of arithmetic, and then later he proceeds as if the reader is quite comfortable with multi-variable calculus.

Third, he only occasionally bothers to communicate the essence of the relevant proofs he discusses. Often, he simply states that a given proof is beyond the scope of the book and moves on, as if without the complete mathematical proof there is nothing more to discuss.

The result is a book that sheds surprisingly little insight on the story of the number e. I doubt that this is because the story is not worth a book, but rather because of the failures of this particular effort. Reading this book only deepened my respect for books like "Journey Through Genius" and "What is Quantum Mechanics?" which pass through similarly treacherous territory with grace and vitality.

Of the numerical histories you'll find via the links on this page, this is the easiest to read. Beckman's "pi," while more a history of ideas through one idea, contains more difficult math than this book, though not much of it. Nahin's book on i is very heavy on math, though some of it is very simple and a lot of it is repetitive in nature. All three are very good, and well worth reading. I enjoyed all in slightly different ways, but this one was the most, well, fun. The books, by the way, should be read in the order pi,e,i, as that is the order in which they were "discovered" and their use popularized. After reading them all, I am not sure that e is not a more subtle concept than i. See what you think.

I know calculus but I didn't know much about some of the history of mathematics. How easy is to learn a complex theme in the words of this author. A fascinating book about an important number, browsing the history of logarithms, then some of the history of calculus and finally the history of Leonhard Euler, and the first appearance of "e". Obviously you find mathematics in this book, but presented in a easy-to-understand way.