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Summary: e: a fundamental number
Review: Almost everyone is familiar with pi. It has been studied for thousands of years and will be a central aspect of a course in geometry. However, few people know about the number represented by e or its mathematical function. Eli Maor's book fills the gap.e, despite its lack of fame among the general population, is one of the fundamental numbers. It ranks in importance with pi, 0, 1, and 2^1/2. e is commonly found in problems associated with natural growth and decay. e ~ 2.71828 , but it is a nonrepeating, irrational number. Eli Maor traces the earliest use of e to the world of finance, and particularly the effect of interest on loans. Maor then cites other examples of where this number is employed. To develop his thesis on the importance of e in higher mathematics, Maor delves into the development of calculus and the discovery of irrational and imaginary numbers, all of which contributed to the recognition of e's importance. For example, Leonard Euler's carefree manipulations led him to revelations about e that forced mathematicians to recognize its role in mathematics apart from its complement, the natural logarithm. His discoveries concerning e also linked various branches of mathematics: arithmetic, geometry, algebra, and analysis. Although e is overshadowed by pi in the minds of nonmathematicians, its role is no less important than that of pi.
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Summary: Never a boring moment.
Review: How much have computers changed our lives? John Napier spend 20 years from 1594 to 1614 performing calculations for his logarithm tables. Today, that entire body of work is easily reproduced in minutes, using Microsoft Excel. But Napier's invention quickly spread around the world, creating a calculation revolution that empowered grateful scientists with speed they could only imagine before. I suppose it was the greatest computation breakthrough since the abacus.
From Napier forward, the story of e proceeds, eloquently recounted by Maor. There is not a boring moment in the book.
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Summary: the calculus dolt speaks
Review: I am taking analytical geometry right now, and this book is a lot of what the course leaves out. Which is the why and what lead someone to do what they did, which makes for an enlightening read.
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Summary: A story well told
Review: I read the book several times. It is an easy reading book for a begining math major (like me).The author did a very good and kind job to us. The other side of the my experience is that I had also tried to read the book "Euler, master of us all" (by a different author) and had found Euler's originals were easier to follow than that "expository" book.
I had always been wondering how people calculated logarithms initially and how logarithm was originated. Well, the story explained to me from the very beginning. Each chapter it tells me something interesting and beautiful that I did not know before. While most textbooks rarely spare the ink to tell the reader how and where some of the most important math ideas and formulas had come along, this book tells me in a gentle and lucid way. I consider this book to be a good friend and I suspect that perhaps even advanced learners may find it a enjoyable read as well. Well, I also think it will be very nice if calculus professors use this book as one of their references.
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Summary: Magical Description of Natural Log and Math History
Review: It would be a great side reading for Real Analysis and Multivariable Calculus class. The greatest wonder I had during the first half of Real Analysis (Rudin's book) class was answered. I thought 'e' should rather be called "magical log" rather than natural log because of all the fascinating features it has. Maor discusses every aspect of this magic number in a language that is friendly to non-mathematicians. I liked the real world application of natural log such as music and finance, and the derivation of Euler's Formula e^(i*pi)+1=0.
Excellent.
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Summary: Unbelievably Interesting, though last chapters are challengi
Review: Many textbooks state very little or nothing about the background behind the history mathematics. Maor describes the thought processes of the mathematicians in such a way that one can appreciate Mathematics more. e: The Story of a Number speaks to a broad audience so that, regardless of your mathematical experience, you will understand mathematics better.
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Summary: Interesting but Unevenly Paced
Review: 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.
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Summary: Unbelievably Interesting, though last chapters are challengi
Review: This is one of the best history of mathematics books I have read. It was comprehensive and easy to follow even for the uninitiated. However the last two or so chapters are challenging even for those who have a strong math background. This is not in anyway saying that the book is not worth reading, just that one might take care to put off reading the last few chapters until later in one's math carreer or seek the guidence of a wiser more experienced mathematician. Additionally I have actually taken a history or math course with Dr. Maor and would like to add he is a great lecturer as well... I highly recomend this book.
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Summary: More than the story of the second-most famous number
Review: 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.
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Summary: If Not Satisfied with Pi Try e
Review: This writer had written three books, as far as I know, the one before you, "Trigonometric Delights" and "To Infinity and Beyond." If this does not convince you to read this book, then nothing will. I read this after Dr Beckmann's "A History of Pi," and found it to be the right choice. (for a review of that book click on the blue "a_mathematician" above to see it in its proper place). As Pi is closely associated with the circle, then so is e associated to logarithms and differential equations. It is even called the natural base! Maor does an excellent job in accounting the historical points about e. In the beginning of the book we see e associated with the logarithms and then we see its characterization as the limit of some sequence of numbers. Then we go into some summary of the origin of calculus to see that the exponential function with base e is its own derivative. If you find all this difficult to grasp right now, do not worry, because it is explained in the book in simple language. The book then goes into spirals, and then to hyperbolic trig functions which are defined in terms of the exponential functions base e, and the book is concluded with the Euler's equation, the equation containing the four most important numbers in mathematics: Pi, e, 1, and zero. I can't emphasize this enough, but the language of the book is understandable by anyone. Maybe you would be convinced if I tell you that my native language is not English, yet I could understand the insights of the writer. If you liked the book try a better one, Singh's "Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem." (For a review please click on the blue a_mathematician above)
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