e: The Story of a Number (Princeton Science Library, 41) by Eli Maor (PDF)

Description

The interest earned on a bank account, the arrangement of seeds in a sunflower, and the shape of the Gateway Arch in St. Louis are all intimately connected with the mysterious number e. In this informal and engaging history, Eli Maor portrays the curious characters and the elegant mathematics that lie behind the number. Designed for a reader with only a modest mathematical background, this biography brings out the central importance of e to mathematics and illuminates a golden era in the age of science.

User’s Reviews

Editorial Reviews: Review “Honorable Mention for the 1994 Award for Best Professional/Scholarly Book in Mathematics, Association of American Publishers””This is a gently paced, elegantly composed book, and it will bring its readers much pleasure. . . . Maor has written an excellent book that should be in every public and school library.”—Ian Stewart, New Scientist”Maor wonderfully tells the story of e. The chronological history allows excursions into the lives of people involved with the development of this fascinating number. Maor hangs his story on a string of people stretching from Archimedes to David Hilbert. And by presenting mathematics in terms of the humans who produced it, he places the subject where it belongs–squarely in the centre of the humanities.”—Jerry P. King, Nature”Maor has succeeded in writing a short, readable mathematical story. He has interspersed a variety of anecdotes, excursions, and essays to lighten the flow…. [The book] is like the voyages of Columbus as told by the first mate.”—Peter Borwein, Science”Maor attempts to give the irrational number e its rightful standing alongside pi as a fundamental constant in science and nature; he succeeds very well…. Maor writes so that both mathematical newcomers and long-time professionals alike can thoroughly enjoy his book, learn something new, and witness the ubiquity of mathematical ideas in Western culture.” ― Choice”It can be recommended to readers who want to learn about mathematics and its history, who want to be inspired and who want to understand important mathematical ideas more deeply.” ― EMS Newsletter”[A] very interesting story about the history of e, logarithms, and related matters, especially the history of calculus. . . . [A] useful complement to a course in calculus and analysis, shedding light on some fundamental topics.”—Mehdi Hassani, MAA Reviews About the Author Eli Maor is the author of Beautiful Geometry (with Eugen Jost), Venus in Transit, Trigonometric Delights, To Infinity and Beyond, and The Pythagorean Theorem: A 4,000-Year History (all Princeton).

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The number e is interesting beast. In this book, the author shows us how this constant is found in a variety of phenomena, such as physics, biology, art, and music. This number was actually known about half a century before the invention of calculus. It was referred to in Edward Wright’s English translation of John Napier’s work on logarithms, published in 1618. It also seems to have appeared in connection of a formula for compound interest. However, work on the area under a hyperbola seems to have led independently to the same number. Its origin appears shrouded in mystery. In the eighteenth century, Leonhard Euler’s work gave it a central role in calculus.We start with an introduction into logarithms, and then segue into what e equates to. We learn that the number e is equal to the limit of (1 + 1/n) ^n as n tends toward infinity. This converges to 2.71828 rounded off. The concept of the limit is interesting. A sequence of numbers can approach a limit as closely as one wishes, but here’s the kicker: it never actually reaches it. Taking a step back in time we learn of the work of Archimedes and his method of exhaustion, where he successfully applied it to the parabola. In time, think 16th century, we see the development of the use of an infinite process written as a mathematical formula. Along the way, we are introduced to the various participants in the development of mathematics. An eventual successor to the method of exhaustion, is something called the method of indivisibles. This involves dividing something into infinitely smaller divisions. Kepler used this method to find volumes of solids of revolution. He came within one step of our modern integral calculus. Fermat discovered the quadrature of an entire family of curves (y = x ^n) with an equation that just happens to be the integration formula we now use in calculus. By the time of Newton with his fluents and fluxions, we see a move toward a general procedure or algorithm for finding the rate of change. The Fundamental Theorem of the Calculus shows the inverse relation between finding the area under a curve and finding the derivative. Ultimately, it was Leibniz’s notation that survived down to our day, e.g., dy/dx.By chapter ten, we revisit the limit formula and its derivation that equals the number e, and that e is an exponential function. We also see how the equations y = e^x and y=ln x are inverse functions. I must mention here that an exposure to limits and some basic calculus theory would be beneficial to a greater understanding and appreciation of the text. We learn of the contributions of the famous Bernoulli family and their achievement of mathematical prominence. Actually, an entire book would be necessary to relate all their contributions. Another interesting concept involving e is the logarithmic spiral, some of the properties of which depend on the fact that e^x is equal to its own derivative. Another interesting curve related to e is the catenary, which turns out to be transcendental. The value e can also be expressed as an infinite series and also by a continued fraction. Many concepts relating to e and PI were discovered by the great mathematician Euler. He was the person to obtain the famous formula that connects the five most important constants of mathematics – e being one of them. In the section on imaginary numbers, there is a discussion of complex variables, which was a tad bit confusing, but e figures in here as well. We are then given a primer on prime numbers, and here we learn that the “Prime Number Theorem shows that the number e is indirectly connected with prime numbers.” The author finds this remarkable as primes are in the domain of integers, the domain of discrete mathematics, whereas e belongs to the realm of analysis, the domain of limits and continuity.This book involves a bit of mathematical derivation, but understanding all the details is not entirely necessary to appreciate the material discussed. In conclusion, the author notes that of the infinity of real numbers, the most important to mathematics – 0, 1, square root of 2, e and PI – are within four units of each other on the number line. “A remarkable coincidence? A mere detail in the Creator’s grand design?” The author will let the reader decide.

⭐This was a good book for someone who likes math and is willing to work a little. You should have had (and enjoyed at some point) a little algebra, geometry, and calculus. Even if your math is rusty like mine, you will be able to follow this book well enough. I was surprised how much of it came back to me. (I wouldn’t want to be tested on it though!)The most fascinating thing to me was the brainpower that thought this stuff up! How they could have pumped so much out of the natural logarithm (e) was simply amazing to me, things such as the elegant infinite series of fractions and continued fractions, continued exponentials, sometimes with factorials. Perhaps the most amazing thing was the totally unintuitive formula e raised to the power of the product of i and pi = -1; imagine e, i, and pi contained in one short,neat, little formula! This book is also about the history of math, how calculus was invented, and how imaginary numbers found their place in math. Fortunately for me, Eli Maor goes slow enough and skips enough of the details and the proofs to make this book readable. He also gives neat short biographies of the main characters in the history of mathematics to break the hard math up. The one that was most fascinating to me was an 18th century mathametician named Leonhard Euler (who came up with e raised to the product of pi and i = 1), whom Eli Maor called “unquestionably the Mozart of math”. He is relatively unknown simply because he was bracketed in time between Newton and Galileo. I do, however, have to confess I got a bit lost near the end of the book with his dissertation on complex variables (imaginary and real). The math there was a bit too dense for me (or maybe I was too dense).I can’t figure out how e raised to the power of the product of pi and i can come out to a real number (-1) since it is about a real number raised to an imaginary power. How is that even possible? How in the world did Euler come up with the formula! Maor says he’ll leave it to the reader to decide if this remarkable formula is a part of “the Creator’s grand scheme”.It was also a relief to read a math book without having to be graded. That was a first for me.

⭐The range of coverage of mathematics in this book is astounding. As a retired professional engineer, I found the explanations to be entirely readable, informative and incredibly thorough. I would recommend this book to anyone who seeks an in-depth understanding of the foundations of mathematics that go far beyond the concepts of the base of the natural logarithms.

⭐A very detailed and not too difficult overview of the second most famous number in mathematics. I loved it and refer to it often. Oneof the joys is the copious graphs that so well illustrate the ideas. Good explanations of how e entered into maths and has comealmost to dominate it, and how e is related to the trigonometrical functions we learnt at school and to the square root of minus one.This book is a valuable addition to other portraits of individual numbers, for example “Zero” by Charles Seife, or “An Imaginary Tale, TheStory of root minus one” by Paul Nahin. For those who wish to pursue e to a high technical level, I recommend “Dr Euler’s FamousFormula” also by Paul Nahin (university level). Can you guess what this famous formuls is? Don’t look immediately at the next line!It is of course e^i.pi = -1

⭐I loved this book. It goes through the history of e in chronological order, starting with the discovery / invention of logarithms, and proceeding to all the other contexts in which e kept cropping up (limit of compound interest paid continuously, the curve which differentiates to itself, the equation of a hanging chain, its relation to pi in the context of imaginary numbers).The main proofs were sent to appendices at the back, so you can go through the details of the algebra of you want, but if you don’t want to then you can skip the proofs without disrupting the flow of the story. I’d say an A-level knowledge of maths is required to understand what it’s on about, but if you don’t have the equivalent of an A-level then you probably aren’t considering buying this book!One of the last chapters discusses e^i pi + 1 = 0 in a philosophical way, which was nice. I knew the results already but it was great to see how the story of e unravelled over the centuries, get to know some of the mathematicians who were involved, and the material is well presented in an interesting way.

⭐Anyone with modest mathematical knowledge (‘AS’ – level for example)and an interest in mathematics would benefit from reading this delightful book. Apart from the material directly related to ‘e’ there are also highly readable accounts of the dispute between Leibniz and Newton, the rivalry of the Bernoullis and the genius of Euler. I particularly enjoyed the chapters ‘Squaring the Hyperbola’ and ‘The Imaginary becomes Real’. The latter builds to simple mappings of the complex plane, illustrating analyticity, the Cauchy – Riemann Equations and their link to Laplace’s equation.The historical aspect to the book is entertaining and the mathematics explained well. Highly recommended.

⭐An enjoyable journey through the history of e. As a one-time user of Log tables and slide-rules user I can but wonder at the remarkable mathematics behind them. As enjoyable a history of e as I have come across. It gives an interesting insight into need powering break-thoughs, followed by development of the notation, then the development of the fundamental theoretic concepts.

⭐Although this is very much a popular introduction it still contained lots of fascinating information and history that I was not aware of. Enjoyed it a lot.

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