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Euler : The Master of Us All by William Dunham (4-Mar-1999)

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  • William Dunham(Author)
  • The Mathematical Association of America (4 Mar. 1999) (1600)
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  • By Path Integral Master on August 3, 2012

    Math can be weird. Imagine this:a product, not of a finite quantity of numbers, but of an INFINITE number. A combination between calculus and number theory. And finally,a fundamental constant that mathematicians haven't even proved to be irrational. You're in the bizarre realm of mathematics, and to survey this vast area, Dunham takes us to the work of Euler.Euler. What a great man. Pursuing against nearly insurmountable obstacles, this mathematical hero founded entirely new fields of the subject, reconstructing the very bedrock of mathematical thought. He brought new light to diverse subjects in Science and Math, and due to his outstanding achievements, it is arguable that he is among the most influencial people in creating our modern way of life. After all, the very foundation of modern life is modern technology; and Euler's deluge of elegant mathematics was incredibly influencial on the creation of much modern technology. From radio to TV, from modern antiobiotics to nuclear power stations, Euler lives with us every day.This book is undoubtedly the single best guide for understanding the nearly unbelievable achievements of Euler. In wonderfully written, witty prose, Dunham provides us with a perfectly chosen sampling from Euler's elegant work. From analytic number theory to p-series, from perfect numbers to complex variables, Euler revolutionized the way mathematicians think about the world; and this book is the best way to understand how that revolution occurred.The book requires as little mathematical knowelege as possible to understand its contents. The reader does not need a PhD in fifty-eight dimensional topology: rather, all that is needed is an understanding of single-variable Calculus.Dunham has succeded where others have undoubtedly failed, in providing a simple, straightforward account of Euler's work accessible to many mathematically literate readers.The book begins with a subject I had (surprisingly) never heard about, let alone seen in connection with Euler: perfect numbers. Perfect numbers are a simple concept: they are just quantities that equal the sum of their proper divisors(the proper divisors of 6, for example, are 1,2, and 3; and since 1+2+3=6, then 6 is a perfect number). Euler discovered a remarkable property of even perfect numbers: all such quantities can be written in the form 2^(k-1) (2^k-1) in which the second factor is a prime number. Euler's proof of this is absolutely ingenious, and requires only an understanding of elementary number theory (due to the nature of this assignment I can not include the proof here; buy the book if you are interested). I was surprised that there lurked such mathematical treasures in elementary number theory, a topic that, until now, I thought I had entirely understood.Then the book travels to the bizarre land of the logarithm, a mathematical function whose properties were wonderfully revealed by Euler. The logarithm function is simple to understand: in the same way that division is the opposite of multiplication, logarithms are exponents "in reverse". Specifically, if x^y=z, then we say that y is the logarithm of z to base x. Euler revolutionized logarithms by writing them as infinite series; that is, infinite sums of simpler functions. His argument is a beautiful demonstration revolving around the properties of infinitely tiny "numbers". I had never before seen a derivation for logarithmic series that was this elegant.Next, the book uses this newfound knowledge on logarithms to describe the Euler constant,a number that appears oftentimes in mathematics. See the book for details, but the number comes from the problem of finding the partial sums of the harmonic series(a well-known infinite series that should be included in any calculus textbook).Dunham next considers Euler's wonderful work in solving the basel problem, a widely-known and tricky puzzle in the study of infinite sums. The problem sounds simple: to calculate the infinite sum1 +1/22 +1/32 +1/42 ...Euler solved this problem ingeniously: by factoring an infinite-degree polynomial, he was able to arrive at the answer! The next port of call the book stops at is the consideration of analytic number theory, a seemingly bizarre topic that is all Euler's. The idea seems strange: analysis considers continuous curves and functions, while number theory studies the discrete and choppy (so to speak) whole numbers. At a first glance, these two concepts seem utterly incompatable. But first impressions can be deceiving: this field has given beautiful proofs for the infinitude of the primes and their relative abundance among the whole numbers. Euler was a genious-and his new field reflects that. If I would want to study anything in further detail, it would be the field of complex variables, in which Euler played a critical, if not central, role. Complex numbers, the square roots of negatives, have fascinated mathematicians for centuries. I would love to learn more about the history of complex numbers-how they came about. In conclusion, Leonhard Euler was a mathematical genius of the upmost quality. He made outstanding achievements to the fields of analysis, imaginary (i.e, complex)numbers, number theory, and more. He also created entirely new fields like analytic number theory. We owe much to this genius who revolutionized the foundational pillars of mathematics!

  • By A customer on August 25, 1999

    I had never read any of William Dunham's many books before. Now I want to read them all. In a scant 173 pages he describes in great detail how Leonhard Euler, arguably the greatest mathematician ever, solved the most difficult mathematical problems of his day.The style in this book is both unusual and clever. Each of the eight chapters covers a different branch of mathematics and each begins with a prologue, then follows with some of Euler's contributions, and finishes with an epilogue. The prologues present the history of mathematics up to Euler's time, so the reader gets a feel of what this great mathematician had to work with. And the epilogues tell where we have come since Euler.This book is full of equations and expects some work (but not much mathematical background) from the reader. If you like mathematics or ever wondered how some of the great discoveries in this field were derived, do yourself a favor and buy, then carefully read, this wonderful book.

  • By Wan Koon Yat on September 15, 2001

    " Analysis incarnate " , no other more suitable words probably can describe the incomparable power of Euler, as his contemparies called him. Concerning the usual style of Dunham to write this stimulating book, other readers have made many comments and I think there is no need to repeat that. What I want is that Dunham to write another book, perhaps volume 2,3 etc and also write a thorough biography of Euler, one the greatest mathematicians in the history. ( To me, for mathematical ability, his should be at the same rank with Newton, Archaemedes, and Gauss, even Einstein concerning the mathematical and theroetical aspect, is below par compared with Euler )

  • By A customer on March 1, 2000

    As with his other books, William Dunham puts mathematics in an historical (and sometime political) context. This time he takes this kind of look at a few narrow slices of the huge volume of works by Euler. Each chapter focuses on a different branch of mathematics touched by Euler and each could probably be expanded to fill a book of its own. Very interesting but it requires a strong mathematical background on the part of the reader. I would not recommend it to someone who has not taken some calculus courses.

  • By Lyn J. on October 11, 2017

    William Dunham’s brilliant analysis of the life of Leonhard Euler provides us not only with a close look into the the work of Euler, but also a message that still can resonate with readers today. Learning about the life and work of someone as intricate as Leonhard Euler from the intellectual, yet humorous lense of Dunham was an absolute pleasure and I highly recommend this book to everyone, not just math-minded individuals. Upon a quick flip-through, this book can be daunting in appearance with the frequent presentations of complex mathematical notation, but Dunham does a great job at summarizing and extracting only the most parts and presenting them in a clear manner. Euler was an important figure who gets little credit for many advancements in many fields such as optics, mechanics, and mathematics and Dunham conveys his life and work in a meaningful manner. When studying the variety and depth at which Euler presented his work, the modern reader can extrapolate their own interpretations and messages about the legacy they want to leave on this earth. This work easily could have of been a dense analysis on the work of Euler, but Dunham took it a step further when creating a truly insightful and engaging work that I recommend for everyone.


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