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Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills [Paperback]

Paul J. Nahin
3.8 out of 5 stars  See all reviews (4 customer reviews)
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Book Description

15 May 2011 0691150370 978-0691150376 Reprint

I used to think math was no fun

'Cause I couldn't see how it was done

Now Euler's my hero

For I now see why zero

Equals e[pi] i+1

--Paul Nahin, electrical engineer

In the mid-eighteenth century, Swiss-born mathematician Leonhard Euler developed a formula so innovative and complex that it continues to inspire research, discussion, and even the occasional limerick. Dr. Euler's Fabulous Formula shares the fascinating story of this groundbreaking formula--long regarded as the gold standard for mathematical beauty--and shows why it still lies at the heart of complex number theory.

This book is the sequel to Paul Nahin's An Imaginary Tale: The Story of I [the square root of -1], which chronicled the events leading up to the discovery of one of mathematics' most elusive numbers, the square root of minus one. Unlike the earlier book, which devoted a significant amount of space to the historical development of complex numbers, Dr. Euler begins with discussions of many sophisticated applications of complex numbers in pure and applied mathematics, and to electronic technology. The topics covered span a huge range, from a never-before-told tale of an encounter between the famous mathematician G. H. Hardy and the physicist Arthur Schuster, to a discussion of the theoretical basis for single-sideband AM radio, to the design of chase-and-escape problems.

The book is accessible to any reader with the equivalent of the first two years of college mathematics (calculus and differential equations), and it promises to inspire new applications for years to come. Or as Nahin writes in the book's preface: To mathematicians ten thousand years hence, "Euler's formula will still be beautiful and stunning and untarnished by time."

Frequently Bought Together

Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills + An Imaginary Tale: The Story of i [the square root of minus one] (Princeton Library Science Edition) (Princeton Science Library) + Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
Price For All Three: 28.24

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Product details

  • Paperback: 416 pages
  • Publisher: Princeton University Press; Reprint edition (15 May 2011)
  • Language: English
  • ISBN-10: 0691150370
  • ISBN-13: 978-0691150376
  • Product Dimensions: 23.4 x 15.5 x 2.7 cm
  • Average Customer Review: 3.8 out of 5 stars  See all reviews (4 customer reviews)
  • Amazon Bestsellers Rank: 397,243 in Books (See Top 100 in Books)

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"Nahin includes gems from all over mathematics, ranging from engineering applications to beautiful pure-mathematical identities. Most of his topics lie just beyond the periphery of a typical mathematics course: they are facts, such as the irrationality of pi, that you may have heard of but never had explained in detail. It would be good to have more books like this."--Timothy Gowers,Nature

"Nahin's tale of the formula e[pi] i+1=0, which links five of the most important numbers in mathematics, is remarkable. With a plethora of historical and anecdotal material and a knack for linking events and facts, he gives the reader a strong sense of what drove mathematicians like Euler."--Matthew Killeya, New Scientist

"It is very difficult to sum up the greatness of Euler. . . . This excellent book goes a long way to explaining the kind of mathematician he really was."--Mathematics Today

"What a treasure of a book this is! This is the fourth enthusiastic, informative, and delightful book Paul Nahin has written about the beauties of various areas of mathematics. . . . This book is a marvelous tribute to Euler's genius and those who built upon it and would make a great present for students of mathematics, physics, and engineering and their professors. Paul Nahin's name has been added to my list of those with whom I wouldn't mind being stranded on a desert island--not only would he be informative and entertaining, but he would probably be able to rig a signaling device from sea water and materials strewn along the beach."--Henry Ricardo, MAA Reviews

"The heart and soul of the book are the final three chapters on Fourier series, Fourier integrals, and related engineering. One can recommend them to all applied math students for their historical development and sensible content."--Robert E. O'Malley, Jr., SIAM Review

"The author conducts a fascinating tour through pure and applied mathematics, physics, and engineering, from the ethereal heights of number theory to the earthiness of constructing speech scramblers. . . . [T]his is a marvelous book that will illuminate the mathematical landscape of complex numbers and their many applications."--Henry Ricardo, Mathematics Teacher

"This is a book for mathematicians who enjoy historically motivated mathematical explanations on a high mathematical level."--Eberhard Knobloch, Mathematical Reviews

"It is a 'popular' book, written for a general reader with some mathematical background equivalent to a first-year undergraduate course in the UK."--Robin Wilson, London Mathematical Society Newsletter

From the Inside Flap

"If you ever wondered about the beauties and powers of mathematics, this book is a treasure trove. Paul Nahin uses Euler's formula as the magic key to unlock a wealth of surprising consequences, ranging from number theory to electronics, presented clearly, carefully, and with verve."--Peter Pesic, St. John's College

"The range and variety of topics covered here is impressive. I found many little gems that I have never seen before in books of this type. Moreover, the writing is lively and enthusiastic and the book is highly readable."--Des Higham, University of Strathclyde, Glasgow

--This text refers to the Hardcover edition.

Inside This Book (Learn More)
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Front Cover | Copyright | Table of Contents | Excerpt | Index
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Customer Reviews

3.8 out of 5 stars
3.8 out of 5 stars
Most Helpful Customer Reviews
13 of 13 people found the following review helpful
4.0 out of 5 stars not for the fearful 14 Aug 2007
Every writer of 'popular' books on math must make a difficult decision about the balance between formulas and stories. Nahin has chosen to take the formulas serious and to reduce the story to witty comments and historical facts about mathematicians accompanying the hard stuff. I must admit it was a bit too daunting for me. So I skipped a lot of the formal treatment and read what I could follow. I finally got an idea why a Fourier transform is so important and for the first time in my life I understood why a Maclaurin or Taylor series is useful and how they came into life.
My friends, who were with me during a week of vacation in the South of France, think I'm raving mad to read this kind of stuff while they were beside the swimming pool. But at least, I won't get melanoma! Thanks Paul!
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5.0 out of 5 stars A Marvellous Writer on Mathematics. 27 July 2014
Format:Paperback|Verified Purchase
I have not read all of this yet, only a part of it. That it is by an engineer would not appeal to the snobbish, the disciples of GH Hardy, for example (perhaps). It is so clearly brilliant! Formulas and proofs are what mathematics is about. They seem within my grasp wherever I open the book. I know that time spent here will be well spent.
How interesting that Euler could recite the whole of the Aeneid. So could Prof AJ Aitken of Edinburgh, my first teacher there. Now I see why he bothered. I do not much care for it myself. And why Prof John Conway of Princeton could recite pie to 500 decimals (and more!) like Aitken. It is all homage to Euler (well, mostly).
I have found the book very clear and it is full of wonders and very accessible. I am greatly indebted to Paul Nahin. He has written something very important. He is an enthusiast and a scholar who can explain anything clearly. He is, for example, in a different league altogether from someone like Prof Stewart of Warwick. Imagine if I had read this before going up? It is miles better than Hardy's book. My best students would have been devouring it before they went up had it been available then.
This is a very well published book by Princeton with a beautiful cover.
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4.0 out of 5 stars A stimulating read, but rather disjointed. 24 April 2013
By Mr R
Nahin's style of writing is gripping, lucid and exciting: he brings the subject to life and gives it real vibrancy, so one can not fail to enjoy this book. But it requires a fair amount of mathematical background if it is to be read fluently, in particular with the handling of Fourier Transforms, convolutions etc.
I was surprised by the detailed handling of the Gibbs phenomenon, and the fact that it should by rights be called the Wilbraham phenomenon. Nahin writes as though Wilbrahim is a new addition to the mathematical firmament, but he isn't. I was able to reference the matter straight away in 2 of my own books (Grattan-Guinness, From Calculus to Set Theory, p129; and G H Hardy, Divergent Series, p20).
Nahin's applied mathematical leanings are apparent from his choice of chapter headings (Vector Trips, and Electronics). I should have liked to see something about the Cauchy-Riemann equations and conformal mappings: I'm sure that Nahin would work wonders on the possibilities here.
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0 of 2 people found the following review helpful
2.0 out of 5 stars Disappointing. Full of formulas 17 Jun 2013
As another reviewer has pointed out this book is a long series of formulas and equations. I had hoped for more description and explanation along the way, but the author thinks that everyone will fully understand any formula he uses, so the book is little more than a long series of formulas one after the other. The various different topics covered are all lists of formulas as he transforms an initial formula into another one, and so on, showing that some "classic" formula or other is true.

I've read a number of other maths books for the ordinary person on e and pi and the golden ratio, and had no problems and enjoyed the discussions of how those numbers crop up in other aspects of life, and how they can be used in various ways. There is none of that here. I think in the Introduction he states something about the maths being enough in itself, so he does not need to add additional explanation or examples of practical uses.

The maths itself within the book may well be correct, but I could not be bothered trying to follow it all. And it is some pretty advanced stuff, though the author claims anyone who has done any advanced level maths courses would be okay. There is no introduction to any maths techniques he uses - he just dives in and gets on with it. I was shocked by the use of matrices early on within the first chapter, when he is supposed to be still discussing just "numbers". He throws in a lot of other maths stuff in that first chapter, which just made my head spin. If that was just the first chapter, then what would the others be like? Too much for me, that is certain.
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Amazon.com: 4.3 out of 5 stars  28 reviews
87 of 90 people found the following review helpful
5.0 out of 5 stars Sequel to An Imaginary Tale 27 April 2006
By T. J. Shortridge - Published on Amazon.com
The reviews of An Imaginary Tale capture much of what will be said of Dr. Euler's Fabulous Formula. I happen to like Paul Nahin's books very much ever since reading The Science of Radio, one of my favorite books of all time. If you didn't like Imaginary, you won't like Dr. Euler's . If you like the earlier book, this one is a must.

Chapter One starts with an introduction to complex numbers. This would make nice supplemental material for an introduction to complex numbers. The chapter is not the standard treatment. It gives a very clear introduction to Gauss' proof of the construction of the regular heptadecagon . The chapter goes on to factoring complex numbers in the context of Fermat's last theorem, with a very clear discussion of Lame's proof for n=7 . Earlier in the chapter Nahin uses the Cayley-Hamilton theorem to get De Moivre's theorem in matrix form without any mention of physical rotations.

Fourier series and integrals comprise most of the book which ends with applications to single side band radio. This last topic is a nice inclusion for folks like me who liked Nahin's early book The Science of Radio. There is a story about G.H. Hardy and Arthur Schuster, that I had never seen elsewhere.

I would recommend this book to anyone who likes undergraduate calculus and has some exposure to linear algebra, maybe a second or third year undergraduate. The material is idiosyncratic enough to be entertaining for anyone who has had courses in complex analysis and number theory. It is a good introduction and supplemental reading for such courses, but not as a primary text.
43 of 44 people found the following review helpful
5.0 out of 5 stars Another fabulous book from Paul Nahin 30 Aug 2006
By David S. Mazel - Published on Amazon.com
Format:Hardcover|Verified Purchase
Here is a book that is a delight to read. It is well-written and the text flows marvelously between each page and around the many formulas that are so carefully presented and worked out. I rate this book as 5-stars for presenting ever more mathematics relating to complex numbers in a clear and detailed manner.

The book is, as the author notes, a continuation of his book, An Imaginary Tale, where Nahin discusses the square root of -1. (If you haven't read that book, read it first because many of the footnotes refer to it.) In this book, we see more of complex numbers and, in particular, we see many applications of Euler's Identity that "e^{i theta} = cos(theta)+ i sin(theta)." This simple looking indentity is rich in applications and explorations. Nahin takes you on a journey to these topics and does so in an easy to follow way.

There are interesting stories as you go such as the one where we find the Gibbs did not, contrary to almost all textbooks, discover what is call Gibbs Phenomena. There are other stories and anecdotes but I'll let you enjoy them on your own.

That said, I must also say that the book assumes you have a good understanding of complex numbers and are comfortable manipulating them. A solid undergraduate understanding is all that's needed and if you have done graduate work, all the better. If you're considering the book at all, and have the math background, read it.

If you don't know anything about complex numbers, well, this book may not be as good as it could be for you.
67 of 72 people found the following review helpful
5.0 out of 5 stars Errata please 13 Feb 2007
By John W. Fuqua - Published on Amazon.com
Like all of Paul Nahin's books, I really like this one.

However, as with so many books an Errata would help. Mathematical and mathematical finance books are getting so expensive, that unless authors or publishers have a URL for Errata, readers esp. of mathematical books will wait for [sometimes years] for a second corrected edition of books.

I could be wrong about these but it seems these are typos:

p. 30 lines 5 & 6 curly bracket should only be around the 2 * cos(x/2) term

p. 121 second equation should be t=(v+u)/(2*c)

p. 121 '* (1/(2*c)' missing at end of the line

p. 123 line 17, first word should be 'bother' not 'other'

p. 127 line 3 and 4, it seems that the 'icnPI/l' [not the ones in the cos() or sin() terms] term after the 'B' and before the '2*cos' respectively, should not be there. Or am I missing something ?

p. 128 4th line from bottom should be 1753 not 1733

p. 143 2nd line before last equation should be '... (x- i * y)...'

p. 144 equation under 'In summary, then...' cases are reversed

p. 216 seems 1/(2*PI) is missing from right side of first equation, i.e. from "...G(u)G(omega-u)...du"
11 of 11 people found the following review helpful
5.0 out of 5 stars Excellent expository book 24 Mar 2007
By Aristarchus - Published on Amazon.com
Paul Nahin's book, "Dr. Euler's Fabulous Formula," is an excellent expository treatment of Euler's formula (you say, "which one?") e^i*theta = cos(theta) + i*sin(theta) and its profound, and far-reaching, ramifications. Dr. Nahin also gives an extensive informal discussion of Fourier series, Fourier transforms, the Dirac Delta Function, and what electrical engineers would call "signals and systems theory." Some mathematical purists may criticize the lack of pure rigor. However, this book is an "expository" book, not a rigorous "textbook." Ideally, I recommend that you read Dr. Nahin's book in conjunction with your standard college textbook. That way, you will get the best of both worlds. Your textbook will give you the disciplined rigor. Dr. Nahin's book will give you the "Aha... insight!" I read Dr. Nahin's book before taking a graduate level course in electrical engineering (EE) Signals and Systems. I breezed through the EE course with perfect scores on my exams, and I give a lot of credit to Dr. Nahin. When you study mathematics, you really need BOTH disciplined mathematical rigor AND intuitive insight and understanding. Beware, however, that this book has LOTS of mathematics in it. The book is loaded with serious mathematics. Don't read this book if you want something for the intelligent layperson. Read this book if you love mathematics, if you are an engineering or mathematics student, or if you like industrial-strength mathematics. Paul Nahin may single-handedly save Americans from mathematical illiteracy. He does something that the mathematical community does not do well... "market and sell" mathematics.
9 of 9 people found the following review helpful
4.0 out of 5 stars excellent for fourrier series and fourrier transform exposition 29 Mar 2007
By Arzi - Published on Amazon.com
A very readable book. Many concepts developed around Euler's magic formula are clearly explained. Including a lucid exposition on the calculus of the sum of classical series such as the value of zeta function for several positive integer values of its argument. Paul Nahin excels in describing the origin and the development of fourrier series and fourrier integrals from Bernoulli to Fourrier and more. Anyone interested in this field will find something interesting in this book to learn. The reason I didn't rank it five stars is that I found explanations often too lengthy while the addition of a chapter on distribution theory could fill the gaps in mathematical rigor and make the transition from fourrier series to fourrier integrals more logical. I should add that the lack of rigor in transition from fourrier series to fourrier integrals, as described by P. Nahin, is inherent to the more fundamental problem of transition from discrete to continuous. Indeed, in mathematics, this is a very slippery terrain. In functional analysis, mathematicians go round this problem by introducing distribution theory. P. Nahin mentions only the name of distribution theory without any decription. I think a chapter on this theory would make the book a must have.
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