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Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills First Edition

by Paul Nahin (Author)
4.4 4.4 out of 5 stars 109 ratings
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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."

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  1. ISBN-10
    0691118221
  2. ISBN-13
    978-0691118222
  3. Edition
    First Edition
  4. Publisher
    Princeton University Press
  5. Publication date
    April 30, 2006
  6. Language
    English
  7. Dimensions
    6.25 x 1.25 x 9.5 inches
  8. Print length
    416 pages
  9. See all details
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Editorial Reviews

Review

"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

"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

"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

"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

Review

"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

Product details

  • Publisher ‏ : ‎ Princeton University Press; First Edition (April 30, 2006)
  • Language ‏ : ‎ English
  • Hardcover ‏ : ‎ 416 pages
  • ISBN-10 ‏ : ‎ 0691118221
  • ISBN-13 ‏ : ‎ 978-0691118222
  • Item Weight ‏ : ‎ 1.6 pounds
  • Dimensions ‏ : ‎ 6.25 x 1.25 x 9.5 inches
  • Customer Reviews:
    4.4 4.4 out of 5 stars 109 ratings

About the author

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Paul J. Nahin

Paul J. Nahin

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Paul Nahin was born in California, and did all his schooling there (Brea-Olinda High 1958, Stanford BS 1962, Caltech MS 1963, and - as a Howard Hughes Staff Doctoral Fellow - UC/Irvine PhD 1972, with all degrees in electrical engineering). He worked as a digital logic designer and radar systems engineer in the Southern California aerospace industry until 1971, when he started his academic career. He has taught at Harvey Mudd College, the Naval Postgraduate School, and the Universities of New Hampshire (where he is now emeritus professor of electrical engineering) and Virginia. In between and here-and-there he spent a post-doctoral year at the Naval Research Laboratory, and a summer and a year at the Center for Naval Analyses and the Institute for Defense Analyses as a weapon systems analyst, all in Washington, DC. He has published a couple dozen short science fiction stories in ANALOG, OMNI, and TWILIGHT ZONE magazines, and has written 24 books on mathematics and physics, published by IEEE Press, Springer, and the university presses of Johns Hopkins and Princeton. Translations of his books in Chinese, Russian, Arabic, Greek, Korean, Spanish, Romanian, and Japanese have appeared. The book THE MATHEMATICAL RADIO was published by Princeton in January 2024 and the book THE PROBABILITY INTEGRAL was published by Springer in October 2023. He has given invited talks on mathematics at the Anja Greer Math and Technology Conference at Phillips Exeter Academy (twice, in 2008 and 2018), as well as at Bowdoin College, the Claremont Graduate School, the University of Tennessee, and Caltech, has appeared on National Public Radio's "Science Friday" show (discussing time travel) as well as on New Hampshire Public Radio's "The Front Porch" show (discussing imaginary numbers), and advised Boston's WGBH Public Television's "Nova" program on the script for their time travel episode. He gave the invited Sampson Lectures for 2011 in Mathematics at Bates College (Lewiston, Maine). He received the 2017 Chandler Davis Prize for Excellence in Expository Writing in Mathematics.

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4.4 out of 5 stars
4.4 out of 5
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109 global ratings

Top reviews from the United States

David S. Mazel
5.0 out of 5 stars Another fabulous book from Paul Nahin
Reviewed in the United States on August 30, 2006
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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.
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Bwanaman
5.0 out of 5 stars A Well-Written Grab Bag
Reviewed in the United States on November 19, 2010
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I'm a fan of Prof. Nahin's writing. I think he's one of the best writers on "pop mathematics" for those who actually know a little math. Unlike many pop-sci and pop-math writers pitching to a more general audience who, when a difficult subject comes along, either punt, write airy nothings, or say something WRONG, Nahin explains well and accurately. Enlightening proofs and insights abound. This book has a "big tent" subject: the ubiquity of Euler's formula. And hence, the book is a bit of a "grab bag" with a very loose unity. You could almost read any chapter without reference to the others. But, as usual, it's well-written, full of historical facts and side-stories, and witty. As with most books coming out of modern publishing houses, it's not well proof-read. But there are no glaring inaccuracies. I'd recommend it to anyone with some basic calculus under his belt. It will be particularly enlightening and entertaining to scientists and engineers who have seen most of the math in here, but perhaps never quite as presented here. Even if you have a strong background in Fourier analysis, you'll still see something in a new light between these covers.
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Marc
5.0 out of 5 stars Great study guide for engineering students
Reviewed in the United States on February 5, 2012
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I bought this book when my daughter was home from college, she is an electrical engineering major and minoring in physics. At the time she was starting to learn Fourier, and she paged through this book and it took me 8 months to get it back. This provided an excellent study resource in Fourier series.

My point is that this book is not really a companion to his other book, which is more of a historical text which discusses the math, but is really a popular math book.

This book is a study of simply the math, and it is instructional on how euler is applied from the standpoint of e^piI. So it ends up being an excellent resource on Fourier, which encompasses most of the text. It is useful to study away from the typical textbooks, which are simply using it as a solution, and not as a goal of instruction.

This is not light reading, so it is for math types, and it the author as always writes in a clear style.
6 people found this helpful
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Gerald J. Desantis
5.0 out of 5 stars Superbly Written
Reviewed in the United States on June 20, 2021
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I purchased this book because I love Nahin's clear and logical writing style and how technical material is presented. The book is a continuation of an earlier book (An Imaginary Tale) and like all Nahin books, the topics are extremely well thought out and superbly organized. Additionally, I have always enjoyed reading about the life and mathematical works of Euler who hold in the highest regard.
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P. Toche
4.0 out of 5 stars Great stuff, but what's the point?
Reviewed in the United States on July 20, 2008
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I came to this book because I enjoyed The Story of the Square Root of Minus One, another book by Paul Nahin. This book is of a very different nature: unlike that other book, this one is light on concepts and heavy on calculations.

I enjoyed it quite a bit, however, hence the 4 stars, because I like complicated-looking integrals, but let me be frank I could not help thinking throughout: what's the point? (and do I deserve to be treated to so many typos?)

What's the point? It shows many uses of Euler's formula, but without explaining why we should care. A couple of chapters are devoted to Fourier series and transforms: again, what's the point? Towards the end, Nahin writes something to the effect that he has "avoided giving physical interpretations to the mathematical calculations" and that's precisely the problem: until the end of the book where there are clear references to things like electricity and other waves, we are never told (or reminded) why these clever manipulations are important.

It was shocking not to see any reference to the Riemann hypothesis and zeta function, which are perhaps the most beautiful example of the use of Euler's formula.

To Nahin's credit, he goes through the calculations step by step, so that if you do care (for some reason) then you can follow pretty much the whole thing without breaking a sweat (Nahin did the hard work). But I will confess that I did skip a few pages here and there: my eyes and brains got tired and the nagging thought came, well, what's the point?

Thoroughly recommended, however.
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Dr. T.
5.0 out of 5 stars „When Did Math Become Sexy“ (The Bosten Globe)
Reviewed in Germany on October 5, 2014
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Im Vorwort zitiert Paul. J. Nahin einen Artikel des Boston Globe „When Did Math Become Sexy“, in dem der 'Einzug' echter Mathematik etwa in solche Theater Stücke wie 'Copenhagen' oder 'Proof'' und Filme, etwa 'A Beautiful Mind', beleuchtet wird; die Eulersche Formel dürfte eine ebensolche öffentlich Aufmerksamkeit wert sein.

Mit „Dr, Eulers Fabulous Formula“ setzt der Autor seine Geschichte der komplexen Zahlen – „An Imaginary Tale“ (1998) – fort; zwar ist auch dieser zweite Teil nicht als Lehrbuch gedacht, setzt aber beim Leser einige einfache mathematischen Vorkenntnis (etwa Differential- und Integralrechnung und Lineare Algebra) voraus. Es enthält das fortgeschrittenere Material, das der Autor aus dem ersten Buch aussparen musste, um dessen Umfang nicht zu sprengen.

In diesem Band werden in interessanter Art und Weise, neben den Grundlagen, Anwendungen komplexer Zahlen in der Zahlentheorie und bei der Beschreibung von 'Vector Walks', dem Beweis der Irrationalität von \pi^2; ferner für Fouier Reihen und Integrale, und deren Anwendung in der Elektronik, betrachtet. Das Buch schließt mit einer kurzen Darstellung des Leben und Werks von Leonhard Euler, dem großen Schweizer Mathematiker, der ein Meisters des unbekümmerten Umgangs der Analysis des 'Unendlichen' war; die nach ihm benannte Formel, die diesem Buch den Titel gab, ist dabei nur ein Beispiel seines geschickten Jonglieren mit unendlichen Reihen.

Die für sich genommenen schon höchst faszinierenden und oft trickreichen mathematischen Miniaturen illustrieren das zentrale Thema des Buches: Mathematische Schönheit. Eulers Formel e^i\pi + 1 = 0 ist dafür ein Paradebeispiel, sie setzt die beiden transzendenten Konstanten \pi und e, die aus zwei sehr verschieden mathematischen Gebieten stammen, über die imaginäre Einheit i miteinander in Beziehung. Es ist nicht selten, dass solche unerwarteten Berührungen Ausgangspunkt neuer Erkenntnisse oder sogar Anlass zum Entstehen neuer mathematischer Theorien sind. Schönheit liegt dabei natürlich im Auge des Betrachters, schöne mathematische Beziehungen haben aber – wenn man David Wells folgt – eigne Gemeinsamkeiten: sie sind einfach, kurz, wichtig und überraschend; und insofern ist die Eulersche Formel so eine Art Goldener Standard.

Fazit: Paul. Nahins Buch ist ein beeindruckende Kombination von interessanten Anwendungen komplexer Zahlen in einer Vielzahl von konkreten Beispielen, mit historischen Bezügen und Hintergründen, etwas, das übliche Einführungs- Lehrbücher in der Regel ausklammern,
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W. Scott
5.0 out of 5 stars A Marvellous Writer on Mathematics.
Reviewed in the United Kingdom on July 27, 2014
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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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catenary
5.0 out of 5 stars 本格的数学読み物
Reviewed in Japan on April 1, 2013
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同じ著者によるAn Imaginary Tale The Story of √-１のいわば続編ともいうべきもの。オイラーの公式の証明は出てこない。証明されたものとしてこれを使う数学となっている（物理学者ファインマンが１５歳のころの手書きの数学ノートの写真があり、そこにサイン・コサインの無限級数展開による「証明」が書かれているのみ）。この本をひとことで紹介するとすれば、「オイラーの公式を使って初等数学（大学１・２年次の教養課程数学）では出てこない、複雑ではあるが答えがきれいな積分などの定理を、基本的には初等数学の知識を使って導く楽しさを提供する本」ということになる。知識の積み上げを前提としながら読まねばならない教科書的な本ではなく、各章ごとに独立している。ある章を最後まで理解しないと次の章がまったく理解できない、という構成にはなっていない。ただ、中心になる数学知識はフーリエ変換であり、これはかなり詳しく解説と「証明」が出てきてこの知識は繰り返し用いられ、その後の各章でこれを使って様々な知識を導く構成となっている。
「初等数学の範囲を超える定理」ということは、「初等数学」はこの本を読む知識として必要になる。著者がペーパーバック版に書いた前書きによれば、To read this book you should have a mathematical background equivalent to what a beginning third-year college undergraduate in an engineering or physics program of study would have completed.つまり、理工系大学教養課程修了時点の数学知識が必要ということ。そういう意味ではちょっとハードルが高いかもしれない。加えて、著者は不要な知識とは言っているが、電気（「電子」でない）工学分野に用いられる数学に話が及ぶ。確かに内容は単純で、AMラジオの原理にフーリエ変換と√-1が関係している、その原理的な分野の数学ということなのだが、電気工学分野にアレルギーとは言わないが、あまり興味がない、いわゆる「数学おたく」の者にとっては、数学の難しさにチャレンジする気持がどうしても萎えてしまうため、第６章の最後の約３０ページはかなり難しい（私は途中でギブアップした）。
とはいえ、よくできているいい本だと思う。まず、全体が６つの章で構成され、そのおのおのの章が全体としてあるテーマを追いかけた構成となっている。けっしてバラバラの知識のエッセイ的な本ではないのが「数学のしっかりした本」という読みがいを感じる理由の一つ。そして、途中（４章以降）からはフーリエ変換から導く数式や定理を著者が示す通りにたどっていくとき、結果として驚くようなきれいな定理や数式が出てくるのがもう一つの楽しみ。そして、残念ながら私には感じることはできなかったが、もし電気工学の興味がある者が読むときには、数学の新たな「地平」が見えるであろうということは大いに予想される。
数学知識の前提条件を満たす者にとっては非常に面白い本といえる。
なお、英語は数学を解説する部分については極めてやさしい。最終章のオイラーの伝記部分になると本格的な英語の文章になるので、辞書なしでは読みづらい。
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Kevin Morgan
5.0 out of 5 stars I'f you've studied maths to uni level, great.
Reviewed in the United Kingdom on May 30, 2017
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I've started it. Enjoying it. If you don't have university maths, it'll be very difficult for most. I have uni maths from umpteen years ago and am slowly going through the ideas... but I enjoy that kind of thing. If you've struggled with A level maths, it'll be difficult to follow.
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A. Nonymus
5.0 out of 5 stars Eulers Formula!!
Reviewed in Germany on January 3, 2013
Verified Purchase
Das ist die unglaublichste Formel aller Zeiten, die schönste und unvergesslichste...!!
Unbedingt lesen! Toll, profund, amüsant geschrieben von einem wirklichen Kenner!
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