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

Paul J. Nahin (Author)

Book Description

Publication Date: April 10, 2006

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."

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. -- othy 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

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

See all Editorial Reviews

Product Details

• Hardcover: 404 pages
• Publisher: Princeton University Press; First Edition edition (April 10, 2006)
• Language: English
• ISBN-10: 0691118221
• ISBN-13: 978-0691118222
• Product Dimensions: 9.2 x 6.6 x 1.3 inches
• Shipping Weight: 12 ounces (View shipping rates and policies)
• Average Customer Review: 4.2 out of 5 stars  See all reviews (20 customer reviews)
• Amazon Best Sellers Rank: #239,916 in Books (See Top 100 in Books)

More About the Author

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). (The lovely lady in the photo is his wife of 49 years, Patricia.) 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 11 books on mathematics and physics, published by IEEE Press, Springer, and the university presses of Johns Hopkins and Princeton. His new book NUMBER-CRUNCHING was published by Princeton in September 2011, and Johns Hopkins recently reprinted his 1997 book TIME TRAVEL FOR WRITERS. Princeton has just released the corrected paperback edition of his 2006 book DR. EULER'S FABULOUS FORMULA. He has just completed (for Princeton) his next book, ELECTRIC LOGIC, that treats the works of George Boole and Claude Shannon and how they created the information age (to be published 2012). Two of his other Princeton math books, CHASES & ESCAPES and DUELLING IDIOTS, are scheduled to be reprinted in 2012, each with a new Preface (the one in CHASES includes an analysis of the B-29 Enola Gay's escape maneuver from the blast wave of the atomic bomb drop on Hiroshima). He has given invited talks on mathematics at Bowdoin College, the Claremont Graduate School, the University of Tennessee, and Caltech, has appeared on National Public Radio's "Science Friday" show, and advised Boston's WGBH Public Television's "Nova" program on the script for their time travel episode. He recently gave the invited Sampson Lectures for 2011 in Mathematics at Bates College (Lewiston, Maine). When he isn't writing he is battling evil-doers on his Xbox360S and, now and then, he even wins. (And he gives a big thumb's up to Valve's terrific PORTAL 2, as well as to the oldie-but-still-goodie original Xbox greats RETURN TO CASTLE WOLFENSTEIN:TIDES OF WAR and THIEF:DEADLY SHADOWS!)

FINALLY - readers have written asking about the solutions manual to THE SCIENCE OF RADIO. I now have the pdf file (3 MB) for the solutions, and if you write to me I'll send you a copy. [e-mail]

Customer Reviews

Most Helpful Customer Reviews

82 of 85 people found the following review helpful:

5.0 out of 5 stars Sequel to An Imaginary Tale, April 26, 2006

By 

T. J. Shortridge "T.J. Shortridge" - See all my reviews
(REAL NAME)   

This review is from: Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills (Hardcover)

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.

40 of 41 people found the following review helpful:

5.0 out of 5 stars Another fabulous book from Paul Nahin, August 29, 2006

By 

David S. Mazel "www.bytesofscience.com" (Washington, DC) - See all my reviews
(REAL NAME)   

Amazon Verified Purchase

This review is from: Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills (Hardcover)

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.

59 of 63 people found the following review helpful:

5.0 out of 5 stars Errata please, February 13, 2007

By 

John W. Fuqua - See all my reviews
(REAL NAME)   

This review is from: Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills (Hardcover)

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"