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An Imaginary Tale: The Story of √-1 (Princeton Science Library, 42) Paperback – Box set, March 15, 2016
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Today complex numbers have such widespread practical use--from electrical engineering to aeronautics--that few people would expect the story behind their derivation to be filled with adventure and enigma. In An Imaginary Tale, Paul Nahin tells the 2000-year-old history of one of mathematics' most elusive numbers, the square root of minus one, also known as i. He recreates the baffling mathematical problems that conjured it up, and the colorful characters who tried to solve them.
In 1878, when two brothers stole a mathematical papyrus from the ancient Egyptian burial site in the Valley of Kings, they led scholars to the earliest known occurrence of the square root of a negative number. The papyrus offered a specific numerical example of how to calculate the volume of a truncated square pyramid, which implied the need for i. In the first century, the mathematician-engineer Heron of Alexandria encountered I in a separate project, but fudged the arithmetic; medieval mathematicians stumbled upon the concept while grappling with the meaning of negative numbers, but dismissed their square roots as nonsense. By the time of Descartes, a theoretical use for these elusive square roots--now called "imaginary numbers"--was suspected, but efforts to solve them led to intense, bitter debates. The notorious i finally won acceptance and was put to use in complex analysis and theoretical physics in Napoleonic times.
Addressing readers with both a general and scholarly interest in mathematics, Nahin weaves into this narrative entertaining historical facts and mathematical discussions, including the application of complex numbers and functions to important problems, such as Kepler's laws of planetary motion and ac electrical circuits. This book can be read as an engaging history, almost a biography, of one of the most evasive and pervasive "numbers" in all of mathematics.
- Print length296 pages
- LanguageEnglish
- PublisherPrinceton University Press
- Publication dateMarch 15, 2016
- Dimensions5.5 x 0.62 x 8.5 inches
- ISBN-100691169241
- ISBN-13978-0691169248
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Editorial Reviews
Review
"Honorable Mention for the 1998 Award for Best Professional/Scholarly Book in Mathematics, Association of American Publishers"
"A book-length hymn of praise to the square root of minus one."---Brian Rotman, Times Literary Supplement
"An Imaginary Tale is marvelous reading and hard to put down. Readers will find that Nahin has cleared up many of the mysteries surrounding the use of complex numbers."---Victor J. Katz, Science
"[An Imaginary Tale] can be read for fun and profit by anyone who has taken courses in introductory calculus, plane geometry and trigonometry."---William Thompson, American Scientist
"Someone has finally delivered a definitive history of this 'imaginary' number. . . . A must read for anyone interested in mathematics and its history."---D. S. Larson, Choice
"Attempting to explain imaginary numbers to a non-mathematician can be a frustrating experience. . . . On such occasions, it would be most useful to have a copy of Paul Nahin's excellent book at hand."---A. Rice, Mathematical Gazette
"Imaginary numbers! Threeve! Ninety-fifteen! No, not those kind of imaginary numbers. If you have any interest in where the concept of imaginary numbers comes from, you will be drawn into the wonderful stories of how i was discovered."---Rebecca Russ, Math Horizons
"There will be something of reward in this book for everyone."---R.G. Keesing, Contemporary Physics
"Nahin has given us a fine addition to the family of books about particular numbers. It is interesting to speculate what the next member of the family will be about. Zero? The Euler constant? The square root of two? While we are waiting, we can enjoy An Imaginary Tale."---Ed Sandifer, MAA Online
"Paul Nahin's book is a delightful romp through the development of imaginary numbers."---Robin J. Wilson, London Mathematical Society Newsletter
"You will definitely enjoy it. In fact it clearly reflects the the joy and delight that the author experienced when he was confronted with complex analysis during his engineering studies."---Adhemar Bultheel, European Mathematical Society
Review
From the Back Cover
"Dispelling many common myths about the origin of the mystic 'imaginary' unit, Nahin tells the story of i from a historic as well as human perspective. His enthusiasm and informal style easily catch on to the reader. An Imaginary Tale is a must for anyone curious about the evolution of our number concept."--Eli Maor, author of Trigonometric Delights, e: The Story of a Number, and To Infinity and Beyond
About the Author
Product details
- Publisher : Princeton University Press; Revised edition (March 15, 2016)
- Language : English
- Paperback : 296 pages
- ISBN-10 : 0691169241
- ISBN-13 : 978-0691169248
- Item Weight : 9 ounces
- Dimensions : 5.5 x 0.62 x 8.5 inches
- Best Sellers Rank: #157,811 in Books (See Top 100 in Books)
- #93 in Mathematics History
- #112 in Calculus (Books)
- Customer Reviews:
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About the author

(The image at the right is of the author's wonderful cat, Tigger, who has been rendered senseless after reading the stunning prose of one of the author's books-in-progress. Or, perhaps, and far more likely, he just ate too much for lunch.) 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 23 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. A new book from Princeton, IN PURSUIT OF ZETA-3, appeared in October 2021 (as did the paperback edition of HOW TO FALL SLOWER THAN GRAVITY). The book THE MATHEMATICAL RADIO is under contract with Princeton and is scheduled to appear in late 2023 or early 2024. The book THE PROBABILITY INTEGRAL is under contract with Springer and will appear in late 2024. 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.
FINALLY - numerous readers have written over the years asking about the solutions manual to my Springer book, THE SCIENCE OF RADIO. Springer has kindly made it available in pdf format (3 MB), and if you write to me I'll send you a copy. paul.nahin@unh.edu
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I didn’t care much for the historical aspects nor the the proofs. Although I did find some proofs conceptually useful. I thoroughly enjoyed the chapter on Complex Functions. I was afraid it would be too theoretical given my background but the author did a great job in simplifying it enough. Outstanding. I would have loved to take taken Complex Variables in school after reading this book. Having taken that course would have allowed to have deeper understanding and appreciation for Signals and Systems analysis.
You see, electrical engineers have to know a lot of math (due to Maxwell's equations), and it turns out that complex numbers are an uncannily perfect match. I have never heard of an explanation of, or even a speculation, why there is this correspondence. It also so happens that geometry in a plane, but bafflingly not in three dimensions, has this correspondence. Here we can speculate it has something to do with the norm of a complex number having the same form as the square of the distance of a point from the origin.
This far from exhausts the amazing correspondences for complex numbers, but every one of them to all appearances is a coincidence. It would be a bit much to ask the author to explain this, when no one else has a clue either. But I had thought this book would remedy a lack in my education concerning exactly what, from a mathematical standpoint, a complex number is, or rather what exactly we are assuming. Instead, my educators attempted to persuade me that complex numbers were nothing to worry about, and then proceeded to make claims about them, which were easy enough to memorize. I never disbelieved, or believed, or worried, just so long as I could get the problems right.
Hauling in geometry to demonstrate something about complex numbers, which actually were only defined as being the root of an equation is a logical fallacy. What you would need to show is that complex numbers so defined did indeed have a consequence that they represented something in geometry correctly. But not to a physicist. To a physicist, things just happen to be what they happen to be. You can show something about complex numbers by claiming they are points in plane. Then show something by algebraically multiplying out an expression. Whatever.
Maybe you would say it's wrong to judge a book called "The Story of the Square Root of Minus One" this way. A story could be anything. It's a story, right. But not to portray the main character of the story at all, when in mathematics, as opposed to physics, everything about the entities that you need to know is known, is exasperating to the mathematically inclined.
Anyway, there is plenty of interesting material in itself. A rhapsody or fantasy on i would be more like what it is. It is miscellaneous, and leaves off just where you would like to see it go deeper, like a magician that is satisfied to mystify, and leave you wondering what made it work.
I would not have bought this book knowing what it was. I was carried away by the enthusiasm of the reviewers.
I myself bought it in a search for material to motivate a bright 11-year-old that I am tutoring. I introduced imaginary and complex numbers to him, but all of the actual applications seemed far out of his reach. So now when I mention imaginary numbers he screws up his face and asks for more boolean algebra instead. But with this book, I now have a number of examples and historical anecdotes to motivate and fascinate him, particularly geometric interpretations and applications.
Here, for example, is one extremely elementary application that I did not know about. Prove: the product of two sums of squares is itself the sum of two squares in two different ways. Symbolically, given any integers a, b, c, d, there are integers p, q, r, s with...
(a^2 + b^2)(c^2 + d^2) = p^2 + q^2 = r^2 + s^2
This was demonstrated by mathematicians a long time ago, but not particularly easily. Using complex numbers, it's almost trivial to see, however, certainly within reach of a student of Algebra I. (There's an even simpler version of the proof that Nahin presents, but it's a bit messy to write without properly typeset mathematics.) This also makes the important point that complex numbers are very useful to help understand non-complex mathematical phenomena, a point Nahin makes throughout the book.
This also illustrates that this is a real math book, not simply a popularization piece ~about~ mathematics and mathematicians. It's really too bad that reviewers who expected the latter are downgrading their ratings of the book, because if you understand and accept what it is trying to be, it's a gem!
Much of this material is, of course, available by searching the internet. But it's not easy to find, and of highly variable quality. So Nahin's book is a real service to teachers and students at all levels.
Top reviews from other countries
単なる数学読み物ではない点も魅力の一つ。著者は、” By a little algebra you can see ・・・” などと書いて、読者の数学力をためしに来る。確かにこれは私に解けるはず、というレベルであり、符号や数式展開のミスなど、注意力の散漫さをやっとのことで克服して著者の書いた式に到達するのもまた楽しからずやということになる。数式を使ってきちんと説明しているだけに、高校数学の基礎的なことを理解していないとつらいが、趣味として数学を学ぶには最適の本である。
最後に「この本は英語で読むべきだ」ということを付け加えたい。その理由は、(1)日本語の翻訳のひどさである。翻訳者K氏の本にはひどい目にあってきた。常に誤植や手抜きが多いという程度ではない。誤植の内容から判断すると、数学を理解して翻訳していないのみならず、翻訳を見る限り日本語も変だ。(2)英語で読むことのメリットは、翻訳されていない数学の本も含めアマゾンで安価に手に入ること。英語は負担という人もいるだろうが、数式に英語も日本語もない。数学特有の言い回しを覚えれば小説を原語で読むよりははるかに楽である。
Yes, this book isn't for the faint-hearted, but if you do work at it and work through the maths, it is amazing what you pick up.
The updated paperback book does still have a few typo's and 'missing' values in equations, but that's part of the fun isn't it ? (lol)
I would totally recommend this book to undergraduate and graduate students, and probably a lot of academics too! Loved it. 5/5
Having said all that, it really is a very good book. It is just that I have been spoiled by Eli Maor's books, which cover similar ground (trigonometry, e) in a similar way (history, characters, mathematical ideas, related concepts), but manage to make it an effortless joy for the reader. This book somehow never became a joy to read.
Thebook is fine but Kindle mangles it. MAny of the equations do not come up astext but as images too small to read and impossibel to enlarge. These that do come up in the text frequently do not reproduce the greek characters. I am trying to get my money back. This ort ofthing is a frequent occurrence on Kindle editions of tecnical books . Amazon should issue a health warning.
BBy all means buy the book but get the print edition.
















