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Paul J. Nahin
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An Imaginary Tale: The Story of √-1 (Princeton Science Library, 42) Paperback – March 15, 2016

by Paul Nahin (Author)
4.3 4.3 out of 5 stars 287 ratings
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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.

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  1. Print length
    296 pages
  2. Language
    English
  3. Publisher
    Princeton University Press
  4. Publication date
    March 15, 2016
  5. Dimensions
    5.4 x 0.9 x 8.4 inches
  6. ISBN-10
    0691169241
  7. ISBN-13
    978-0691169248
  8. See all details
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Editorial Reviews

Review

"One of Choice's Outstanding Academic Titles for 1999"

"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

"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

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 ‏ : ‎ 5.4 x 0.9 x 8.4 inches
  • Customer Reviews:
    4.3 4.3 out of 5 stars 287 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.

Customer reviews

4.3 out of 5 stars
4.3 out of 5
287 global ratings
I’m enjoying this book (plus a warning).
5 Stars
I’m enjoying this book (plus a warning).
This book is a really informative history of the number i. Though I’m going to have to warn you first before reading this book because you need to know at least multi-variable calculus level math to understand what the author is saying; and maybe some linear algebra and differential equations because it contains many proofs as well as some exercises the author puts at the end of the book.
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Top reviews from the United States

Estefania Silva
5.0 out of 5 stars Nothing at all imaginary about i
Reviewed in the United States on May 28, 2023
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As someone who has a BS in BioMedical Engineering; I found this book quite insightful in regards to some of the coursework I took in school. It allowed me to better understand the Fourier Transform, Bold Plots (Signals) and Series solutions to differential equations (singularity problems). Additionally, I very much enjoyed the chapter on Circuits.
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.
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Jon
4.0 out of 5 stars Welcome reading I'm an interesting path through history. the appendices need editorial review.
Reviewed in the United States on April 14, 2024
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I enjoy this book and 4 stars covers that sentiment. Two things that I would have like to see improvement are, firstly, some more care in formulation: showing a bit more work; but that also reflects some of my lack of experience. Secondly, the appendices are helpful, but I feel editorial and that the slight against Bell in appendix A was based on an uncharitble, if not stark misinterpretation, of his comments about Leibniz
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David J. Lewis
5.0 out of 5 stars Perfect enrichment for math students & teachers
Reviewed in the United States on July 21, 2008
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Teaching mathematics is often an uphill battle against the forces of abstraction and dullness. This delightful book is a perfect antidote, weaving as it does the history, applications and actual mathematics surrounding the concept of "imaginary" and "complex" numbers. But don't get the wrong expectation -- it's a real math book, with equations, proofs, etc, varying in level from high-school algebra and geometry to college calculus and physics.

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.
8 people found this helpful
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Roy E. Kimbrell
5.0 out of 5 stars Thank you Amazon!
Reviewed in the United States on April 7, 2009
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Below you'll see the initial review I had written of the Kindle edition of this wonderful book. I had been sorely disappointed in its formatting (as you might see if you read what I had written). I got a notice a couple of days ago from Amazon telling me that they had reformatted the book to correct the errors. I've started rereading the book, and I am delighted! All of the magic is there. I am not a mathematician (nor do I play one on TV - old joke), but I do like math. And I do like this book. Paul Nahin tells an engrossing history of imaginary numbers, which by itself is very cool. And then he shows us all the ways it is used today. You do need a little understanding of math to really get what he says: some high school math. A little freshman college calculus wouldn't hurt as well. If you're like me, someone who likes math but isn't really a mathematician, you'll love this book.

 An Imaginary Tale: The Story of i [the square root of minus one ]

This delightful book, full of really cool mathematics, was ruined by Amazon when they scanned it for distribution to Kindle users. I believe they used an optical character reader to scan a hard-copy of the book, and then didn't bother to check what they had. Quality control of this product is egregious! There are errors on nearly every page: large letter V's in the place of square root radicals. wildly varying character sizes when reproducing mathematical notation, missing parts of equations (for example, primes missing from functions indicating second derivatives), all exponents tagged as footnotes, and more.

All of this makes reading the book a chore - when it could be enjoyable and instructive.

Shame on you Amazon!
23 people found this helpful
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Top reviews from other countries

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lim_bus
3.0 out of 5 stars no puedo opinar, me censuran los censores de amazon.es
Reviewed in Spain on November 9, 2019
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no puedo opinar, me censuran los censores de amazon.es
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Matthewie0pi
5.0 out of 5 stars The explanation of the i (or j) operator.
Reviewed in Canada on October 22, 2016
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Arrived in excellent condition. The book is accessible to those with a high school mathematics education.
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A user
5.0 out of 5 stars Well researched, detailed
Reviewed in Germany on October 2, 2015
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This is, of course, not just a history of i ( = sqrt(-1) ), but of many consequences that the discoveries of its properties have engendered.
Nahin's text is easily readable, and the step-by-step derivations of remarkable algebraic identities is well don. This implies, however, that the reader has experience (and knows how to) read texts that include mathematical exposes and mathematics as part of the text. If not, the text will probably be overwhelming. That said: there should be books like Nahin's, and his is an excellent one about the topic.
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Vilas P. Bokil
4.0 out of 5 stars It is great that such titles on Basics are Available...!
Reviewed in India on April 8, 2014
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It is nice to see a beautifully presented actual view of an "Imaginary thing")... Sq rt of -1.
I am an mathematis amatuer interested in basics, basically a Diploma & B.E Mech ,passed out with Ranking 50 years ago
Searched throgh the Book store .I want to buy some titles which are available at pracically throw away prices ,on amezon.com but by sea-mail since I do not mind sacrificing time ,the present shipment rates are too much to bear.
Thanks a lot. Look forward to more purchases through you. Best Wishes!
2 people found this helpful
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Peter
5.0 out of 5 stars Excellent book but beware of the Kindle edition
Reviewed in the United Kingdom on January 10, 2011
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As with most of the other reviews I found this book very interesting with lots of intriguing detail and no desire to avoid the maths.

Nahin's book "Dr Euler's Fabulous Formula" can be similarly recommended.

A word of warning. I made the mistake of buying the Kindle edition which is poorly typeset and has very low quality graphics used to render the equations. Typical of examples I have found so far are a multiplication sign replaced by a minus sign, square root symbol replaced by a "V" and so on. All of this interferes with the flow of thought while reading and I am reasonably sure does not exist in the paper version.

The 5 star rating is for the author's intended version of the book and certainly not for the Kindle production of it.
12 people found this helpful
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