le 11 mars 2003
This marvelous book fulfills a long-standing need for a history of how "i" (the square-root-of-minus-one) went from a disreputable construct, to an indispensable tool in the mathematician's toolbox. The author, Paul J. Nahin, is an electrical engineer with an unmistakable flair for mathematics. He is also a good writer who has done his homework. The result is an outstanding book covering an important chapter of mathematical history.
The book has something to offer to a broad cross-section of readers: from bright high-school students, to professional mathematicians, to historians. For the professional mathematician, Nahin offers many arcane tidbits, such as how Euler first summed the reciprocals of the integers-squared. (Such information is usually not found in text books.)
The book is a case study of how important mathematical concepts arrive at maturity. The history of "i" may be divided into six phases: 1) initial recognition of the "impossibility" of taking the square-root of minus one; 2) need to reconsider "i" in connection with the equations for the solution of the cubic (the delFerro-Tartaglia-Cardano equations); 3) Euler introduces the notation "i", and publishes his celebrated formula connecting the circular and exponential functions; 4) Wessel, Argand, and Gauss independently discover the correct geometric interpretation of complex numbers, 5) Cauchy introduces the theory of complex functions, 6) complex numbers are recognized as special instances of abstract fields. The author correctly points out that - contrary to what is taught in introductory courses - the deciding impetus to take "imaginary" numbers seriously came not from quadratic equations, but from cubics.
On a larger scale the book raises a fascinating question: why do some concepts (such as the zero, or "i") produce boundless fruit, while others (e.g., "perfect numbers"), upon final analysis, appear sterile.
le 2 décembre 2002
I loved reading this book. It is exactly what it states that it is, a story of imaginary numbers. A loving story. Imaginary numbers have a facinating history of very slow adoption through the centuries, a history that wonderfully facilitates a certain love and joy of mathematics and better understanding of our struggles as humans to improve ourselves and better understand the language of the physical universe: mathematics.
I did not find this book too tedious at all. Nothing run into the ground at all. If you encounter sections of this book with math too tedious for you, or if you are simply a more casual reader or don't have the time to go deeper, then do as I did, skip those sections. The vast majority of the book is text. The author is a mathematician, so he used mathematical examples, that is all. I assert that the only way to do justice to math history is to include some math.
Understanding imaginary numbers by the broader historical view offered in this book allowed me deeper insight and the ability to see deeper parallels with other areas of matahematics. Just as there were eons where people had no use for negative numbers, but where negative numbers were found convenient for arithmetic operations and so put into common everyday usage, so it goes for imaginary numbers.
One of the reviewers wrote that this book is an excellent introductory treatment of complex analysis. I believe that reviewer to be a mathematician. But I really want to emphasize that this book is unlike any text book that I encountered while learning complex number algebra and engineering usage. This book is great for a fun casual read by any curious person.
There was lots of new and insightful stuff in this book for me. Highly recommended. A fun read.
le 9 septembre 2002
This book will introduce you to complex numbers, complex variables, and complex functions and you _will_ be able to make the journey. You'll need a little familiarity with algebra but, like all these modern mathematical expositories, you can completely grasp the subject with diligence. The hard or clever parts are spelled out for you.
Perhaps there are some typos but I wasn't hampered appreciably by them. Some beautiful and elegant mathematics is exposed very sensitively in this book and with a great appreciation for the chronology and history of the process. The demonstration bears out Hadamard's comment, "The shortest distance two points in the real plane oftens passes through the complex plane."
This book really spurred on my interest in complex variables. The continued study of complex math can take you to some stunning and unexpected connections in mathematics. I encourage interested readers to consider this book as a starting place for that journey.
le 4 septembre 2015
An Imaginary Tale covers the number i from two main perspectives: the historical development of complex numbers, and some interesting proofs and real-world examples involving complex variables.
In the historical sections, Nahin identifies two somewhat distinct routes academics arrived at the concept of i. While some mathematicians were spurred by geometric problems, others found that complex numbers were required to solve certain polynomial equations. In both cases, Nahin includes figures and equations from the original papers of the 1500s through 1700's, adding a high degree of authenticity.
More interesting however are the sections in which Nahin shows the reader a series of neat proofs involving complex numbers, including bizarre infinite series for pi and e, integrals that are unsolvable using only the real numbers, and the evaluation of at-first-glance confusing quantities like i to the ith power. Having just TA'd a course on electronics, I found Nahin's reproduction of the original Hewlett-Packard oscillator circuit solution using complex differential equations to be particularly noteworthy.
As far as popular science books go, Nahin's An Imaginary Tale is pretty good, mainly because (obviously) it doesn't shy away from all the math. Some of the historical sections (especially the geometrical ones) could get fairly dry, but the book overall rates four stars.
le 5 mars 2004
This is a well-written and researched book. The author offers a historical perspective of the development of complex numbers, with very interesting examples. This is not a textbook, but I have found it to be helpful in getting a gut-level understanding of some of the concepts covered in undergraduate complex analysis; it is an excellent supplement to the incomplete text I am currently using. I like the book mostly because it manages to cover a lot of ground in a small amount of pages, and does so without getting bogged down in a lot of details (anyone who has paid attention in freshman calculus/physics will be able to fill in the gaps). The main reasons for my five star rating are: the material on 'wizard mathematics' and complex function theory (chapters 6 and 7), and also the way the author stresses the geometrical interpretation of 'i' as the rotation operator in the plane. In fact, once I had realized what the number 'i' really was, this was the first book in which I actually found the equation 'i = 1 (angle) pi/2'.
I highly recommend this book to anyone who wants to see unique examples of the usefulness of complex numbers from throughout history, and who maybe doesn't like to read textbooks.
le 21 septembre 2003
This was an incredible book. I'm an electrical engineer by degree and a physicist by hobby, so I'm pretty familiar with imaginary numbers. While a lot of the concepts were a review to me, the book also introduced me to a lot of new and fascinating territory. But besides the pure math, it also introduced me to a lot of the history and personalities behind it all. Putting it in perspective and historical context helps breathe new life into it.
I must strongly disagree with the reviewers who said that the math was not rigorous enough, and that the presentation was lacking in personality (two opposite viewpoints).
The style had way more personality than any textbook on mathematics. And anyone with a high-school math background can get through most of the book (not all of it - they may need to skip the bits involving calculus). And whoever says the presentation lacks rigor is missing the point entirely, because this is NOT a textbook and was never meant to be. The author never intended to scare away the casual reader with lenghty proofs - he wants to explain in accessible terms, not alienate.
le 27 avril 2004
In high school and college mathematics courses it is generally stated that, since the square root of -1 cannot be expressed as any real number, it must be a so-called imaginary number, usually designated as i. Furthermore, any number multiplied by i, say 2i, is also said to be imaginary. So-called imaginary numbers generally cause even very bright students some discomfiture, as well they should.
But, in fact, i is not an imaginary number (whatever an imaginary number would mean); rather it is something quite real: a 90 degree rotational operator. Mathematical operators -- including rotational operators -- are beyond the average person's knowledge (or interest) of mathematics, but at least they are real. And they are also quite useful, not only in mathematics but in various fields on science and engineering.
In this fascinating book Nahin traces the history of the centuries-long struggles which the concept of negative numbers and, eventually, of their square roots caused both mathematicians and philosophers until an obscure Norwegian surveyor discovered the true meaning of i in 1797.
As a scientist who spent decades using i -- but never really accepted the traditional view that it is an imaginary number -- I was overjoyed when I finally discovered its real meaning.
Clearly this book is not for everyone; but it should be quite interesting to anyone who, like I, never full accepted the concept of an imaginary number.
le 8 juillet 2002
I purcheased this book in high school and I have read it over and over again as I gain a better understanding of mathematics. Paul J. Nahin provides a great set up of how complex numbers work, how they originated in history, and their current use today. I was amazed at the different applications it was used for and I even let a math teacher I know barrow it to read. I recommomend this book to anyone who wants to see a bit of history about mathematics or anyone who doesn't believe that math is the most perfect human invention.
le 25 mai 2002
Nahin's text on the history of i is an exciting, comprehensive look into the origins of i and its elementary theoretical applications. It rightfully has been compared to Eli Maor's wonderful book "e: The Story of A Number", which deserves five stars in its own right. I do have to take issue with some of the other reviews posted here. For instance, a few have said that you have to have a "graduate math" background to fully appreciate this book?!? Who are they kidding? Nahin actually *sacrifices* mathematical rigor in order to improve his exposition. Anyone with a real mathematics background knows that complex analysis gets far more complicated than the basic material Nahin presents in his book. To get an idea, you can peruse Walter Rudin's fine text "Real and Complex Analysis". To be fair, I agree with the reviewer who wrote that Nahin should not have omitted material on Klein groups, Julia and Mandelbrot sets. However, I can understand why he did. It is difficult to write on such subjects as groups and fractals to an audience intended to have a (motivated) high school or freshman calculus background. I read this book, understood it, and loved it, long before I had any idea what groups or fractals were. Nahin gives fair warning in the introduction to his book that it is not a "mathematical lightweight". I do think that a solid background in (single variable) calculus, including power series, is crucial to a true appreciation of the book. In particular, one must know these things to value the genius of Euler and others in the section on "Wizard Mathematics". Nahin does tread lightly into other topics, such as differential equations and (advanced) algebra, but to say these are a prerequisite to reading the book is ridiculous. I think even if the reader has never encountered ideas such as the Fundamental Theorem of Algebra before, they serve to enrich, not detract from, the material. In any case, the reader should be pleased to see a leisurely treatment of something so blown out of proportion as FTA, as an understanding of it is basic to anything beyond calculus. Proofs of it are rich in variety, ranging from topology to geometry to complex variables (using the theorem of Liouville and properties of entire functions). One criticism that is entirely justified is the typographical errors that regrettably plague the book. In particular, the theorem of Green, relating double integrals to single contour integrals, a result that is surprising and illuminating. However, the careful reader can usually spot and correct such errors, and he or she should be delighted in their own astuteness, rather than blame the author. He does a wonderful job explaining the conceptual basis of i, and I think this overrides any of the books minor flaws. The book does seem to end rather abruptly, however, and I hope that if the author chooses to revise his work, he will expand upon the material, in particular, a (brief?) treatment of the Residue Theorem, the crowning jewel of complex integration. Perhaps even a section on conformal mapping? I do realize though that this may place the book too far out of reach of his intended audience.
The bottom line: if you want a storybook, this is not for you. If you like mathematics, and have a historical bent, this book will satisfy you. Those with a mathematical background will realize that Nahin has the perfect background to write this book: electrical engineers have a *much better* idea of what's going on with complex variables in terms of getting their hands dirty than mathematicians themselves. This is because most mathematicians insist on strict formalism and rigor, but engineers think more freely, and in any case they are the ones that discovered half of the applications of complex variables. E.g., imagine Laplace transforms even existing without Oliver Heaviside, who was thought to be a fool by the mathematical community in his day!
For those that are curious, I only have a B.A. in math, and no graduate education, though I do pursue math study in my free time. So I think I am in a position to make the above arguments.
le 8 décembre 2003
Paul Nahin has taken a text on complex variables and made something of a historical novel out of it. Don't be fooled, this is not for the math weak. I will wager many bought it because of the title (and the super cover art, reminded me of Hesse' Magister Ludi) and it now lies gathering dust somewhere. If you can read it, it is a great story and lucid mathematical development.