An Imaginary Tale: The Story of i [the square root of minus one] (Princeton Library Science Edition) [Paperback]

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.

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.

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.