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

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.

Nahin is a professor of electrical engineering at the University of New Hampshire; he has also written a number of science fiction short stories. His style is far more lively and humane than a mathematics textbook while covering much of the same ground. Readers will end up with a good sense for the mathematics of i and for its applications in physics and engineering. --Mary Ellen Curtin

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All Customer Reviews
Average Customer Review:
0 of 0 people found the following review helpful:

The square root of minus one and the complex plane, 2008-10-22
As electrical engineer and after reading the excellent reviews above, I just can add that

An Imaginary Tale the Story of the sqr(-1),

by professor Nahin, is really a historical proof of how difficult is the evolution of truth, even in science and mathematics, and in this case I mean the complex plane. Without it, for sure, complex numbers would not have the acceptance and applications they have today.

Argand, Buée and Wessel they all recognized the

"geometric understanding of the sqr(-1)"

as it is

"the sign of perpendicularity"",

and professor Nahin wrote:

"This is so important a statement that it is the only mathematical expression in the entire book that I have enclosed"pag53.

Let me add, please that, when dealing with the electromagnetic field, its wave nature, the rotation concept or its correspondent 90 degrees concept are so inherent to it, that we can understand, why complex numbers are for real and are so essential for understanding and representing the physical world, and why it so is important to understand why the sqr(-1) is called the rotation operator, as is recalled by professor Nahin.

This book, the same as Oliver Heaviside is another must-read for all those interested in the evolution of science and physics, which normally have not evolved by what we call today mainstream.



0 of 0 people found the following review helpful:

A Wonderful Book, 2008-10-19
This is a wonderful book for those who have studied at least some Maths at a university level. One enjoys delightful and elegant maths without the plodding drudgery of the usual textbooks. Adding to the fun is the historical context in which the results are set.


1 of 1 people found the following review helpful:

First-Rate Introduction to complex numbers, 2008-08-05
This is a first-rate introduction to complex numbers. Little background is required to start reading, though a decent high-school background will help.

If you ever wondered what complex numbers were for, if you have forgotten why the number i was created, if you want to learn the art of equation solving, if you are curious about Gauss and Euler, this is the book.

I've read several other books by Paul Nahin, and this one is among his finest.


1 of 1 people found the following review helpful:

Perfect enrichment for math students & teachers, 2008-07-21
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.


1 of 1 people found the following review helpful:

Wonderful book -- very highly recommend it, 2008-05-02
A fantastic resource for anyone who has an inclination to learn math with history of how it really developed. I truly felt sorry that I didn't have this book when I was learning Trignometry in high school -- would have used De Moivre's theorem to derive the interesting identities without having to resort to painful coordinate geometry proofs.

ps: this book is not bedtime reading

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