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An Imaginary Tale: The Story of \(\sqrt{-1}\)

Paul J. Nahin
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Ed Sandifer
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People used to write about families of numbers. They would write about Number Theory, or about Complex Variables. The cast of characters in these books is so large that only a few numbers, usually 0 and 1, ever assume individual identities.

Recently, though, there have been a few books about individual numbers. Some years ago, there was a book about the golden ratio. Other than that, the genre was empty until in the last few years there have been well received books about pi and e. With the newest element of the sequence, electrical engineering professor Paul Nahin breaks the confines of the real numbers and devotes an entire book to \(\sqrt{-1}\).

The book follows a roughly historical trail, opening with a story of how ancient mathematicians Heron and Diophantus missed a chance to discover imaginary numbers. Both knew a formula for the volume of a truncated square pyramid in terms of the sides of the upper and lower square surfaces and the length of the edge connecting those sides. In one of his examples, Heron picked a length, 15, for the edge that wasn't long enough to reach the corners of the squares, of sides 28 and 4. Perhaps he was teaching too many classes that semester, for when Heron reached a point where he was to take the square root of a negative value, he just took the root of a positive instead, and thus missed a chance to discover imaginary numbers. It took over a thousand years until del Ferro and Cardano actually made the discovery in their pursuit of roots of cubic equations. Nahin tells this familiar story with delightful enthusiasm.

It is one of the only widely familiar stories Nahin relates. Most of his other stories are rare gems. For example, he gives us considerable detail of the discovery in 1797 by Norwegian cartographer Caspar Wessel of the now familiar geometric description of the complex numbers as a plane. This was a considerable improvement on Wallis' obscure characterizations involving perpendicularity, a story also well related by Nahin. Nahin also shows us a fascinating way to find the complex roots of a cubic or quadratic equation using a graph and a straight edge.

Nahin's willingness to use calculators is refreshing. Early on, he urges the reader to use a calculator to evaluate a nasty expression involving radicals and arc cosines, to find that the value really is close to 4, as promised. Later, he uses a calculator analogy to explain how the multivalued nature of the arctangent function complicates converting between the Cartesian form and the polar form of complex numbers.

As we get deeper into the book, the difficulty of the material increases considerably. Readers who found the history and geometry a little non-technical for their taste will revel in the details and nuances of convergence theorems and contour integrals that fill the second half of the book.

We get a slick application of complex variables to orbital mechanics. There is a remarkably short explanation of why planets sometimes appear to be travelling the "wrong" direction and how the inverse square law of gravitation is related to Kepler's laws and elliptical orbits.

On page 107, we see a clever application of geometric series of complex numbers. Take a step in any direction. Turn 45 degrees, and take half a step. Continue turning 45 degrees, and taking each step half as long as the previous one. Where do you end up? A moment's reflection will show that the problem is an infinite series of complex numbers, with ratio \(r=\frac12 e^{it}\), \(t=45\) degrees. A few weeks ago, I used this example with great success before an audience of high school precalculus students.

We learn Kirchoff's laws for electrical circuits, and how Euler's ideas about Riemann's zeta function are illuminated by complex variables. We learn about complex exponents and about Green's theorem. It is quite an exposition of the virtuosity of the complex plane.

Like complex numbers themselves, this book has two parts. The first half of Nahin's book is a pleasant and anecdotal introduction to complex numbers, full of ideas and stories that are seldom seen in the popular literature. The second half requires a good deal more concentration, and, appropriately, leads us into some greater complexities.

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 ( is professor of mathematics at Western Connecticut State University and has run the Boston Marathon 27 times.


List of Illustrations  
  Introduction 3
Ch. 1 The Puzzles of Imaginary Numbers 8
Ch. 2 A First Try at Understanding the Geometry of [the square root of] -1 31
Ch. 3 The Puzzles Start to Clear 48
Ch. 4 Using Complex Numbers 84
Ch. 5 More Uses of Complex Numbers 105
Ch. 6 Wizard Mathematics 142
Ch. 7 The Nineteenth Century, Cauchy, and the Beginning of Complex Function Theory 187
App. A The Fundamental Theorem of Algebra 227
App. B The Complex Roots of a Transcendental Equation 230
App. C ([the square root of] -1)[superscript [square root of] -1] to 135 Decimal Places, and How It Was Computed 235
  Notes 239
  Name Index 251
  Subject Index 255
  Acknowledgments 259