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

Paul J. Nahin
Pages: 296
https://www.jstor.org/stable/j.ctt7sxsj

1. Front Matter
(pp. i-viii)
(pp. ix-x)
3. List of Illustrations
(pp. xi-xii)
4. Preface to the Paperback Edition
(pp. xiii-xx)
5. Preface
(pp. xxi-2)
6. Introduction
(pp. 3-7)

In 1878 a pair of brothers, the soon-to-become-infamous thieves Ahmed and Mohammed Abd er-Rassul, stumbled upon the ancient Egyptian burial site in the Valley of Kings, at Deir el-Bahri. They quickly had a thriving business going selling stolen relics, one of which was a mathematical papyrus; one of the brothers sold it to the Russian Egyptologist V. S. Golenishchev in 1893, who in turn gave it to the Museum of Fine Arts in Moscow in 1912.¹ There it remained, a mystery until its complete translation in 1930, at which time the scholarly world learned just how mathematically advanced the ancient...

7. CHAPTER ONE The Puzzles of Imaginary Numbers
(pp. 8-30)

At the end of his 1494 bookSumma de Arithmetica, Geometria, Proportioni et Proportionalita, summarizing all the knowledge of that time on arithmetic, algebra (including quadratic equations), and trigonometry, the Franciscan friar Luca Pacioli (circa 1445–1514) made a bold assertion. He declared that the solution of the cubic equation is “as impossible at the present state of science as the quadrature of the circle.” The latter problem had been around in mathematics ever since the time of the Greek mathematician Hippocrates, circa 440 B.C. The quadrature of a circle, the construction by straightedge and compass alone of the square...

8. CHAPTER TWO A First Try at Understanding the Geometry of $\sqrt { - 1}$
(pp. 31-47)

Despite the success of Bombelli in giving formal meaning to$\sqrt { - 1}$when it appeared in the answers given by Cardan’s formula, there still lacked a physical interpretation. Mathematicians of the sixteenth century were very much tied to the Greek tradition of geometry, and they felt uncomfortable with concepts to which they could not give a geometric meaning. This is why, two centuries after Bombelli’sAlgebra, we find Euler writing in hisAlgebraof 1770

All such expressions as$\sqrt { - 1}$,$\sqrt { - 2}$, etc., are consequently impossible or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly...

9. CHAPTER THREE The Puzzles Start to Clear
(pp. 48-83)

More than a hundred years after Wallis’ valiant but flawed attempt to tame complex numbers geometrically, the problem was suddenly and quite undramatically solved by the Norwegian¹ Caspar Wessel (1745–1818). This is both remarkable and, ironically, understandable, when you consider that Wessel was not a professional mathematician but a surveyor. Wessel’s breakthrough on a problem that had stumped a lot of brilliant minds was, in fact, motivated by the practical problems he faced every day in making maps, i.e., by the survey data he regularly encountered in the form of plane and spherical polygons. There was no family tradition...

10. CHAPTER FOUR Using Complex Numbers
(pp. 84-104)

In this chapter, and in the next, I will show you some specific examples or case studies of the application of complex numbers to the solution of interesting problems in mathematics and applied science. Most of the underlying theory in this chapter will be based on the elementary idea that complex numbers can represent vectors, i.e., quantities with magnitude and direction, in the complex plane. Indeed, complex number arithmetic can be interpreted as sequences of vector manipulations.

The addition and subtraction of vectors is routinely taught in high school physics and, as shown in figure 4.1, the ideas are quite...

11. CHAPTER FIVE More Uses of Complex Numbers
(pp. 105-141)

I ended the last chapter in spacetime, and this next example of using complex numbers remains at least a little bit connected with that part of mathematical physics. Watchers of science fiction movies are well acquainted with the idea ofhyperspace wormholesas spacetime shortcuts from one point to another, paths that are traversable in less time than it takes light to make the straight line trip; for example, see the 1994 filmStargate.¹ For the first example of this chapter, I want to show you that mathematicians long ago gave a hint at how such a thing might happen,...

12. CHAPTER SIX Wizard Mathematics
(pp. 142-186)

While it is general practice today to date the beginning of the modern theory of complex numbers from the appearance of Wessel’s paper, it is a fact that many of the particular properties of$\sqrt { - 1}$were understood long before Wessel. The Swiss genius Leonhard Euler (1707–83), for example, knew of the exponential connection to complex numbers. The son of a rural pastor, he originally trained for the ministry at the University of Basel, receiving, at age seventeen, a graduate degree from the Faculty of Theology. Mathematics, however, soon became his life’s passion. He remained a pious man, but there was...

13. CHAPTER SEVEN The Nineteenth Century, Cauchy, and the Beginning of Complex Function Theory
(pp. 187-226)

With the completion of the previous chapter we have really, I think, done pretty much everything we can do with just the imaginary$\sqrt { - 1}$itself, and its extension to complex numbers. To continue on, the next logical step is to considerfunctionsof variables that are complex valued, i. e., functionsf (z )wherez = x + iy. But then the question is, where do I stop? There is, today, an absolutely enormous literature in complex function theory, much of it purely mathematical, and just as much again of a practical, applications-oriented nature. Physicists and engineers, I believe,...

14. APPENDIX A The Fundamental Theorem of Algebra
(pp. 227-229)
15. APPENDIX B The Complex Roots of a Transcendental Equation
(pp. 230-234)
16. APPENDIX C ${\left( {\sqrt { - 1} } \right)^{\left( {\sqrt { - 1} } \right)}}$ to 135 Decimal Places, and How It Was Computed
(pp. 235-237)
17. APPENDIX D Solving Clausen’s Puzzle
(pp. 238-239)
18. APPENDIX E Deriving the Differential Equation for the Phase-Shift Oscillator
(pp. 240-243)
19. APPENDIX F The Value of the Gamma Function on the Critical Line
(pp. 244-246)
20. Notes
(pp. 247-260)
21. Name Index
(pp. 261-264)
22. Subject Index
(pp. 265-268)
23. Acknowledgments
(pp. 269-269)