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.

͙Ϫ1. Indeed, it was that geometrical knowledge that led him on to his next mathematical quest, one that was to obsess him for the rest of his life. Just as ͙Ϫ1 rotates vectors in the complex plane, Hamilton wondered what would rotate vectors in three-dimensional space. This led him to his discovery of quaternions or hypercomplex numbers, a story I will not tell here.13 3.7 Gauss By the time Hamilton published his work on couples the geometric interpretation had already been given the stamp of

millihenrys (500 10Ϫ3 H). The first case means any electrical device that has the property of conducting twenty amperes of current at a particular instant of time if, at that instant, the voltage drop across the device’s terminals is changing at the rate of one million volts per second, and such high rates of change are not at all uncommon in certain electronic circuits for very brief intervals of time. Figure 5.4. The three standard electrical components used in circuits. 127 CHAPTER FIVE

The second case means any electrical device that has the property of producing a one volt potential difference between its terminals if the current through it is changing at the rate of two amperes per second. It is important to notice in figure 5.4 that the current i flows in the direction of the voltage drop, i.e., in the direction of going from the ϩ terminal to the Ϫ terminal. The plus/minus symbology can be confusing, as the symbols do not really mean plus and minus. Rather, the ϩ terminal

by the way, provides the proof to a statement I asked you just to accept back in section 3.2. Thus we have sin( x ) 1 1 = 1 − x2 + x4 . . . x 3! 5! and so, with x ϭ (1/2n) , we have 1 sin⎛ n θ⎞ ⎝2 ⎠ sin( x ) lim = lim = 1, x→0 n →∞ ⎛ 1 ⎞ x θ ⎝ 2n ⎠ as claimed. The power series expansion of ey was used by both Bernoulli and Euler in some breathtaking calculations. In 1697, for example, John Bernoulli used it to evaluate the mysterious-looking integral ͐l0 xxdx. Here’s how he did it. First, using

⎛1 − 1 ⎞ ζ( z ) = 1 + 1 + 1 + 1 + . . . . ⎝ 2 z ⎠ ⎝ 3z ⎠ 5 z 7 z 11z Next, multiply this last result by 1/5z and so on . . . well, you see the pattern now, I’m sure. As we repeat this process over and over, multiplying through our last result by 1/pz, where p denotes successive primes, we relentlessly subtract out all the multiples of the primes. You may recognize this as essentially the technique called Eratosthenes’ sieve, developed by the third century b.c. Greek mathematician Eratosthenes of