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.

Copenhagen’s examination in Roman law in 1778—and by 1798 he had risen to a supervisory position. He “retired” in 1805 but continued to work for several more years before rheumatism, a bad ailment for anyone but particularly so for a surveyor, forced him to really stop. He was highly regarded as a surveyor, receiving a silver medal from the Royal Danish Academy of Sciences for the mapping he did for the French government. Still, respected as he was as a surveyor, he was not the most obvious

+ tan −1 (1 / 2 )} i{( nπ / 4 ) + tan −1 (1 / 2 )} un = 2 3n / 2 − 2 5 e +e , but this can be greatly simplified. Using Euler’s identity we have 1 nπ un = 2 3n / 2 − 2 5 2 cos + tan −1 . 4 2 Now, recalling the formula cos(␣ ϩ ) ϭ cos(␣)cos( ) Ϫ sin(␣)sin( ) derived in section 3.1, and observing that 1 2 costan −1 = , 2 5 1 1 sin tan −1 = , 2 5 it then quickly follows that nπ nπ un = 2 3n / 2 −1 2 cos − sin , n = 0, 1, 2,

have the closest approach occur when the orbit crosses the real axis, i.e., we can assume o ϭ 0 with no loss of generality, and I will do that now. Notice that if we are to have a closed, repeating orbit we must always have r finite and so c1c2 /k Ͻ 1, or else r will, for some values of , become arbitrarily large and/or negative. The value of c1c2 /k ϭ E is called the eccentricity of the elliptical orbit, and obviously if E ϭ 0 the ellipse degenerates into a circle. More precisely, the orbit is a

Next, change variables to x ϭ tan(t), which says dx ϭ dt/cos2(t), and so L= tan( θ ) ∫ 0 158 dx . 1 + x2 WIZARD MATHEMATICS ϭ /2, then Since L equals one-fourth of the circle’s circumference when π ∞ dx . = 2 ∫0 1 + x 2 This result is considered elementary today as the indefinite integral is commonly shown in freshman calculus to be tanϪ1(x), and of course tanϪ1(x)͉0ϱ ϭ /2. The weird manipulations I just went through are a way to arrive at the definite integral without having to know

readers up for an “explanation” of the astounding properties of contra-polar energy, the very next sentence makes the following assertion, hilarious now, as I read it decades later, but quite logical to me in 1955: “One of the reasons why atomic energy has not yet become popular among home xxii PREFACE experimenters is that an understanding of its production requires a knowledge of very advanced mathematics.” Just algebra, however, would strip bare contra-polar energy, or so claimed the