The Writing of a New Book

by Paul J. Nahi(The following has been adapted from the Preface of Professor Nahin's new book-in-progress, An Imaginary Tale: the story of, to be published by Princeton University Press. During the fall 1996 semester Professor Nahin will be using the final typescript of the book as the course text in an INCO 404 General Education Freshman Honors Seminar titled "Complex Numbers".)

At the beginning of George Gamow's beautiful little 1947 book of popularized science, One Two Three . . . Infinity, there's the following limerick to give the reader both a flavor of what is coming next, and of the author's playful sense of humor:

"There was a young fellow from Trinity

Who took .

But the number of digits

Gave him the fidgets;

He dropped math and took up Divinity."

An Imaginary Tale is not about the truly monumental task of taking the square root of infinity, but rather of another task that a great many very clever mathematicians of the past thought an even more absurd one that of figuring out the meaning of the square root of minus one.

The matter of is one of great antiquity, and by the end of the book readers will see how the centuries-long search for an understanding of what such a thing could possibly mean has led to the creation of mathematics that is especially beautiful and elegant, and that is so in a discipline in which beauty and elegance are demanded as a routine matter of course. Rather than telling the majestic story of here, however, I want to tell you a much more personal one, a story that explains my fascination with for over forty years.

Long ago, in a year so far in the past (1954) that my life then as a high school freshman now seems like a dream, my father gave me the gift of a subscription to a new magazine called Popular Electronics. He did this because he was a scientist, and his oldest son seemed to have talents in science and mathematics that were in danger of being subverted by the evil (in his conservative opinion) of science fiction. I had, in fact, given him plenty of reason for such concern. I devoured science fiction in those days, you see, often sitting in the kitchen at eleven at night eating a huge sandwich and reading a novel set on Mars a million years in the future. Dad, of course, would have preferred that I be reading a book on algebra or physics.

Being a clever man, he decided not to simply forbid the science fiction, but rather to outflank the science fiction stories by getting me to read technical stories, like the 'Carl-and-Jerry' tales by John Frye that appeared each month in Popular Electronics. Carl and Jerry were two high school electronics whiz-kids (nerds in today's unattractive term) who managed each month to get involved in some exciting adventure in which their technical knowledge saved the day. They were a 1950s amalgamation of the Hardy Boys and Tom Swift. They were teenage 'MacGyvers' long before television discovered the concept (all together, Frye wrote 119 of these stories and each was worth, by itself, the 25 cent cover price of the magazine). My father's plan was to get me to 'identify' with Carl and Jerry, instead of with Robert Heinlein's neurotic time travelers.

Well, Dad's devious plan worked (although I never completely gave up the science fiction), and I got hooked not only on Carl and Jerry but also on the electronic construction projects the magazine featured each issue. I learned how to read electrical schematics from the magazine, whose editors used the same exploded-view, pictorial wiring diagrams that became so well-known to all who ever built an electronic, mail-order HeathKit. I constructed a home workshop in the garage behind the house, and a lot of amazing gadgets were built there (not all of them worked, or at least not in the way the originial designers had intended!) I have many fond memories of the hundreds of hours I spent in that garage during my high school years 1954 to 1958.

My high school was torn down years ago to make way for a huge shopping mall, and my father died in 1990, but my mother still lives in the same house in Brea, California, and the garage with my work bench is also still there. When I visit (twenty years older now than my father was in 1954, and I can hardly believe that even as I write these words), and sit out in the garage by myself for a while, I can almost imagine it's 1954 again and Steven Spielberg is just a seven year old kid somewhere and Elvis is just getting started and . . . well, I digress! (As you might imagine, the original 1985 Back to the Future film had a special meaning for me.)

I built a primitive analog computer in that garage, and some simple digital games (using toggle switches and relays as memory elements), and once scared the absolute living daylights out of both my chemist father and myself when I fired up a 50,000-volt, high-power transformer (not just a puny induction coil) and flashed a hissing, wiggling flame of dragon's breath across the space between two thick electrodes. (I was lucky - to say nothing of my father's luck! - that I didn't vaporize the both of us.) More happily, the magazine taught me how to make a relaxation oscillator out of a 90-volt battery, a resistor, a capacitor, and a cheap Ne-2 neon lamp. I used a soldering iron to melt a bunch of those flashing lights into the branches of a plastic Christmas tree to make a pretty sharp-looking, sparkling display. I have nice memories of how that particular project really wowed all the girls in my high school . . .but, again, I digress!

My greatest success was an 'applause meter' that was used by the judges at the high school talent show one year (a loudspeaker pick-up, an audio amplifier, and a 500 microampere meter wired into the amplifier output was all it really was). But what really had the greatest impact on me wasn't that gadget, or any of the others that I built during my high school years. It was one, in a burst of youthful enthusiasm exceeded only by my enormous ignorance of theory, that I didn't realize is impossible to build.

When the April 1955 issue of Popular Electronics arrived in the mail, one of the inside photographs displayed an incredible sight - a desk lamp emitting not a cone of light, but instead, a cone of darkness! My eyes bugged-out when I saw that. What wondrous science was at work here, I gasped (metaphorically speaking, of course, because what fourteen year old kid do you know, other than in a TV sitcom, who actually talks like that?) The secret, according to the accompanying article, was that the lamp was not plugged into a normal power outlet, but rather into an outlet delivering contra-polar power. Another photograph showed a soldering iron plugged into the contra-polar power outlet it was covered with ice! And another displayed a frozen ice tray on a hot plate, except it was now a cold plate because it was plugged into contra-polar power! I looked at those three photographs and I remember my pulse rate elevated and I felt a momentary spell of faintness. This was simply wonderful! If that was what contra-polar power could do, I wanted contra-polar power!

Well, of course it was all just a huge editorial joke, aided by some nifty photographic retouching. When I showed the article to my father after he came into the room to see what all the excitement was about, he glanced at it and then looked at me with what I now know was a mixture of pity and amusement. Dad wasn't an electrical engineer or a physicist, but with a PhD in chemistry he wasn't totally ignorant of technical matters that fell outside the realm of benzene rings and molecular bonds. He immediately suspected that 'contra-polar power' probably violated perhaps seven different fundamental principles of physics. Rather than laughing at me, however, he simply said, "Son, look at the date on the cover." I had not noticed the 'April' before, or even the sub-title "In keeping with the first day of April", but I quickly understood the significance. I still remember my enormous embarrassment at having been so completely taken in.

In retrospect, I can now see that at that moment I came to a crossroads. I could have reacted to the spoof with anger, and simply have tossed that issue of Popular Electronics (and all that came after) into the trash. Perhaps I would have become an accountant, or a TV weatherman, but certainly not an electrical engineer. Or I could, which I did thank the Lord, have sat right down and read through the spoof carefully to see what it was all about. Like any good spoof, it had lots of tantalizing truths in it, but presented in a slightly goofy way. To give you a sample of the tone of the article, here's a typical passage: "When 'contra-polar energy' is applied to an ordinary table lamp, light is not produced, but taken away, and the area affected by the lamp becomes dark. (Editor's Note: This phenomenon should not be confused with 'black light', so-called, which actually is merely light without any visible elements. As far as the human eye is concerned, 'black light' is equivalent to zero light; the light produced by contra-polar energy might be designated 'negative light', since it subtracts from light already present.)"

To set 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 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 article. Contra-polar power 'worked' by simply using the negative square root (instead of the positive root) in calculating the resonant frequency in a circuit containing both inductance and capacitance. The idea of negative frequency was intriguing to my mind (and I tell readers of my book how electrical engineers have actually made sense out of it when combined with ), but then the editors played a few more clever tricks and came up with negative resistance!

As everybody knows who likes to curl up in a warm bed at night under an electric blanket, or who likes a crunchy piece of toast in the morning, resistors (positive resistors) get hot when conducting a current. 'Obviously', then, a negative resistor should cool down when conducting a current hence the soldering iron and the ice cube tray photographs. (The 'logic', if you can call it that, behind the cone of darkness out of a desk lamp still eludes me, however!) Now, there really is such a thing as negative resistance, and it has long been known by electrical engineers to occur in the operation (under certain specific conditions) of electric arcs. Such arcs were used, for example, in the very early pre-electronic days of radio to build extremely powerful transmitters that were able to broadcast music and human speech, rather than just the on-off telegraph code signals that were all the Hertz and Marconi spark-gap transmitters could send. (My last book, The Science of Radio, AIP Press 1996, tells that incredible story, one that most electrical engineers have never heard of.) Later, in college, I would learn that the operation of radio is impossible to understand, at a deep theoretical level, without an understanding of .

All of this was more fascinating to my young mind than I can tell you, even forty years later (with an increased vocabulary!) It showed me that there were big, exciting ideas in the world of electronics, bigger than I had ever imagined while out in the garage tinkering with twinkling Christmas trees. Two months later, when Forest H. Frantz, Sr.'s article "Planning to be an Electrical Engineer?" appeared in the June 1955 issue of Popular Electronics, I eagerly read it through, several times. And later, when in my high school algebra classes I was introduced to complex numbers as the solutions to certain quadratic equations, I knew (unlike my mostly perplexed classmates) that they were not just part of a sterile intellectual game. I already knew that was important to electrical engineers, and to their ability to construct truly amazing gadgets.

Three and half years after reading about contra-polar power I was sitting in an early morning train out of Los Angeles' Union Station, heading north to Palo Alto on my way to join the Stanford University freshman class of 1958. Over the years that Carl and Jerry appeared in Popular Electronics, the tales chronicled their progression from high school freshmen to electrical engineering students at the fictional "Parvoo University" and, like them, I was taking the first step on the career path I've trod ever since, as an electrical engineer. Once at Stanford I had more than enough to fill my days with reading and so I quickly drifted away from Popular Electronics, but it had been there at just the right time for me; Dad's plan had worked better than he could have possibly hoped. In a certain sense, then, my whole professional life has been the result of my youthful fascination with the mystery of , and that is why I am writing An Imaginary Tale.

And, I must admit, another big reason for the book is simply that I love the history of science and mathematics. My earlier attempt at writing it, a biography of the 19^th century English eccentric Oliver Heaviside, was largely prompted by my wanting to learn more about the often nasty battles he waged with Cambridge mathematicians over the many liberties Heaviside took with his operational calculus when solving electrical problems. The same motivation is at work for An Imaginary Tale. In a letter (dated January 13, 1852) to his English friend Augustus De Morgan, the Irish mathematician William Rowan Hamilton wrote, "I see that either you or I but I hope it will be you must write, some time or other, a history of ". Five days later De Morgan replied, "As to a history of , it would be no small job to do it well from the Hindoos downwards." Well, neither Hamilton or De Morgan ever wrote that history and, as far as I know, nobody else has either. And so that's another reason for why I am writing my book. I simply want to learn more!

My one great regret is that Dad isn't here to read it. But if he were, I hope he would be pleased at the result of his investment in a magazine subscription forty-two years ago.