Paul Nahin is the author of many books we’ve proudly published over the years, including An Imaginary Tale, Dr. Euler’s Fabulous Formula, and Number Crunching. For today’s installment in our Math Awareness Month series, he writes about his first encounter with √-1.
Electrical Engineering and √-1
It won’t come as a surprise to very many to learn that mathematics is central to electrical engineering. Probably more surprising is that the cornerstone of that mathematical foundation is the mysterious (some even think mystical) square-root of minus one. Every electrical engineer almost surely has a story to tell about their first encounter with √-1, and in this essay I’ll tell you mine.
Lots of different kinds of mathematics have been important in my personal career at different times; in particular, Boolean algebra (when I worked as a digital logic designer), and probability theory (when I wore the label of radar system engineer). But it’s the mathematics of √-1 that has been the most important. My introduction to √-1 came when I was still in high school. In my freshman year (1954) my father gave me the gift of a subscription to a new magazine called Popular Electronics. From it I learned how to read electrical schematics from the projects that appeared in each issue, but my most important lesson came when I opened the April 1955 issue.
It had an article in it about something called contra-polar power: a desk lamp plugged into a contra-polar outlet plug would emit not a cone of light, but a cone of darkness! There was even a photograph of this, and my eyes bugged-out when I saw that: What wondrous science was at work here?, I gasped to myself —I really was a naive 14-year old kid! It was, of course, all a huge editorial joke, along with some nifty photo-retouching, but the lead sentence had me hooked: “One of the reasons why atomic energy has not yet become popular among home experimenters is that an understanding of its production requires knowledge of very advanced mathematics.” Just algebra, however, was all that was required to understand contra-polar power.
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 me (and electrical engineers have actually made sense of it when combined with √-1, but then the editors played a few more clever math tricks and came up with negative resistance. Now, there really is such a thing as negative resistance, and it has long been known by electrical engineers to occur in the operation of electric arcs. Such arcs were used, in the very early, pre-electronic days of radio, to build powerful AM transmitters that could broadcast music and human speech, and not just the on-off telegraph code signals that were all the Marconi transmitters could send. I eventually came to appreciate that the operation of AM/FM radio is impossible to understand, at a deep, theoretical level, without √-1.
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, but that √-1 was important to electrical engineers, and to their ability to construct truly amazing devices. That early, teenage fascination with mathematics in general, and √-1 in particular, was the start of my entire professional life. I wish my dad was still alive, so I could once again thank him for that long-ago subscription.