Math Drives Careers: Paul Nahin on Electrical Engineering and √-1

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 power scan

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.

In Celebration of Mathematicians

This week San Diego, California is home to the largest mathematics meeting in the world. Hosted by the Mathematical Association of America (MAA) and the American Mathematical Society (AMS), the 2013 Joint Mathematics Meeting is more than just panels and presentations—it is a mass gathering of people who are passionate about mathematics.

Mathematicians come from diverse backgrounds, maintain varying interests, and have their own unique journeys. In Fascinating Mathematical People: Interviews and Memoirs, Fern Hunt describes what it was like to be among the first black women to earn a PhD in mathematics, Harold Bacon makes trips to Alcatraz to help a prisoner learn calculus, and Thomas Banchoff, who first became interested in the fourth dimension while reading a Captain Marvel comic, relates his fascinating friendship with Salvador Dalí and their shared passion for art, mathematics, and the profound connection between the two. But whether they view mathematics as reason, art, or something else, all mathematicians are in search of truth.

This week is not only an endeavor in furthering the pursuit of knowledge, but a celebration of the gifted mathematical intellectuals who shape society, culture, and our awareness and understanding of ourselves and the world in which we live. Browse our website or latest mathematics catalog to see more by and about mathematicians, such as Paul J. Nahin’s The Logician and the Engineer: How George Boole and Claude Shannon Created the Information Age. If you’re at the Joint Mathematics Meeting, you may even visit us at booth 311. As Underwood Dudley wrote in “What Is Mathematics For?” included in The Best Writing on Mathematics: 2011 (The Best Writing on Mathematics: 2012 also available.), “What mathematics education is for is not for jobs. It is to teach the race to reason,” and we’ve all got room to learn.