I was browsing some of the features of a popular computer program for doing mathematics. Wow, I thought! What I would have given for this years ago!

But suddenly I was overcome with sadness. I don’t need this anymore, I realized. In fact, it has been many years since I worked with ”real” mathematics. I just never really thought about that loss before. It was as if my profession had slipped away when I wasn’t looking.

I commiserated with several engineering friends. Two of them weren’t concerned at all. That’s what happens when you move along in your career, they said, and it doesn’t make you any less of an engineer. The other, a researcher like me, shared my nostalgia and pain. It made him think of what he had been—and was no more.

I wonder how many engineers use advanced math in their jobs and whether fewer do so, now that computers have consumed so much of our work. Has mathematics disappeared behind the screens of our monitors, as have so many other subjects since engineering began to center increasingly on writing software?

Yet mathematics is a way of thought that binds us to our profession. Maxwell’s equations are inscribed in the entrance foyer of the National Academy of Engineering as the very symbol of what we do. I look at them as the scripture of engineering—a concise and elegant description of the laws that govern electromagnetism. But I also wonder: How many engineers have actually used Maxwell’s equations in their work? Alas, I’ve never had the pleasure myself.

Our journals are still full of mathematics. If you want to publish and have your work inscribed in stone for eternity, you must code your work in mathematical symbolism. If you want to parade among the elite of the profession, you must cloak yourself in mathematics. This is the way it has always been. Now, if math is disappearing from our practice, this would make me sad.

I remember well the day in high school algebra class when I was first introduced to imaginary numbers. The teacher said that because the square root of a negative number didn’t actually exist, it was called imaginary. That bothered me a lot. I asked, If it didn’t exist, why give it a name and study it? Unfortunately, the teacher had no answers for these questions.

As with much of the math that we’ve all studied, understanding comes only much later. We’ve all had the experience of learning mathematical principles before we had any idea what they were good for. If I could go back to that day in high school, how would I have explained matters?

I can think of two approaches, although somehow I doubt that my younger self would have been happy with either. The first is to say that mathematics is beautiful in itself, a study of consistent rules of logic that can be appreciated as an art form, quite apart from any application it may have to everyday problems. The second is to note that this square root of minus one is actually useful (in problems that my younger self didn’t know about yet). It opens the door to two-dimensional thinking—a dimension that gets you off the line of real numbers. So whether or not this imaginary number exists in your world of arithmetic training, it’s useful. In real world problems, it works.

I’m reminded of a famous saying in physics, variously attributed to Paul Dirac and Richard Feynman: ”Shut up and calculate.” It was a response to a class of problems in quantum mechanics in which the Shrödinger wave equation often contradicts common perception, yet it always provides the right answers. So don’t worry about it: quit complaining and just calculate. Like using the square root of minus one, it works.

Since that first introduction to imaginary numbers, I’ve just about come full circle. I learned to appreciate math, and I found imaginary numbers useful. But now I’m thinking that, though the appreciation remains, the usefulness to me has faded.

The more I think about this as I write, the sadder I get. I’m going to go back and look at the features of that mathematics program again.

## About the Author

ROBERT W. LUCKY (IEEE Fellow), now retired, was vice president for applied research at Telcordia Technology in Red Bank, N.J. (rlucky@telcordia.com).

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