# That *A-Ha* Moment

Have you ever had an a-ha moment? Sure, you have. The Merriam-Webster dictionary defines it as "a moment of sudden realization, inspiration, insight, recognition, or comprehension." But I'm taking it one step further -- *a-ha!* with an exclamation point. This is a more dramatic realization. It's the moment when you discover a great truth, when something that was complicated or unpredictable suddenly becomes clear. As engineers, I'm sure we've had many.

One of my first a-ha!s came in grade school. I was a hobbyist, and I enjoyed creating things from the local Radio Shack, even though I didn't know why they worked. I had hoarded quite the collection of resistors, capacitors, tubes, and speakers. I could read the color bands on a resistor to get its value, and I had rudimentary soldering skills, but I didn't know how to design anything. One day, my older brother, who had been a Navy technician, explained Ohm's Law to me. A lightning bolt ignited in my head. You mean, there's a relationship between voltage, current, and resistance? That makes a lot of sense. And the world became a little bit more understandable. Building a circuit was a little less about pleasing the electron gods and a little more like engineering.

I've had several of these moments since then. Calculus was a key ingredient to many. Though I knew formulas from high school physics, calculus enabled me to *derive* the formulas. As a ham radio operator, I knew how to calculate the resonate frequency of an LC network.

But I didn't know *why* that was the resonant frequency. With calculus, I was able to derive it myself, and I discovered why it also explained the resonant frequency of a pendulum or a spring and mass. Calculus explained that the current through a capacitor was proportional to the first derivative of the voltage. Suddenly, first-order differential equations explained time-domain and frequency-domain phenomena. This was another a-ha! moment.

I've had many since then. Boolean logic explained digital circuits to me -- no longer a mystery. A more recent a-ha! moment came when I learned how WCDMA worked. In retrospect, all these things seem obvious, but I can recall the very day that each of these a-ha! moments came.

Science and engineering aren't the only subjects that have created these moments. An Economist article about international trade led the reader through a simplified two-party, two-industry model, where one party had a productivity advantage for both industries, and the other party had inferior productivity for both. Much to my surprise, simple arithmetic in the example showed that both parties produced and acquired more goods than either one could have done by itself if trading could occur between them. Until I had done the math, I had assumed trade was only advantageous if each had an absolute advantage in some industry. This was the principle of comparative advantage -- and another a-ha! moment for me. I remember the day I did the surprising arithmetic.

What about you? I'd like to hear about your a-ha! moments.

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