In the early 1800s, Joseph Fourier found that every periodic function is made up of (a possibly infinite series of) sine and cosine functions of various frequencies and magnitudes. Just add the right sine waves together and you'll get the desired function. Any function. This collection of waves is called the Fourier series, and it would soon propel the complex numbers from the ivory tower of pure math onto the mad merry-go-round of technology.
Mathematicians used the Fourier series to shift difficult problems to an easier battleground, by transforming a complicated function into an infinite sum of very simple ones. This was the beginning of frequency-domain analysis. It was soon discovered that—thanks to Cotes's discovery—the Fourier series was much simpler if you used complex numbers. Other such transformations were discovered too, notably the Fourier transform and the Laplace transform. Both are based on the complex numbers.
Frequency-domain analysis was the killer app for complex numbers. And then came electricity. As it happens, most of electrical engineering would be practically impossible without frequency-domain analysis. Beginning problems in circuits—problems that in the time domain would require two or three semesters of college-level calculus to tackle—can be solved in the frequency domain with basic high-school algebra and a few complex numbers.
There is just one more application of the complex numbers I want to talk about, by far the weirdest, probably the most controversial, and just maybe the most beautiful of them all.
(Concluded in part 4.)