As the title says, how come the value e is special? it has functions and rules just dedicated to it, but what sets it apart from another random number?
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Ok. Everyone has completely missed why e is important. Other people have already mentioned that the derivative of e^x is e^x. In engineering, we learn that systems are described by differential equations, and all but the most simple differential equations are essentially unsolvable except for an extremely limited set of input functions—essentially functions that differentiate to themselves, which means we can’t tell if our amplifier will have a flat frequency response in our target frequency range or our bridge is going to fall down. What to do? Well euler’s formula tells us that an arbitrary single frequency function can be represented by e^jomega where j is the sqrt of -1 and omega is the frequency, and Fourier tells us that an arbitrary sinusoid can be modeled as an infinite sum of single frequency sinusoids, so (hold on to your hats) we can decompose our arbitrary input sinusoid an infinite sum of single frequency sinusoids, represent those as e^jomega, and since the derivative of e^jomega is e^jomega, our we can now determine important things about our system—specifically whether our bridge or skyscraper is going to fall down, whether our jet is going to fall out of the sky, whether our chemical reaction is going to explode, or whether our amplifier is going to have a flat 20-24KHz frequency respose
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