How is Euler’s Formula/Identity practical?

Question

For those who don’t know, and correct me if I’m wrong, Euler’s Formula is e^(xi) = cos(x) + isin(x) and Euler’s Identity is where “x” is equal to pi, therefore e^(πi) = -1. It’s fascinating to me how something so complex can be equal to something so simple (talking about Euler’s Identity). But what I don’t get is how they’re practical. What are they used for, and why are they important?

In: 3

Answer 1

The reason Euler’s formula is so important is because of its use in decomposing waveforms–i.e. fourier transforms.

Fourier transforms allow you form almost any realizable waveform into an infinite series of sines and cosines. Using Euler’s formula you compact those two functions into a single complex exponential function. Integrations and derivations of a simple exponential is MUCH easier than that of trigonometric functions.