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Q: What is a Fourier transform? What is it used for?

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This fact is the central philosophy behind Fourier transforms (Fourier was very French, so his name is pronounced a little wonky: “4 E yay”). The buffeting movement of the air is the signal, and the tone is the Fourier transform of that signal. For example, for not-terribly-obvious reasons, in quantum mechanics the Fourier transform of the position a particle (or anything really) is the momentum of that particle. Mathematicians tend to be more excited by the abstract mathematical properties of Fourier transforms than by the more intuitive properties. A lot of problems that are difficult/nearly impossible to solve directly become easy after a Fourier transform.