Disclaimer: I wrote this, but it's a little more from an engineering POV than a Math POV, and covers some applications: jezzamon.com/fourier Hacker news folks

Disclaimer: I wrote this, but it's a little more from an engineering POV than a Math POV, and covers some applications: jezzamon.com/fourier Hacker news folks seem to enjoy it :) Another thought on the "why": many things in the real world operate on frequencies, sound being the most obvious but also electrical signals, mechanical systems like springs, etc. Doing the analysis representing a signal as a bunch of frequencies makes more sense, and Fourier transforms lets us do that