Math Files(@Math_files):Fourier showed that any periodic function can be built by adding sine waves. For example take a square wave: it follows a repeating up-down pattern over a fixed period. Even though sine waves are smooth, we can stretch, scale, and shift them. By adding multiple sine waves with different properties, we start to approximate the square wave. At first, it’s rough—but as we include more terms, the sum becomes increasingly accurate, eventually closely matching the square wave.

2026.05.16 02:30

Fourier showed that any periodic function can be built by adding sine waves. For example take a square wave: it follows a repeating up-down pattern over a fixed period. Even though sine waves are smooth, we can stretch, scale, and shift them. By adding multiple sine waves with different properties, we start to approximate the square wave. At first, it’s rough—but as we include more terms, the sum becomes increasingly accurate, eventually closely matching the square wave.