Fourier transform says that any pattern in space and time can be considered a superposition of sinusoidal patterns with different frequencies. This formula is important because the component frequencies can be used to analyze the patterns, create them to order, extract important features, and remove random noise. Fourier transform led us to image processing and quantum mechanics. It also helped us structure large biological molecules like DNA, do image compression in digital photography, cleaning up old or damaged audio recordings, analyzing earthquakes.
The Formula to Get 42 Billion Digits of π
While writing "7 Utterly Well-written Math Books About Pi," I found a very interesting math formula that will give you 42 consecutive digits of π accurately but is still wrong.