Joseph Fourier
Is every complicated wave a sum of simple ones?
That time, that place
Fourier went with Napoleon to Egypt, came back, and studied how heat spreads through a body.
The trouble was shape. Heat one end of an iron bar and the temperature traces an awkward curve. How do you handle an arbitrary curve with equations?
Why this question
His claim: any curve, however awkward, can be built by adding up simple sine waves.
Think of sound. An orchestra makes a complicated waveform, but it is really the simple tones of each instrument overlaid. Run it backwards and you can pull out which tones are present, and how much of each.
When he said even a square-cornered step could be built from sine waves, mathematicians objected: adding smooth things can never produce a corner.
What was found
He was right: infinitely many terms do produce a corner. Proving it rigorously took another century, and in the process the foundations of mathematics were rebuilt.
And the tool is inside nearly everything we use. MP3 drops the components you cannot hear; JPEG drops the ones you barely see. Wi-Fi and mobile phones divide signals into frequencies; MRI turns signal back into an image.
It is also the first step by which a machine hears speech.
Complicated waveforms had no handle in equations
Heat diffusion solved as a sum of sines, matching measured profiles
It became the ground of every technology that splits signal into frequency