Leonhard Euler
Do unrelated-looking numbers sit inside one equation?
That time, that place
Eighteenth-century mathematics ran in separate channels: circles and triangles here, exponentials and logarithms there, imaginary numbers somewhere else.
Euler connected them. He wrote 866 papers, and kept writing after going blind — calculating in his head and dictating.
Why this question
One relation he found is especially famous: exponentials and trigonometric functions are two faces of one thing.
In plain words: growing without stopping and going round in circles look like different behaviours, but turn the direction onto an imaginary axis and growing becomes turning.
What was found
Put one particular angle into that relation and five constants line up in a single sentence: zero, one, pi, the base of natural logarithms, and the imaginary unit — found in different eras for different reasons.
It is not merely pretty. Every calculation about things that rotate sits on it: alternating current, signal processing, the wave function of quantum mechanics.
Euler also left us notation. Writing a function as f(x), a sum as sigma, the circle constant as π — these hardened out of his habits.
Exponentials, trigonometry and imaginary numbers were unrelated tools
Series expansions showing the three grow from one root
It became the ground of every calculation about rotation and oscillation