The Pythagorean theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle.If you add the areas of the squares on the legs, you get the area of the square on the hypotenuse. Simple geometry: walk 3 meters east and 4 meters north — you are exactly 5 meters from the starting point.
The theorem has been known since ancient times. Babylonian mathematicians around 1800 BC used triples of integers satisfying the relation to construct right angles. The school of Pythagoras (6th–5th centuries BC) is credited with the first proof and giving the theorem philosophical significance. Euclid in the 'Elements' (c. 300 BC, Book I, Proposition 47) gave a classical geometric proof. In the non-Euclidean geometries of Lobachevsky and Bolyai, the theorem no longer holds, and in the 20th century it was generalized to curved manifolds and became the foundation of the metric tensor in general relativity.
The theorem is constantly applied in physics: distance calculations in mechanics, determining the interval in special relativity, constructing the metric in general relativity. In quantum mechanics, the norm of state vectors, calculated Pythagorean-way, ensures conservation of total probability. Limits of applicability: it holds strictly only in flat Euclidean space; in curved manifolds it works locally in infinitesimal neighborhoods. In Minkowski spacetime, the signs in front of squares differ, leading to null intervals for lightlike paths.
The Babylonian clay tablet Plimpton 322 (1800 BC) contains a list of Pythagorean triples — the theorem was known a thousand years before Pythagoras.
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