Pythagorean theoremtheorem

# tag · @ author · ! scientist

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.

How it works

The theorem is indispensable in distance calculations: from determining the length of a ladder to computing orbital trajectories. In special relativity, the spacetime interval unites time and space with a minus sign — this is a pseudo-Pythagorean structure. In the Hilbert space of quantum mechanics, the probability of finding a particle relies on the square of the modulus of the wave function, which follows directly from the Pythagorean norm.

💡 The Babylonian clay tablet Plimpton 322 (1800 BC) contains a list of Pythagorean triples — the theorem was known a thousand years before Pythagoras.

The Pythagorean theorem is a geometric theorem stating that in the Euclidean plane, for a right triangle with legs a, b and hypotenuse c, the relation a² + b² = c² holds. In n-dimensional Euclidean space, the squared distance ds² = dx₁² + ... + dx_n² defines the metric. In Riemannian geometry and general relativity, the metric is generalized to ds² = g_ij dx^i dx^j, where g_ij is the metric tensor describing curved spacetime. In Hilbert space, the inner product that defines probability amplitudes is based on the same quadratic structure.

Discovery

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.

How it works

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.

Caveats

In non-Euclidean geometries, the theorem does not hold; its role is taken by local metric relations.; In spaces with indefinite metric (as in special relativity), the sum of squares can be negative, requiring the concept of signature.; Historical priority is disputed: ancient Egyptians and Babylonians knew triples long before Pythagoras.

c^2 = a^2 + b^2

a, b are the lengths of the legs, c is the length of the hypotenuse of a right triangle

|\mathbf{v}|^2 = \sum_{i=1}^n v_i^2

|v| is the length (norm) of vector v in Euclidean space, v_i are its components

ds^2 = g_{ij} dx^i dx^j

ds is the interval element, g_{ij} is the metric tensor (generalization of the identity matrix in Euclidean space), dx^i, dx^j are coordinate differentials; summation over repeated indices is implied

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Pythagorean theoremtheorem

How it works

How it works

Discovery

How it works

Caveats

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