Concept

Pythagorean theorem

Overview

In mathematics, the Pythagorean theorem or Pythagoras's theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.

The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation:

{\displaystyle a^{2}+b^{2}=c^{2}.}

The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proved numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.

When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies the Pythagorean relation: the squared distance between two points equals the sum of squares of the difference in each coordinate between the points.

The theorem can be generalized in various ways: to higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and to objects that are not triangles at all but n-dimensional solids.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

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With the same inequalities of excess or diminution, all the extremes are differenced one from another, and the means from the means, either according to the quantity of the numbers or the measure of geometry; which Aristotle thus explains, observing that nete exceeds mese by a third part of itself, and hypate is exceeded by paramese in the same proportion, so that the excesses stand in proportion. For by the same…
Plutarch, Plutarch's essays and miscellanies, Vol. 1
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Aristotle, that it is formed by the fifth body, and by the mixture of extreme heat and cold. CHAPTER XII. INTO HOW MANY CIRCLES IS THE HEAVEN DISTINGUISHED; OR, OF THE DIVISION OF HEAVEN. Thales, Pythagoras, and the followers of Pythagoras do distribute the universal globe of heaven into five circles, which they denominate zones; one of which is called the arctic circle, which is always conspicuous to us, another is…
Plutarch, Complete Works of Plutarch — Volume 3
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In other words there are alternative metrical geometries which all exist by an equal right so far as the intrinsic theory of space is concerned. Poincare, the great French mathematician, held that our actual choice among these geometries is guided purely by convention, and that the effect of a change of choice would be simply to alter our expression of the physical laws of nature. By 'convention' I understand…
Alfred North Whitehead, The Concept of Nature The Tarner Lectures Delivered in Trinity College, November 1919
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Pythagorean theorem

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