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In mathematics, the Pythagorean theorem or Pythagoras' 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.
Each problem includes an answer and a corresponding arithmetic algorithm. It is an important source on early Chinese cosmology , glossing the ancient idea of a round heaven over a square earth ( 天 圆 地 方 , tiānyuán dìfāng ) as similar to the round parasol suspended over some ancient Chinese chariots [ 10 ] or a Chinese chessboard ...
The theorem for which the greatest number of different proofs have been discovered is possibly the Pythagorean theorem, with hundreds of proofs being published up to date. [3] Another theorem that has been proved in many different ways is the theorem of quadratic reciprocity .
Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic ( addition , subtraction , multiplication , and division ) first appear in the archaeological ...
Triangles based on Pythagorean triples are Heronian, meaning they have integer area as well as integer sides. The possible use of the 3 : 4 : 5 triangle in Ancient Egypt, with the supposed use of a knotted rope to lay out such a triangle, and the question whether Pythagoras' theorem was known at that time, have been much debated. [3]
The spiral is started with an isosceles right triangle, with each leg having unit length.Another right triangle (which is the only automedian right triangle) is formed, with one leg being the hypotenuse of the prior right triangle (with length the square root of 2) and the other leg having length of 1; the length of the hypotenuse of this second right triangle is the square root of 3.
An illustration of Euclid's proof of the Pythagorean theorem. Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly from the 5th century BC to the 6th century AD, around the shores of the Mediterranean.
Pythagorean fields can be used to construct models for some of Hilbert's axioms for geometry (Iyanaga & Kawada 1980, 163 C). The coordinate geometry given by F n {\displaystyle F^{n}} for F {\displaystyle F} a Pythagorean field satisfies many of Hilbert's axioms, such as the incidence axioms, the congruence axioms and the axioms of parallels.
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