The distance formula on the plane follows from the Pythagorean theorem.

which can be viewed as a version of the Pythagorean theorem.

The Pythagorean theorem is proved.

Image: Pythagorean. svg | Pythagoras ' theorem: The sum of the areas of the two squares on the legs ( a and b ) of a right triangle equals the area of the square on the hypotenuse ( c ).

The celebrated Pythagorean theorem ( book I, proposition 47 ) states that in any right triangle, 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 whose sides are the two legs ( the two sides that meet at a right angle ).

In Euclid's original approach, the Pythagorean theorem follows from Euclid's axioms.

In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems.

* Pythagorean theorem

It can be viewed as a form of the Pythagorean theorem.

Animation illustrating Pythagorean theorem | Pythagoras ' rule for a right-angle triangle, which shows the algebraic relationship between the triangle's hypotenuse, and the other two sides.

For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras ; the Egyptians had a correct formula for the volume of a frustum of a square pyramid ;

The Pythagorean theorem was also known to the Babylonians.

Pythagorean theorem: a < sup > 2 </ sup > + b < sup > 2 </ sup > = c < sup > 2 </ sup >

Baudhayana ( c. 8th century BCE ) composed the Baudhayana Sulba Sutra, the best-known Sulba Sutra, which contains examples of simple Pythagorean triples, such as:,,,, and as well as a statement of the Pythagorean theorem for the sides of a square: " The rope which is stretched across the diagonal of a square produces an area double the size of the original square.

" It also contains the general statement of the Pythagorean theorem ( for the sides of a rectangle ): " The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together.

All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.

# REDIRECT Pythagorean theorem

The proof relies on the Pythagorean theorem.

It is known from the Pythagorean theorem that a < sup > 2 </ sup > + b < sup > 2 </ sup > = c < sup > 2 </ sup >.

The latter was a result of the Pythagorean theorem and the former the assumption that a + b ≤ c. The contradiction means that it is impossible for both to be true and it is known that the Pythagorean theorem holds.

At any given x, a right-angled triangle connects x, y and r to the origin, hence it follows from the Pythagorean theorem that:

Then he went to hear a Pythagorean philosopher, who demanded that he first learn music, astronomy and geometry, which he did not wish to do.

In non-Cartesian ( non-square ) or curved coordinate systems, the Pythagorean theorem holds only on infinitesimal length scales and must be augmented with a more general metric tensor g < sub > μν </ sub >, which can vary from place to place and which describes the local geometry in the particular coordinate system.

For example, in the " game " of Euclidean geometry ( which is seen as consisting of some strings called " axioms ", and some " rules of inference " to generate new strings from given ones ), one can prove that the Pythagorean theorem holds ( that is, you can generate the string corresponding to the Pythagorean theorem ).

In Euclidean geometry, for right triangles it is a consequence of the Pythagorean theorem, and for general triangles a consequence of the law of cosines, although it may be proven without these theorems.

In addition to the familiar theorems of geometry, such as the Pythagorean theorem, the Elements includes a proof that the square root of two is irrational and that there are infinitely many prime numbers.

* Book 1 contains Euclid's 10 axioms ( 5 named postulates — including the parallel postulate — and 5 named axioms ) and the basic propositions of geometry: the pons asinorum ( proposition 5 ), the Pythagorean theorem ( Proposition 47 ), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are " equal " ( have the same area ).

Of particular interest is the focus applied to the Platonic Solids derived from the Pythagorean theories of geometry and numbers by Plato.

Figurate numbers were a concern of Pythagorean geometry.

His book ; Ceyuan haijing revolutionized the idea of inscribing a circle into triangles, by turning this geometry problem by algebra instead of the traditional method of using Pythagorean theorem.

Alcott described his sustenance as a " Pythagorean diet ": meat, eggs, butter, cheese, and milk were excluded and drinking was confined to well water.

The Platonist seemed to outweigh the Aristotelian in Alan, but he felt strongly that the divine is all intelligibility and argued this notion through much Aristotelian logic combined with Pythagorean mathematics.

* Pythagorean expectation: estimates a team's expected winning percentage based on runs scored and runs allowed.

Zopyrus has been plausibly equated with a Pythagorean of that name who seems to have flourished in the late 5th century BC.

Zopyrus has been plausibly equated with a Pythagorean of that name who seems to have flourished in the late 5th century BC.

In those oblique coordinate systems the computations of distances and angles is more complicated than in standard Cartesian systems, and many standard formulas ( such as the Pythagorean formula for the distance ) do not hold.

* Manas, John Helen, Divination, ancient and modern, New York, Pythagorean Society, 1947.

< td > For n = 2 there are infinitely many solutions ( x, y, z ): the Pythagorean triples.

According to, the Śulba Sūtras contain " the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians.

They contain lists of Pythagorean triples ,< ref > Pythagorean triples are triples of integers with the property:.

Because the four color theorem is true, this is always possible ; however, because the person drawing the map is focused on the one large region, he fails to notice that the remaining regions can in fact be colored with three colors.

In fact the cardinality of sets fails to be totally ordered ( see Cantor – Bernstein – Schroeder theorem ).

The converse of Fermat's little theorem is not generally true, as it fails for Carmichael numbers.

Counterexamples in which the theorem fails are known in spacetime dimensions higher than four ; in the presence of non-abelian Yang-Mills fields, non-abelian Proca fields, some non-minimally coupled scalar fields, or skyrmions ; or in some theories of gravity other than Einstein ’ s general relativity.

Kummer improved this further in 1857 by showing that for the " first case " of Fermat's Last Theorem ( see Sophie Germain's theorem ) it is sufficient to establish that either or fails to be an irregular pair.

* Dilbert and the Coase Theorem ' The Coase theorem fails in the presence of asymmetric information.

More generally, the theorem fails for equipped with any norm () ( Schwartz 1969, p. 20 ).

However, there are first-order systems in which new inference rules are added for which the deduction theorem fails.

However, an actual overpayment will generally occur only if the winner fails to account for the winner's curse when bidding ( an outcome that, according to the revenue equivalence theorem, need never occur ).

The Penrose / Lucas argument about the implications of Gödel's incompleteness theorem for computational theories of human intelligence has been widely criticized by mathematicians, computer scientists and philosophers, and the consensus among experts in these fields seems to be that the argument fails, though different authors may choose different aspects of the argument to attack.

Otherwise the formulation " the smallest number that fails to satisfy the theorem " would be an internal formula that uniquely defined a non-standard number.

( It is a theorem of ZFC that MA () fails.

Small exotic R < sup > 4 </ sup > s can be constructed by starting with a non-trivial smooth 5-dimensional h-cobordism ( which exists by Donaldson's proof that the h-cobordism theorem fails in this dimension ) and using Freedman's theorem that the topological h-cobordism theorem holds in this dimension.