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Pythagorean theorem

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	Topics Pythagorean theorem Topic Home In mathematicsMathematics Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.... , the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometryEuclidean geometry Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria.... among the three sides of a right triangleTriangle A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line seg... . The theoremTheorem A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions.... is named after the GreekGreeks The Greeks are an ethnic group mostly found in the southern Balkan peninsula of southeastern Europe and are primarily assoc... mathematicianMathematician A mathematician is a person whose primary area of study and research is the field of mathematics.... PythagorasPythagoras Pythagoras of Samos was an Ionian mathematician and philosopher, founder of the mystic, religious and scientific society c... , who by tradition is credited with its discovery and proof, although knowledge of the theorem almost certainly predates him. The theorem is as follows: In any , the area of the squareSquare (geometry) In plane geometry, a square is a polygon with four equal sides, four right angles, and parallel opposite sides.... whose side is the hypotenuseHypotenuse A hypotenuse is the longest side of a right triangle, the side opposite of the right angle.... (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). This is usually summarized as follows: The square of the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides. If we let c be the lengthLength Length is the long dimension of any object.... of the hypotenuse and a and b be the lengths of the other two sides, the theorem can be expressed as the equation: or, solved for c: If c is already given, and the length of one of the legs must be found, the following equations can be used (The following equations are simply the converse of the original equation): or This equation provides a simple relation among the three sides of a right triangle so that if the lengths of any two sides are known, the length of the third side can be found. A generalization of this theorem is the law of cosinesLaw of cosines n trigonometry, the law of cosines is a statement about a general triangle which relates the lengths of its sides to the co... , which allows the computation of the length of the third side of any triangle, given the lengths of two sides and the size of the angle between them. If the angle between the sides is a right angle it reduces to the Pythagorean theorem. History The history of the theorem can be divided into four parts: knowledge of Pythagorean triples, knowledge of the relationship between the sides of a right triangle, knowledge of the relationship between adjacent angles, and proofs of the theorem. Megalithic monuments from circa 2500 BC in EgyptEgypt Overview Egypt , officially the Arab Republic of Egypt, is a Middle Eastern country in North Africa.... , and in Northern EuropeNorthern Europe Northern Europe is the northern part of the European continent.... , incorporate right triangles with integer sides. Bartel Leendert van der WaerdenBartel Leendert van der Waerden Bartel Leendert van der Waerden was a Dutch mathematician.... conjectures that these Pythagorean triples were discovered algebraAlgebra Algebra is a branch of mathematics concerning the study of structure, relation and quantity.... ically. Written between 2000 and 1786 BC, the Middle KingdomMiddle Kingdom of Egypt Overview The Middle Kingdom is a period in the history of ancient Egypt stretching from the establishment of the Eleventh Dynasty to ... EgyptEgypt Egypt , officially the Arab Republic of Egypt, is a Middle Eastern country in North Africa.... ian papyrus Berlin 6619Berlin papyrus The Berlin papyrus is an ancient Egyptian papyrus document from the 19th dynasty circa 1300 - 1200 BCE.... includes a problem whose solution is a Pythagorean triplePythagorean triple A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 =... . During the reign of HammurabiHammurabi Hammurabi was the sixth king of Babylon.... the Great, the MesopotamiaMesopotamia Mesopotamia refers to the region now occupied by modern Iraq, eastern Syria, and southeastern Turkey.... n tablet Plimpton 322Plimpton 322 Of the approximately half million clay tablets excavated at the beginning of the 19th century, about 400 are of a mathematical nat... , written between 179018th century BC ----... and 1750 BC, contains many entries closely related to Pythagorean triples. The BaudhayanaBaudhayana Baudhayana, was an Indian mathematician, who... Sulba SutraSulba Sutras The *Sulba Sutras* or *Sulva Sutras* are texts of the Hindu canon dealing with the geometry of altar construction.... , the dates of which are given variously as between the 8th century BC and the 2nd century BC, in IndiaFacts About History of India The history of India can be traced in fragments to as far back as 9500 years ago.... , contains a list of Pythagorean triples discovered algebraically, a statement of the Pythagorean theorem, and a geometricalGeometry Geometry arose as the field of knowledge dealing with spatial relationships.... proof of the Pythagorean theorem for an isosceles right triangle. The ApastambaApastamba Apastamba, was an Indian priest/scholar associated with the black Yajur Veda.... Sulba Sutra (circa 600 BC) contains a numerical proof of the general Pythagorean theorem, using an area computation. Van der Waerden believes that "it was certainly based on earlier traditions". According to Albert Burk, this is the original proof of the theorem; he further theorizes that Pythagoras visited ArakonamArakonam Arakonam is a city and a municipality in Vellore district in the state of Tamil Nadu, India.... , India, and copied it. PythagorasPythagoras Pythagoras of Samos was an Ionian mathematician and philosopher, founder of the mystic, religious and scientific society c... , whose dates are commonly given as 569–475 BC, used algebraic methods to construct Pythagorean triples, according to Proklos'sProclus Proclus Lycaeus , surnamed "The Successor" or "diadochos" , was a Greek Neoplatonist philosopher, one of the last major Gre... commentary on EuclidEuclid Euclid , a Greek mathematician, who lived in Alexandria, Hellenistic Egypt, almost certainly during the reign of Ptolemy I... . Proklos, however, wrote between 410 and 485 AD. According to Sir Thomas L. HeathT. L. Heath Sir Thomas Little Heath was a British civil servant, mathematician, classical scholar, historian of ancient Greek mathematic... , there is no attribution of the theorem to Pythagoras for five centuries after Pythagoras lived. However, when authors such as PlutarchPlutarch Mestrius Plutarchus , known in English as Plutarch, was a Greek historian, biographer, and essayist.... and CiceroCicero Overview Marcus Tullius Cicero January 3, 106 BC – December 7, 43 BC) was an orator, statesman, political theorist, and philos... attributed the theorem to Pythagoras, they did so in a way which suggests that the attribution was widely known and undoubted. Around 400 BC, according to Proklos, PlatoPlato Plato , whose real name is believed to have been Aristocles, was an immensely influential ancient Greek philosopher, ... gave a method for finding Pythagorean triples that combined algebra and geometry. Circa 300 BC, in Euclid's ElementsEuclid's Elements Euclid's Elements is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathemat... , the oldest extant axiomatic proofMathematics Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.... of the theorem is presented. Written sometime between 500 BC and 200 AD, the ChineseChina China is a cultural region and ancient civilization in East Asia.... text Chou Pei Suan ChingChou Pei Suan Ching Summary The Chou Pei Suan Ching is one of the oldest and most famous Chinese mathematical texts.... (????), (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) gives a visual proof of the Pythagorean theorem — in China it is called the "Gougu Theorem" (????) — for the (3, 4, 5) triangle. During the Han DynastyHan Dynasty The Han Dynasty followed the Qin Dynasty and preceded the Three Kingdoms in China.... , from 202 BC to 220 AD, Pythagorean triples appear in The Nine Chapters on the Mathematical ArtFacts About The Nine Chapters on the Mathematical Art *The Nine Chapters on the Mathematical Art* is a Chinese mathematics book, probably composed by the 1st century AD, but pe... , together with a mention of right triangles. The first recorded use is in ChinaChina China is a cultural region and ancient civilization in East Asia.... , known as the "Gougu theorem" (????) and in IndiaIndia India , officially the Republic of India, is a country in South Asia.... known as the Bhaskara Theorem. There is much debate on whether the Pythagorean theorem was discovered once or many times. Boyer (1991) thinks the elements found in the Shulba Sutras may be of Mesopotamian derivation. Proofs This is a theorem that may have more known proofs than any other (the law of quadratic reciprocityQuadratic reciprocity Overview In number theory, the law of quadratic reciprocity connects the solvability of two related quadratic equations in modular ar... being also a contender for that distinction); the book Pythagorean Proposition, by Elisha Scott Loomis, contains 367 proofs. Some arguments based on trigonometric identities (such as Taylor seriesTaylor series In mathematics, the Taylor series of an infinitely differentiable real function f, defined on an open interval , is the... for sineSiné Maurice Sinet, known as Sin? is a French cartoonist.... and cosine) have been proposed as proofs for the theorem. However, since all the fundamental trigonometric identities are proved using the Pythagorean theorem, there cannot be any trigonometric proof. (See also begging the questionBegging the question In logic, begging the question is the term for a type of fallacy occurring in deductive reasoning in which the proposition t... .) Proof using similar triangles Like most of the proofs of the Pythagorean theorem, this one is based on the proportionalityProportionality (mathematics) In mathematics, two quantities are called proportional if they vary in such a way that one of the quantities is a constant m... of the sides of two similarSimilarity (mathematics) In mathematics, there are several notions of similarity:... triangles. Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. We draw the altitudeAltitude (triangle) In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side or an exte... from point C, and call H its intersection with the side AB. The new triangle ACH is similarSimilarity (geometry) Geometry Two geometrical objects are called similar if one is congruent to the result of a uniform scaling of the other.... to our triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well. By a similar reasoning, the triangle CBH is also similar to ABC. The similarities lead to the two ratios..: As so These can be written as Summing these two equalities, we obtain In other words, the Pythagorean theorem: Euclid's proof In Euclid'sEuclid Euclid , a Greek mathematician, who lived in Alexandria, Hellenistic Egypt, almost certainly during the reign of Ptolemy I... ElementsEuclid's Elements Euclid's Elements is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathemat... , Proposition 47 of Book 1, the Pythagorean theorem is proved by an argument along the following lines. Let A, B, C be the vertices of a right triangle, with a right angle at A. Drop a perpendicular from A to the side opposite the hypotenuse in the square on the hypotenuse. That line divides the square on the hypotenuse into two rectangles, each having the same area as one of the two squares on the legs. For the formal proof, we require four elementary lemmata: If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles included by those sides equal, then the triangles are congruent. (Side - Angle - Side Theorem) The area of a triangle is half the area of any parallelogram on the same base and having the same altitude. The area of any square is equal to the product of two of its sides. The area of any rectangle is equal to the product of two adjacent sides (follows from Lemma 3). The intuitive idea behind this proof, which can make it easier to follow, is that the top squares are morphed into parallelogramParallelogram A parallelogram is a four-sided plane figure that has two sets of opposite parallel sides.... s with the same size, then turned and morphed into the left and right rectangles in the lower square, again at constant area. The proof is as follows: Let ACB be a right-angled triangle with right angle CAB. On each of the sides BC, AB, and CA, squares are drawn, CBDE, BAGF, and ACIH, in that order. From A, draw a line parallel to BD and CE. It will perpendicularly intersect BC and DE at K and L, respectively. Join CF and AD, to form the triangles BCF and BDA. Angles CAB and BAG are both right angles; therefore C, A, and G are collinear. Similarly for B, A, and H. Angles CBD and FBA are both right angles; therefore angle ABD equals angle FBC, since both are the sum of a right angle and angle ABC. Since AB and BD are equal to FB and BC, respectively, triangle ABD must be equal to triangle FBC. Since A is collinear with K and L, rectangle BDLK must be twice in area to triangle ABD. Since C is collinear with A and G, square BAGF must be twice in area to triangle FBC. Therefore rectangle BDLK must have the same area as square BAGF = AB². Similarly, it can be shown that rectangle CKLE must have the same area as square ACIH = AC². Adding these two results, AB² + AC² = BD × BK + KL × KC Since BD = KL, BD* BK + KL × KC = BD(BK + KC) = BD × BC Therefore AB² + AC² = BC², since CBDE is a square. This proof appears in Euclid's Elements as that of Proposition 1.47. Garfield's proof James A. Garfield (later President of the United States) is credited with a novel algebraic proof using a trapezoidTrapezoid A trapezoid or trapezium is a quadrilateral two of whose sides are parallel to each other.... containing two examples of the triangle, the figure comprising one-half of the figure using four triangles enclosing a square shown below. Similarity proof From the same diagram as that in Euclid's proof above, we can see three similarSimilarity (geometry) Summary Geometry Two geometrical objects are called similar if one is congruent to the result of a uniform scaling of the other.... figures, each being "a square with a triangle on top". Since the large triangle is made of the two smaller triangles, its area is the sum of areas of the two smaller ones. By similarity, the three squares are in the same proportions relative to each other as the three triangles, and so likewise the area of the larger square is the sum of the areas of the two smaller squares. Proof by rearrangement A proof by rearrangement is given by the illustration and the animation. In the illustration, the area of each large square is (a + b)². In both, the area of four identical triangles is removed. The remaining areas, a² + b² and c², are equal. Q.E.D.Q.E.D. *Q.E.D.* is an abbreviation of the Latin phrase "quod erat demonstrandum" .... This proof is indeed very simple, but it is not elementary, in the sense that it does not depend solely upon the most basic axioms and theorems of Euclidean geometryEuclidean geometry Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria.... . In particular, while it is quite easy to give a formula for area of triangles and squares, it is not as easy to prove that the area of a square is the sum of areas of its pieces. In fact, proving the necessary properties is harder than proving the Pythagorean theorem itself (see Lebesgue measureLebesgue measure Overview In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space... and Banach-Tarski paradox). Actually, this difficulty affects all simple Euclidean proofs involving area; for instance, deriving the area of a right triangle involves the assumption that it is half the area of a rectangle with the same height and base. For this reason, axiomatic introductions to geometry usually employ another proof based on the similarity of triangles (see above). A third graphic illustration of the Pythagorean theorem (in yellow and blue to the right) fits parts of the sides' squares into the hypotenuse's square. A related proof would show that the repositioned parts are identical with the originals and, since the sum of equals are equal, that the corresponding areas are equal. To show that a square is the result one must show that the length of the new sides equals c. Note that for this proof to work, one must provide a way to handle cutting the small square in more and more slices as the corresponding side gets smaller and smaller. Algebraic proof An algebraic variant of this proof is provided by the following reasoning. Looking at the illustration which is a large square with identical right triangles in its corners, the area of each of these four triangles is given by an angle corresponding with the side of length C. The A-side angle and B-side angle of each of these triangles are complementary anglesComplementary angles A pair of angles are complementary if the sum of their measures is 90 degrees.... , so each of the angles of the blue area in the middle is a right angle, making this area a square with side length C. The area of this square is C². Thus the area of everything together is given by: However, as the large square has sides of length A + B, we can also calculate its area as (A + B)², which expands to A² + 2AB + B². (Distribution of the 4) (Subtraction of 2AB) Proof by differential equations One can arrive at the Pythagorean theorem by studying how changes in a side produce a change in the hypotenuse in the following diagram and employing a little calculusCalculus Calculus is a central branch of mathematics, developed from algebra and geometry.... . As a result of a change in side a, by similar triangles and for differential changes. So upon separation of variablesSeparation of variables In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, i... . which results from adding a second term for changes in side b. IntegratingIntegral In calculus, the integral of a function is an extension of the concept of a sum.... gives When a = 0 then c = b, so the "constant" is b². So As can be seen, the squares are due to the particular proportionProportionality (mathematics) In mathematics, two quantities are called proportional if they vary in such a way that one of the quantities is a constant m... between the changes and the sides while the sum is a result of the independent contributions of the changes in the sides which is not evident from the geometric proofs. From the proportion given it can be shown that the changes in the sides are inversely proportional to the sides. The differential equationDifferential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables.... suggests that the theorem is due to relative changes and its derivation is nearly equivalent to computing a line integralLine integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.... . These quantities da and dc are respectively infinitely small changes in a and c. But we use instead real numbers Δa and Δc, then the limit of their ratio as their sizes approach zero is da/dc, the derivative, and also approaches c/a, the ratio of lengths of sides of triangles, and the differential equation results. Converse The converseConversion (logic) In traditional logic conversion is a "type of immediate inference in which from a given proposition another proposition is i... of the theorem is also true: For any three positive numbers a, b, and c such that a² + b² = c², there exists a triangle with sides a, b and c, and every such triangle has a right angle between the sides of lengths a and b. This converse also appears in Euclid's Elements. It can be proven using the law of cosinesLaw of cosines n trigonometry, the law of cosines is a statement about a general triangle which relates the lengths of its sides to the co... (see below under Generalizations), or by the following proof: Let ABC be a triangle with side lengths a, b, and c, with a² + b² = c². We need to prove that the angle between the a and b sides is a right angle. We construct another triangle with a right angle between sides of lengths a and b. By the Pythagorean theorem, it follows that the hypotenuse of this triangle also has length c. Since both triangles have the same side lengths a, b and c, they are congruentCongruence (geometry) In geometry, two sets are called congruent if one can be transformed into the other by an isometry, i.e., a combination of ... , and so they must have the same angles. Therefore, the angle between the side of lengths a and b in our original triangle is a right angle. A corollaryCorollary A corollary is a statement which follows readily from a previously proven statement.... of the Pythagorean theorem's converse is a simple means of determining whether a triangle is right, obtuse, or acute, as follows. Where c is chosen to be the longest of the three sides: If a² + b² = c², then the triangle is right. If a² + b² > c², then the triangle is acute. If a² + b² < c², then the triangle is obtuse. Consequences and uses of the theorem Pythagorean triples A Pythagorean triple has 3 positive numbers a, b, and c, such that . In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. Evidence from megalithic monuments on the Northern Europe shows that such triples were known before the discovery of writing. Such a triple is commonly written (a, b, c). Some well-known examples are (3, 4, 5) and (5, 12, 13). List of primitive Pythagorean triples up to 100 (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97) The existence of irrational numbers One of the consequences of the Pythagorean theorem is that irrational numberIrrational number In mathematics, an irrational number is any real number that is not a rational number, i.e., it is not of the form ... s, such as the square root of 2, can be constructed. A right triangle with legs both equal to one unit has hypotenuse length square root of 2. The PythagoreansPythagoreanism Pythagoreanism is a term used for the esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagorea... proved that the square root of 2 is irrationalIrrational number In mathematics, an irrational number is any real number that is not a rational number, i.e., it is not of the form ... , and this proof has come down to us even though it flew in the face of their cherished belief that everything was rational. According to the legend, HippasusHippasus Hippasus of Metapontum, born circa 500 B.C.... , who first proved the irrationality of the square root of two, was drowned at sea as a consequence. Distance in Cartesian coordinates The distance formula in Cartesian coordinates is derived from the Pythagorean theorem. If (x₀, y₀) and (x₁, y₁) are points in the plane, then the distance between them, also called the Euclidean distanceEuclidean distance In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between the two points that one ... , is given by More generally, in Euclidean n-spaceEuclidean space Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "plane Euclidean geometry", wh... , the Euclidean distance between two points, and , is defined, using the Pythagorean theorem, as: Generalizations The Pythagorean theorem was generalized by EuclidEuclid Euclid , a Greek mathematician, who lived in Alexandria, Hellenistic Egypt, almost certainly during the reign of Ptolemy I... in his ElementsEuclid's Elements Euclid's Elements is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathemat... : If one erects similar figures (see Euclidean geometryEuclidean geometry Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria.... ) on the sides of a right triangle, then the sum of the areas of the two smaller ones equals the area of the larger one. The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosinesLaw of cosines n trigonometry, the law of cosines is a statement about a general triangle which relates the lengths of its sides to the co... : where ? is the angle between sides a and b. When ? is 90 degrees, then cos(?) = 0, so the formula reduces to the usual Pythagorean theorem. Given two vectorsVector space In mathematics, a vector space is a collection of objects that, informally speaking, may be scaled and added.... v and w in a complexFacts About Complex number In mathematics, a complex number is a number of the form ... inner product spaceInner product space In mathematics, an inner product space is a vector space with additional structure, an inner product , which allows us... , the Pythagorean theorem takes the following form: In particular, \|\|v + w\|\|² = \|\|v\|\|² + \|\|w\|\|² if v and w are orthogonal, although the converse is not necessarily true. Using mathematical inductionMathematical induction Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all n... , the previous result can be extended to any finiteFinite set In mathematics, a set is called *finite* if there is a bijection between the set and some set of the form where is a nat... number of pairwise orthogonal vectors. Let v₁, v₂,…, v_n be vectors in an inner product space such that <v_i, v_j> = 0 for 1 = i < j = n. Then The generalization of this result to infinite-dimensional realReal number In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers.... inner product spaces is known as Parseval's identityParseval's identity In functional analysis, Parseval's identity, also known as Parseval's equality, is the... . When the theorem above about vectors is rewritten in terms of solid geometrySolid geometry In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for pr... , it becomes the following theorem. If lines AB and BC form a right angle at B, and lines BC and CD form a right angle at C, and if CD is perpendicular to the plane containing lines AB and BC, then the sum of the squares of the lengths of AB, BC, and CD is equal to the square of AD. The proof is trivial. Another generalization of the Pythagorean theorem to three dimensions is de Gua's theoremDe Gua's theorem De Gua's theorem is a generalization of the Pythagorean theorem to three dimensions and named for Jean Paul de Gua de Malves... , named for Jean Paul de Gua de MalvesJean Paul de Gua de Malves Jean Paul de Gua de Malves was a French mathematician who published in 1740 a work on analytical geometry in which he applie... : If a tetrahedronTetrahedron Summary A tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex.... has a right angle corner (a corner like a cube), then the square of the area of the face opposite the right angle corner is the sum of the squares of the areas of the other three faces. There are also analogs of these theorems in dimensions four and higher. In a triangle with three acute anglesAngle An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.... , a + ß > ? holds. Therefore, a² + b² > c² holds. In a triangle with an obtuse angleAngle Summary An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.... , a + ß < ? holds. Therefore, a² + b² < c² holds. Edsger DijkstraEdsger Dijkstra Prof Dr Edsger Wybe Dijkstra was a Dutch computer scientist.... has stated this proposition about acute, right, and obtuse triangles in this language: sgnSign function In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real... (a + ß - ?) = sgnSign function Overview In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real... (a² + b² - c²) where a is the angle opposite to side a, ß is the angle opposite to side b and ? is the angle opposite to side c. The Pythagorean theorem in non-Euclidean geometry The Pythagorean theorem is derived from the axioms of Euclidean geometryEuclidean geometry Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria.... , and in fact, the Euclidean form of the Pythagorean theorem given above does not hold in non-Euclidean geometryNon-Euclidean geometry ----The term non-Euclidean geometry describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclid... . (It has been shown in fact to be equivalent to Euclid's Parallel (Fifth) Postulate.) For example, in spherical geometrySpherical geometry Spherical geometry is the geometry of the two-dimensional surface of a sphere.... , all three sides of the right triangle bounding an octant of the unit sphere have length equal to ; this violates the Euclidean Pythagorean theorem because . This means that in non-Euclidean geometry, the Pythagorean theorem must necessarily take a different form from the Euclidean theorem. There are two cases to consider — spherical geometrySpherical geometry Spherical geometry is the geometry of the two-dimensional surface of a sphere.... and hyperbolic plane geometryHyperbolic geometry Hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is rejected.... ; in each case, as in the Euclidean case, the result follows from the appropriate law of cosines: For any right triangle on a sphere of radius R, the Pythagorean theorem takes the form By using the Maclaurin series for the cosine function, it can be shown that as the radius R approaches infinity, the spherical form of the Pythagorean theorem approaches the Euclidean form. For any triangle in the hyperbolic planeHyperbolic geometry Hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is rejected.... (with Gaussian curvatureCurvature Curvature refers to a number of loosely related concepts in different areas of geometry.... -1), the Pythagorean theorem takes the form where cosh is the hyperbolic cosineHyperbolic function Overview In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.... . By using the Maclaurin series for this function, it can be shown that as a hyperbolic triangle becomes very small (i.e., as a, b, and c all approach zero), the hyperbolic form of the Pythagorean theorem approaches the Euclidean form. In hyperbolic geometryHyperbolic geometry Hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is rejected.... , for a right triangle one can also write, where is the angle of parallelismAngle of parallelism In hyperbolic geometry, the angle of parallelism F is the angle at one vertex of a right hyperbolic triangle that has two hy... of the line segment AB that where µ is the multiplicative distanceMultiplicative distance In algebraic geometry, is said to be a multiplicative distance function over a field if it satisfies,... function (see Hilbert's arithmetic of endsHilbert's arithmetic of ends Hilbert's arithmetic of ends is an algebraic approach introduced by German mathematician David Hilbert for Poincar? disk mo... ). In hyperbolic trigonometry, the sine of the angle of parallelismAngle of parallelism In hyperbolic geometry, the angle of parallelism F is the angle at one vertex of a right hyperbolic triangle that has two hy... satisfies Thus, the equation takes the form where a, b, and c are multiplicative distances of the sides of the right triangle (Hartshorne, 2000). See also External links (more than 70 proofs from cut-the-knotCut-the-knot cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variet... ) - An e-bookE-book An e-book is an electronic version of a book.... (in PDFPortable Document Format Portable Document Format is a file format proprietary to Adobe Systems for representing two-dimensional documents in a devi... format) with 367 different proofs of The Pythagorean Theorem Interactive links: in JavaJava style="margin-left: inherit; font-size: medium;" \| Java... of The Pythagorean Theorem in JavaJava style="margin-left: inherit; font-size: medium;" \| Java... of The Pythagorean Theorem with interactive animation Pythagorean Theorem