Pythagorean theorem

• 1. Ancient Origins And Discovery

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  The Berlin Papyrus 6619, written around 1800 BC during Egypt's Middle Kingdom, contains a problem involving two squares whose areas sum to a third square. The solution to this problem is the Pythagorean triple 6:8:10, though the text does not explicitly mention a triangle. Historians of Mesopotamian mathematics have concluded that the Pythagorean rule was in widespread use during the Old Babylonian period between the 20th and 16th centuries BC. This usage occurred over a thousand years before Pythagoras was born around 570 BC. The Mesopotamian tablet Plimpton 322, written near Larsa also around 1800 BC, contains entries that can be interpreted as the sides and diagonals of 15 different Pythagorean triples. Another tablet from a similar time, YBC 7289, calculates the diagonal of a square or equivalently of an isosceles right triangle. In India, the Baudhayana Shulba Sutra, dated variously between the 8th and 5th century BC, contains a list of Pythagorean triples and a statement of the theorem. The Apastamba Shulba Sutra also includes these concepts for both the special case of the isosceles right triangle and the general case. By the 1st century BC, the Chinese text Zhoubi Suanjing gave a reasoning for the theorem for the 3:4:5 triangle. During the Han Dynasty which lasted from 202 BC to 220 AD, Pythagorean triples appear in The Nine Chapters on the Mathematical Art. This work is a compilation of 246 problems, some of which survived the book burning of 213 BC.

• 2. Proofs Through History

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  Euclid's Elements, published around 300 BC, presents the oldest extant axiomatic proof of the theorem alongside Euclid's formula for generating all primitive Pythagorean triples. English mathematician Sir Thomas Heath gives this proof in his commentary on Proposition I.47 in Euclid's Elements. He mentions proposals by German mathematicians Carl Anton Bretschneider and Hermann Hankel that Pythagoras may have known this proof. Albert Einstein gave a proof by dissection in which the pieces do not need to be moved. In Einstein's proof, the shape that includes the hypotenuse is the right triangle itself. The dissection consists of dropping a perpendicular from the vertex of the right angle of the triangle to the hypotenuse. U.S. president James A. Garfield published a related proof before he was elected president while he was a U.S. representative. His proof uses a trapezoid constructed from the square in one of the earlier proofs by bisecting along a diagonal of the inner square. The book The Pythagorean Proposition contains 370 proofs, possibly more than any other mathematical theorem. Recent scholarship has cast increasing doubt on any sort of role for Pythagoras as a creator of mathematics, although debate about this continues. No specific attribution of the theorem to Pythagoras exists in the surviving Greek literature from the five centuries after Pythagoras lived.

• 3. Pythagorean Triples And Number Theory

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  A Pythagorean triple has three positive integers such that their squares satisfy the equation. Such a triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. Some well-known examples are 3:4:5 and 5:12:13. A primitive Pythagorean triple is one in which the greatest common divisor of the three numbers is 1. Euclid's formula is the most well-known method for generating these triples. Given arbitrary positive integers m and n, the formula states that the integers form a Pythagorean triple. Byzantine Neoplatonic philosopher and mathematician Proclus, writing in the fifth century AD, states two arithmetic rules for generating special Pythagorean triples. One rule attributed to Plato starts from an even number and produces a triple with leg and hypotenuse differing by two units. The rule attributed to Pythagoras starts from an odd number and produces a triple with leg and hypotenuse differing by one unit. Kurt von Fritz wrote a careful discussion of Hippasus's contributions regarding incommensurable lengths. According to one legend, Hippasus of Metapontum was drowned at sea for making known the existence of the irrational or incommensurable. His fellows cast him overboard while he was on a voyage.

• 4. Generalizations To Higher Dimensions

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  In terms of solid geometry, Pythagoras's theorem can be applied to three dimensions as follows. Consider the cuboid shown in the figure. The length of face diagonal AC is found from Pythagoras's theorem as the square root of the sum of squares of its sides. Using diagonal AC and the horizontal edge, the length of body diagonal AD then is found by a second application of Pythagoras's theorem. A substantial generalization of the Pythagorean theorem to three dimensions is de Gua's theorem named for Jean Paul de Gua de Malves. If a tetrahedron has a right angle corner like a corner of 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. This result can be generalized as in the n-dimensional Pythagorean theorem. Another generalization applies to Lebesgue-measurable sets of objects in any number of dimensions. Specifically, the square of the measure of an n-dimensional set of objects in one or more parallel k-dimensional flats in m-dimensional Euclidean space is equal to the sum of the squares of the measures of the orthogonal projections of the object onto all k-dimensional coordinate subspaces.

• 5. Non-Euclidean Geometry Applications

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  For any right triangle on a sphere of radius R with sides a, b, and c where c is the hypotenuse, the relation between the sides takes the form cos(c/R) equals cos(a/R) times cos(b/R). This equation can be derived as a special case of the spherical law of cosines that applies to all spherical triangles. For infinitesimal triangles on the sphere or equivalently for finite spherical triangles on a sphere of infinite radius, the spherical relation reduces to the Euclidean form of the Pythagorean theorem. In a hyperbolic space with uniform Gaussian curvature K, for a right triangle with legs a and b and hypotenuse c, the relation between the sides takes the form cosh(sqrt(K)c) equals cosh(sqrt(K)a) times cosh(sqrt(K)b). By using the Maclaurin series for the hyperbolic cosine, it can be shown that as a hyperbolic triangle becomes very small, the hyperbolic relation approaches the form of Pythagoras's theorem. The Pythagorean theorem implies and is implied by Euclid's Parallel Postulate. Thus, right triangles in a non-Euclidean geometry do not satisfy the standard Pythagorean theorem.

• 6. Modern Mathematical Frameworks

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  In an inner product space, the concept of perpendicularity is replaced by the concept of orthogonality. Two vectors u and v are orthogonal if their inner product is zero. The inner product is a generalization of the dot product of vectors. The dot product is called the standard inner product or the Euclidean inner product. However, other inner products are possible. In an inner-product space, the Pythagorean theorem states that for any two orthogonal vectors u and v we have the norm squared of their sum equals the sum of their individual norms squared. A further generalization of the Pythagorean theorem in an inner product space to non-orthogonal vectors is the parallelogram law. This says that twice the sum of the squares of the lengths of the sides of a parallelogram is the sum of the squares of the lengths of the diagonals. Any norm that satisfies this equality is ipso facto a norm corresponding to an inner product. For any complex number z with real part x and imaginary part y, the absolute value or modulus is given by the square root of x squared plus y squared. Geometrically this is the distance from z to the origin 0 in the complex plane.