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# Pythagorean theorem

In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. [1]

141 relations: Absolute value, Acute and obtuse triangles, Albert Einstein, Algebra, Altitude (triangle), American Mathematical Society, Ancient Greece, Apastamba Dharmasutra, Asymptotic expansion, Babylonian mathematics, Bartel Leendert van der Waerden, Baudhayana sutras, Berlin Papyrus 6619, Big O notation, British flag theorem, Calculus, Cartesian coordinate system, Cathetus, China, Chinese mathematics, Cicero, Clearing denominators, Commensurability (mathematics), Compass-and-straightedge construction, Complex number, Complex plane, Congruence (geometry), Coprime integers, Corollary, Cross product, Cube, Curved space, Curvilinear coordinates, Cut-the-Knot, De Gua's theorem, Dick Teresi, Differential equation, Dissection problem, Dot product, Edsger W. Dijkstra, Education Resources Information Center, Egypt, Equation, Euclid, Euclid's Elements, Euclidean distance, Euclidean geometry, Euclidean space, Facet (geometry), Fermat's Last Theorem, ... Expand index (91 more) »

## Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

## Acute and obtuse triangles

An acute triangle is a triangle with all three angles acute (less than 90°).

## Albert Einstein

Albert Einstein (14 March 1879 – 18 April 1955) was a German-born theoretical physicist who developed the theory of relativity, one of the two pillars of modern physics (alongside quantum mechanics).

## Algebra

Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.

## Altitude (triangle)

In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex).

## American Mathematical Society

The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.

## Ancient Greece

Ancient Greece was a civilization belonging to a period of Greek history from the Greek Dark Ages of the 13th–9th centuries BC to the end of antiquity (AD 600).

## Apastamba Dharmasutra

Āpastamba Dharmasūtra is a Sanskrit text and one of the oldest Dharma-related texts of Hinduism that have survived into the modern age from the 1st-millennium BCE.

## Asymptotic expansion

In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point.

## Babylonian mathematics

Babylonian mathematics (also known as Assyro-Babylonian mathematics) was any mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC.

## Bartel Leendert van der Waerden

Bartel Leendert van der Waerden (February 2, 1903 – January 12, 1996) was a Dutch mathematician and historian of mathematics.

## Baudhayana sutras

The Baudhayana sūtras are a group of Vedic Sanskrit texts which cover dharma, daily ritual, mathematics, etc.

## Berlin Papyrus 6619

The Berlin Papyrus 6619, simply called the Berlin Papyrus when the context makes it clear, is an ancient Egyptian papyrus document from the Middle Kingdom, second half of the 12th or 13th dynasty.

## Big O notation

Big O notation is a mathematical notation that describes the limiting behaviour of a function when the argument tends towards a particular value or infinity.

## British flag theorem

In Euclidean geometry, the British flag theorem says that if a point P is chosen inside rectangle ABCD then the sum of the squared Euclidean distances from P to two opposite corners of the rectangle equals the sum to the other two opposite corners.

## Calculus

Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

## Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

## Cathetus

In a right triangle, a cathetus (originally from the Greek word Κάθετος; plural: catheti), commonly known as a leg, is either of the sides that are adjacent to the right angle.

## China

China, officially the People's Republic of China (PRC), is a unitary one-party sovereign state in East Asia and the world's most populous country, with a population of around /1e9 round 3 billion.

## Chinese mathematics

Mathematics in China emerged independently by the 11th century BC.

## Cicero

Marcus Tullius Cicero (3 January 106 BC – 7 December 43 BC) was a Roman statesman, orator, lawyer and philosopher, who served as consul in the year 63 BC.

## Clearing denominators

In mathematics, the method of clearing denominators, also called clearing fractions, is a technique for simplifying an equation equating two expressions that each are a sum of rational expressions – which includes simple fractions.

## Commensurability (mathematics)

In mathematics, two non-zero real numbers a and b are said to be commensurable if their ratio is a rational number; otherwise a and b are called incommensurable.

## Compass-and-straightedge construction

Compass-and-straightedge construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.

## Congruence (geometry)

In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.

## Coprime integers

In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.

## Corollary

A corollary is a statement that follows readily from a previous statement.

## Cross product

In mathematics and vector algebra, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space \left(\mathbb^3\right) and is denoted by the symbol \times.

## Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

## Curved space

Curved space often refers to a spatial geometry which is not "flat" where a flat space is described by Euclidean geometry.

## Curvilinear coordinates

In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved.

## Cut-the-Knot

Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics.

## De Gua's theorem

De Gua's theorem is a three-dimensional analog of the Pythagorean theorem and named after Jean Paul de Gua de Malves.

## Dick Teresi

Dick Teresi is an American writer.

## Differential equation

A differential equation is a mathematical equation that relates some function with its derivatives.

## Dissection problem

In geometry, a dissection problem is the problem of partitioning a geometric figure (such as a polytope or ball) into smaller pieces that may be rearranged into a new figure of equal content.

## Dot product

In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.

## Edsger W. Dijkstra

Edsger Wybe Dijkstra (11 May 1930 – 6 August 2002) was a Dutch systems scientist, programmer, software engineer, science essayist, and early pioneer in computing science.

## Education Resources Information Center

The Education Resources Information Center (ERIC) is an online digital library of education research and information.

## Egypt

Egypt (مِصر, مَصر, Khēmi), officially the Arab Republic of Egypt, is a transcontinental country spanning the northeast corner of Africa and southwest corner of Asia by a land bridge formed by the Sinai Peninsula.

## Equation

In mathematics, an equation is a statement of an equality containing one or more variables.

## Euclid

Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".

## Euclid's Elements

The Elements (Στοιχεῖα Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC.

## Euclidean distance

In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space.

## Euclidean geometry

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## Facet (geometry)

In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself.

## Fermat's Last Theorem

In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers,, and satisfy the equation for any integer value of greater than 2.

## First Babylonian dynasty

The chronology of the first dynasty of Babylonia (also First Babylonian Empire) is debated as there is a Babylonian King List A and a Babylonian King List B. In this chronology, the regnal years of List A are used due to their wide usage.

## Flat (geometry)

In geometry, a flat is a subset of n-dimensional space that is congruent to a Euclidean space of lower dimension.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.

## Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

## Greatest common divisor

In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.

## Hammurabi

Hammurabi was the sixth king of the First Babylonian Dynasty, reigning from 1792 BC to 1750 BC (according to the Middle Chronology).

## Hippasus

Hippasus of Metapontum (Ἵππασος ὁ Μεταποντῖνος, Híppasos; fl. 5th century BC), was a Pythagorean philosopher.

## Hippocrates of Chios

Hippocrates of Chios (Ἱπποκράτης ὁ Χῖος) was an ancient Greek mathematician, geometer, and astronomer who lived c. 470 &ndash; c. 410 BC.

## Hyperbolic function

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.

## Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.

## Hyperbolic law of cosines

In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigonometry.

## Hypotenuse

In geometry, a hypotenuse (rarely: hypothenuse) is the longest side of a right-angled triangle, the side opposite of the right angle.

## India

India (IAST), also called the Republic of India (IAST), is a country in South Asia.

## Indian mathematics

Indian mathematics emerged in the Indian subcontinent from 1200 BC until the end of the 18th century.

## Inner product space

In linear algebra, an inner product space is a vector space with an additional structure called an inner product.

## Institute of Education Sciences

The Institute of Education Sciences (IES) is the independent, non-partisan statistics, research, and evaluation arm of the U.S. Department of Education.

## Ipso facto

Ipso facto is a Latin phrase, directly translated as "by the fact itself", which means that a specific phenomenon is a direct consequence, a resultant effect, of the action in question, instead of being brought about by a previous action.

## Isosceles triangle

In geometry, an isosceles triangle is a triangle that has two sides of equal length.

## James A. Garfield

James Abram Garfield (November 19, 1831 – September 19, 1881) was the 20th President of the United States, serving from March 4, 1881, until his assassination later that year.

## Java (programming language)

Java is a general-purpose computer-programming language that is concurrent, class-based, object-oriented, and specifically designed to have as few implementation dependencies as possible.

## Jean Paul de Gua de Malves

Jean Paul de Gua de Malves (1713, Malves-en-Minervois (Aude) – June 2, 1785, Paris) was a French mathematician who published in 1740 a work on analytical geometry in which he applied it, without the aid of differential calculus, to find the tangents, asymptotes, and various singular points of an algebraic curve.

## Law of cosines

In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles.

## Lebesgue measure

In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space.

## Lemma (mathematics)

In mathematics, a "helping theorem" or lemma (plural lemmas or lemmata) is a proven proposition which is used as a stepping stone to a larger result rather than as a statement of interest by itself.

## Length

In geometric measurements, length is the most extended dimension of an object.

## Leon M. Lederman

Leon Max Lederman (born July 15, 1922) is an American experimental physicist who received the Wolf Prize in Physics in 1982, along with Martin Lewis Perl, for their research on quarks and leptons, and the Nobel Prize for Physics in 1988, along with Melvin Schwartz and Jack Steinberger, for their research on neutrinos.

## Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.

## List of triangle topics

This list of triangle topics includes things related to the geometric shape, either abstractly, as in idealizations studied by geometers, or in triangular arrays such as Pascal's triangle or triangular matrices, or concretely in physical space.

## Loss of significance

Loss of significance is an undesirable effect in calculations using finite-precision arithmetic such as floating-point arithmetic.

## Lp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces.

## Mathematical proof

In mathematics, a proof is an inferential argument for a mathematical statement.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Mesopotamia

Mesopotamia is a historical region in West Asia situated within the Tigris–Euphrates river system, in modern days roughly corresponding to most of Iraq, Kuwait, parts of Northern Saudi Arabia, the eastern parts of Syria, Southeastern Turkey, and regions along the Turkish–Syrian and Iran–Iraq borders.

## Middle Kingdom of Egypt

The Middle Kingdom of Egypt (also known as The Period of Reunification) is the period in the history of ancient Egypt between circa 2050 BC and 1710 BC, stretching from the reunification of Egypt under the impulse of Mentuhotep II of the Eleventh Dynasty to the end of the Twelfth Dynasty.

## Non-Euclidean geometry

In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry.

## Nonhypotenuse number

In mathematics, a nonhypotenuse number is a natural number whose square cannot be written as the sum of two nonzero squares.

## Normed vector space

In mathematics, a normed vector space is a vector space over the real or complex numbers, on which a norm is defined.

## Orthogonality

In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.

## Pappus of Alexandria

Pappus of Alexandria (Πάππος ὁ Ἀλεξανδρεύς; c. 290 &ndash; c. 350 AD) was one of the last great Greek mathematicians of Antiquity, known for his Synagoge (Συναγωγή) or Collection (c. 340), and for Pappus's hexagon theorem in projective geometry.

## Pappus's area theorem

Pappus's area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle.

## Parallel postulate

In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry.

## Parallelogram law

In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry.

## Perpendicular

In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees).

## Plato

Plato (Πλάτων Plátōn, in Classical Attic; 428/427 or 424/423 – 348/347 BC) was a philosopher in Classical Greece and the founder of the Academy in Athens, the first institution of higher learning in the Western world.

## Plimpton 322

Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics.

## Plutarch

Plutarch (Πλούταρχος, Ploútarkhos,; c. CE 46 – CE 120), later named, upon becoming a Roman citizen, Lucius Mestrius Plutarchus, (Λούκιος Μέστριος Πλούταρχος) was a Greek biographer and essayist, known primarily for his Parallel Lives and Moralia.

## Polar coordinate system

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

## Proclus

Proclus Lycaeus (8 February 412 – 17 April 485 AD), called the Successor (Greek Πρόκλος ὁ Διάδοχος, Próklos ho Diádokhos), was a Greek Neoplatonist philosopher, one of the last major classical philosophers (see Damascius).

## Proportionality (mathematics)

In mathematics, two variables are proportional if there is always a constant ratio between them.

## Ptolemy's theorem

In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle).

## Pythagoras

Pythagoras of Samos was an Ionian Greek philosopher and the eponymous founder of the Pythagoreanism movement.

## Pythagorean expectation

Pythagorean expectation is a sports analytics formula devised by Bill James to estimate the percentage of games a baseball team "should" have won based on the number of runs they scored and allowed.

## Pythagorean tiling

A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides.

## Pythagorean triple

A Pythagorean triple consists of three positive integers,, and, such that.

## Q.E.D.

Q.E.D. (also written QED and QED) is an initialism of the Latin phrase quod erat demonstrandum meaning "what was to be demonstrated" or "what was to be shown." Some may also use a less direct translation instead: "thus it has been demonstrated." Traditionally, the phrase is placed in its abbreviated form at the end of a mathematical proof or philosophical argument when the original proposition has been restated exactly, as the conclusion of the demonstration or completion of the proof.

In mathematics, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the set of rational numbers.

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers.

## Ratio

In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second.

## Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

## Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point.

## Right angle

In geometry and trigonometry, a right angle is an angle of exactly 90° (degrees), corresponding to a quarter turn.

## Right triangle

A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle).

## Seven-dimensional cross product

In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space.

## Shulba Sutras

The Shulba Sutras or Śulbasūtras (Sanskrit: "string, cord, rope") are sutra texts belonging to the Śrauta ritual and containing geometry related to fire-altar construction.

## Sign function

In mathematics, the sign function or signum function (from signum, Latin for "sign") is an odd mathematical function that extracts the sign of a real number.

## Similarity (geometry)

Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other.

## Sine

In mathematics, the sine is a trigonometric function of an angle.

## Spherical geometry

Spherical geometry is the geometry of the two-dimensional surface of a sphere.

## Spherical law of cosines

In spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry.

## Square root

In mathematics, a square root of a number a is a number y such that; in other words, a number y whose square (the result of multiplying the number by itself, or) is a. For example, 4 and −4 are square roots of 16 because.

## Stigler's law of eponymy

Stigler's law of eponymy is a process proposed by University of Chicago statistics professor Stephen Stigler in his 1980 publication "Stigler’s law of eponymy".

## Sum of angles of a triangle

In several geometries, a triangle has three vertices and three sides, where three angles of a triangle are formed at each vertex by a pair of adjacent sides.

## Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

## Tetrahedron

In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners.

## Thābit ibn Qurra

(ثابت بن قره, Thebit/Thebith/Tebit; 826 – February 18, 901) was a Syrian Arab Sabian mathematician, physician, astronomer, and translator who lived in Baghdad in the second half of the ninth century during the time of Abbasid Caliphate.

## The Mathematical Gazette

The Mathematical Gazette is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive areas of mathematics." It was established in 1894 by Edward Mann Langley as the successor to the Reports of the Association for the Improvement of Geometrical Teaching.

## The Nine Chapters on the Mathematical Art

The Nine Chapters on the Mathematical Art is a Chinese mathematics book, composed by several generations of scholars from the 10th&ndash;2nd century BCE, its latest stage being from the 2nd century CE.

## Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms.

## Thomas Little Heath

Sir Thomas Little Heath (5 October 1861 – 16 March 1940) was a British civil servant, mathematician, classical scholar, historian of ancient Greek mathematics, translator, and mountaineer.

## Trapezoid

In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezoid in American and Canadian English but as a trapezium in English outside North America.

## Triangle inequality

In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.

## Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.

## Trigonometry

Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles.

## United States Department of Education

The United States Department of Education (ED or DoED), also referred to as the ED for (the) Education Department, is a Cabinet-level department of the United States government.

## United States House of Representatives

The United States House of Representatives is the lower chamber of the United States Congress, the Senate being the upper chamber.

## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

## Vertex (geometry)

In geometry, a vertex (plural: vertices or vertexes) is a point where two or more curves, lines, or edges meet.

## Yale University

Yale University is an American private Ivy League research university in New Haven, Connecticut.

## Zhoubi Suanjing

The Zhoubi Suanjing, or Chou Pei Suan Ching (周髀算经), is one of the oldest Chinese mathematical texts.

## References

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