In mathematics, the Pythagorean theorem (American English) or Pythagoras' theorem (British English) is a relation in Euclidean geometry among the three sides of a right triangle. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Phonology North American English regional phonology In many ways compared to English English, North American English is conservative in its Phonology. British English or UK English ( BrE, BE, en-GB) is the broad term used to distinguish the forms of the English language used in the Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line The theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and proof,^[1] although knowledge of the theorem almost certainly predates him. In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements The Greeks ( Greek: Έλληνες) are a Nation and Ethnic group native to Greece, Cyprus and neighbouring regions A mathematician is a person whose primary area of study and research is the field of Mathematics. "Pythagoras of Samos" redirects here For the Samian statuary of the same name see Pythagoras (sculptor.

The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).

The theorem is as follows:

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). Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides A hypotenuse is the longest side of a Right triangle, the side opposite of the 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 length 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. Length is the long Dimension of any object The length of a thing is the distance between its ends its linear extent as measured from end to end Conversion is a concept in Traditional logic referring to a "type of immediate Inference in which from a given Proposition another proposition is inferred A generalization of this theorem is the law of cosines, 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. In Trigonometry, the law of cosines (also known as Al-Kashi law or the cosine formula or cosine rule) is a statement about a general If the angle between the sides is a right angle it reduces to the Pythagorean theorem.

Visual proof for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC. The Zhou Bi Suan Jing (周髀算经 The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven is one of the oldest and most famous Chinese

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. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. The history of Trigonometry and of Trigonometric functions may span about 4000 years Trigonometry has an enormous variety of applications The ones mentioned explicitly in textbooks and courses on trigonometry are its uses in practical endeavors such as For a more comprehensive list see the List of trigonometry topics. In Mathematics, trigonometric identities are equalities that involve Trigonometric functions that are true for every single value of the occurring variables Exact constant expressions for Trigonometric expressions are sometimes useful mainly for simplifying solutions into radical forms which allow further simplification In Mathematics, tables of Trigonometric functions are useful in a number of areas CORDIC (digit-by-digit method Volder's algorithm (for CO ordinate R otation DI gital C omputer is a simple and efficient Algorithm Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. The law of sines ( sines law sine formula sine rule) in Trigonometry, is a statement about any Triangle in a plane In Trigonometry, the law of cosines (also known as Al-Kashi law or the cosine formula or cosine rule) is a statement about a general In Trigonometry, the law of tangents is a statement about the relationship between the lengths of the three sides of a triangle and the tangents of the angles Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In Mathematics, the trigonometric integrals are a family of Integrals which involve Trigonometric functions A number of the basic trigonometric In Mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions The following is a list of Integrals ( Antiderivative functions of Trigonometric functions. The following is a list of Integrals ( Antiderivative formulas for integrands that contain inverse Trigonometric functions (also known as "arc functions" A Pythagorean triple consists of three positive Integers a, b, and c, such that a 2 +  b 2 =  Two types of special right triangles appear commonly in geometry the "angle based" and the "side based" (or Pythagorean Triangles The former are characterised

Megalithic monuments from circa 2500 BC in Egypt, and in Northern Europe, incorporate right triangles with integer sides. This article is about the country of Egypt For a topic outline on this subject see List of basic Egypt topics. Northern Europe is a term for the northern part of Europe. The United Nations defines Northern Europe as (Finland ^[2] Bartel Leendert van der Waerden conjectures that these Pythagorean triples were discovered algebraically. Bartel Leendert van der Waerden ( February 2 1903, Amsterdam, Netherlands – January 12 1996, Zürich, Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. ^[3]

Written between 2000 and 1786 BC, the Middle Kingdom Egyptian papyrus Berlin 6619 includes a problem whose solution is a Pythagorean triple. The Middle Kingdom is the period in the history of Ancient Egypt stretching from the establishment of the Eleventh Dynasty to the end of the Fourteenth Dynasty This article is about the country of Egypt For a topic outline on this subject see List of basic Egypt topics. The Berlin Papyrus 6619 commonly known as the Berlin Papyrus is an Ancient Egyptian papyrus document from the Middle Kingdom. A Pythagorean triple consists of three positive Integers a, b, and c, such that a 2 +  b 2 = 

During the reign of Hammurabi the Great, the Mesopotamian tablet Plimpton 322, written between 1790 and 1750 BC, contains many entries closely related to Pythagorean triples. Hammurabi ( Akkadian from Amorite ˤAmmurāpi, "the kinsman is a healer" from ˤAmmu, "paternal kinsman" and Rāpi Mesopotamia (from the Greek meaning "land between the rivers" is an area geographically located between the Tigris and Euphrates rivers largely corresponding Of the approximately half million Babylonian Clay tablets excavated since the beginning of the 19th century several thousand are of a mathematical nature The 18th century BC was the Century which lasted from 1800 BC to 1701 BC

The Baudhayana Sulba Sutra, the dates of which are given variously as between the 8th century BC and the 2nd century BC, in India, contains a list of Pythagorean triples discovered algebraically, a statement of the Pythagorean theorem, and a geometrical proof of the Pythagorean theorem for an isosceles right triangle. Baudhāyana, (fl ca 800 BCE was an Indian mathematician whowas most likely also a priest The Shulba Sutras or Śulbasūtras ( Sanskrit śulba: "string cord rope" are Sutra texts belonging to the The 8th century BC started the first day of 800 BC and ended the last day of 701 BC. The 2nd century BC started the first day of 200 BC and ended the last day of 101 BC. This article is about the history of South Asia prior to the Partition of British India in 1947 A Pythagorean triple consists of three positive Integers a, b, and c, such that a 2 +  b 2 =  Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line

The Apastamba 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". Bartel Leendert van der Waerden ( February 2 1903, Amsterdam, Netherlands – January 12 1996, Zürich, According to Albert Bŭrk, this is the original proof of the theorem; he further theorizes that Pythagoras visited Arakonam, India, and copied it. WikipediaWikiProject Indian cities for details --> Arakkonam (also "Arkonam") (Tamil அரக்கோணம் is a mid-sized town in the

Pythagoras, whose dates are commonly given as 569–475 BC, used algebraic methods to construct Pythagorean triples, according to Proklos's commentary on Euclid. "Pythagoras of Samos" redirects here For the Samian statuary of the same name see Pythagoras (sculptor. Proclus Lycaeus ( February 8, c 411 &ndash April 17, 485) called "The Successor" or "Diadochos" ( Greek Próklos Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Proklos, however, wrote between 410 and 485 AD. According to Sir Thomas L. Heath, there is no attribution of the theorem to Pythagoras for five centuries after Pythagoras lived. Sir Thomas Little Heath ( October 5, 1861 &ndash March 16, 1940) was a British civil servant Mathematician, classical However, when authors such as Plutarch and Cicero attributed the theorem to Pythagoras, they did so in a way which suggests that the attribution was widely known and undoubted. Lucius Mestrius Plutarchus ( Greek: Μέστριος Πλούταρχος c Marcus Tullius Cicero ( Classical Latin ˈkikeroː usually ˈsɪsərəʊ in English January 3, 106 BC &ndash December 7, 43 BC was a Roman ^[4]

Around 400 BC, according to Proklos, Plato gave a method for finding Pythagorean triples that combined algebra and geometry. Biography Early life Birth and family Plato was born in Athens Greece Circa 300 BC, in Euclid's Elements, the oldest extant axiomatic proof of the theorem is presented. Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and

Written sometime between 500 BC and 200 AD, the Chinese text Chou Pei Suan Ching (周髀算经), (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" (勾股定理) — for the (3, 4, 5) triangle. China ( Wade-Giles ( Mandarin) Chung¹kuo² is a cultural region, an ancient Civilization, and depending on perspective a National The Zhou Bi Suan Jing (周髀算经 The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven is one of the oldest and most famous Chinese During the Han Dynasty, from 202 BC to 220 AD, Pythagorean triples appear in The Nine Chapters on the Mathematical Art, together with a mention of right triangles. The Han Dynasty ( 206 BC–220 AD followed the Qin Dynasty and preceded the Three Kingdoms in China. The Nine Chapters on the Mathematical Art ( is a Chinese Mathematics book composed by several generations of scholars from the 10th&ndash2nd century BC and ^[5]

The first recorded use is in China, known as the "Gougu theorem" (勾股定理) and in India known as the Bhaskara Theorem. China ( Wade-Giles ( Mandarin) Chung¹kuo² is a cultural region, an ancient Civilization, and depending on perspective a National India, officially the Republic of India (भारत गणराज्य inc-Latn Bhārat Gaṇarājya; see also other Indian languages) is a country

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. ^[6]

Proofs

This is a theorem that may have more known proofs than any other (the law of quadratic reciprocity being also a contender for that distinction); the book Pythagorean Proposition, by Elisha Scott Loomis, contains 367 proofs. The law of quadratic reciprocity is a theorem from Modular arithmetic, a branch of Number theory, which shows a remarkable relationship between the solvability

Some arguments based on trigonometric identities (such as Taylor series for sine and cosine) have been proposed as proofs for the theorem. In Mathematics, trigonometric identities are equalities that involve Trigonometric functions that are true for every single value of the occurring variables In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives However, since all the fundamental trigonometric identities are proved using the Pythagorean theorem, there cannot be any trigonometric proof. (See also begging the question. In Logic, begging the question has traditionally described a type of Logical fallacy (also called petitio principii) in which the proposition )

Proof using similar triangles

Proof using similar triangles

Like most of the proofs of the Pythagorean theorem, this one is based on the proportionality of the sides of two similar triangles. This article is about proportionality the mathematical relation

Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. We draw the altitude from point C, and call H its intersection with the side AB. In Geometry, an altitude of a triangle is a Straight line through a vertex and Perpendicular to (i The new triangle ACH is similar 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. Geometry Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking of the other 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

Proof in Euclid's Elements

In Euclid's Elements, Proposition 47 of Book 1, the Pythagorean theorem is proved by an argument along the following lines. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek 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:

1. 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)
2. The area of a triangle is half the area of any parallelogram on the same base and having the same altitude.
3. The area of any square is equal to the product of two of its sides.
4. 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 parallelograms with the same size, then turned and morphed into the left and right rectangles in the lower square, again at constant area. In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides

The proof is as follows:

1. Let ABC be a right-angled triangle with right angle CAB.
2. On each of the sides BC, AB, and CA, squares are drawn, CBDE, BAGF, and ACIH, in that order.
3. From A, draw a line parallel to BD and CE. It will perpendicularly intersect BC and DE at K and L, respectively.
4. Join CF and AD, to form the triangles BCF and BDA.

   

   Illustration including the new lines

5. Angles CAB and BAG are both right angles; therefore C, A, and G are collinear. Similarly for B, A, and H.
6. 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.
7. Since AB and BD are equal to FB and BC, respectively, triangle ABD must be equal to triangle FBC.
8. Since A is collinear with K and L, rectangle BDLK must be twice in area to triangle ABD.
9. Since C is collinear with A and G, square BAGF must be twice in area to triangle FBC.
10. Therefore rectangle BDLK must have the same area as square BAGF = AB².
11. Similarly, it can be shown that rectangle CKLE must have the same area as square ACIH = AC².
12. Adding these two results, AB² + AC² = BD* BK + KL* KC
13. Since BD = KL, BD* BK + KL* KC = BD(BK + KC) = BD* BC
14. Therefore AB² + AC² = BC², since CBDE is a square.

This proof appears in Euclid's Elements as that of Proposition 1. 47. ^[7]

Garfield's proof

James A. Garfield (later President of the United States) is credited with a novel algebraic proof[1] using a trapezoid containing two examples of the triangle, the figure comprising one-half of the figure using four triangles enclosing a square shown below. James Abram Garfield (November 19 1831 September 19 1881 was the twentieth President of the United States. A trapezoid (in North America or a trapezium (in Britain and elsewhere is a Quadrilateral (a closed plane shape with four linear sides that has at least one

Similarity proof

From the same diagram as that in Euclid's proof above, we can see three similar figures, each being "a square with a triangle on top". Geometry Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking of the other 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 using area subtraction

Proof by rearrangement

Proof of Pythagorean theorem by rearrangement of 4 identical right triangles. Since the total area and the areas of the triangles are all constant, the total black area is constant. But this can be divided into squares delineated by the triangle sides a, b, c, demonstrating that a² + b² = c² .

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.

Elegant animation showing another proof by rearrangement

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 geometry. QED is an abbreviation of the Latin phrase "la '''quod erat demonstrandum'''" which means literally "that which was to be demonstrated" 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 measure and Banach-Tarski paradox). In Mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a Length, Area or Volume to The Banach–Tarski paradox is a Theorem in set theoretic Geometry which states that a solid ball in 3-dimensional space can be split into several 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).

Proof using rearrangement

A third graphic illustration of the Pythagorean theorem (in yellow and blue to the left) 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. ^[8]

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.

A square created by aligning four right angle triangles and a large square.

The A-side angle and B-side angle of each of these triangles are complementary angles, 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. A pair of Angles is complementary if the sum of their measures add up to 90 degrees. 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 calculus. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives ^[9]

Proof using differential equations

As a result of a change in side a,

by similar triangles and for differential changes. So

upon separation of variables. In Mathematics, separation of variables is any of several methods for solving ordinary and partial Differential equations in which algebra allows one to re-write an A more general result is

which results from adding a second term for changes in side b.

Integrating gives

So

As can be seen, the squares are due to the particular proportion 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. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space This article is about proportionality the mathematical relation From the proportion given it can be shown that the changes in the sides are inversely proportional to the sides. The differential equation suggests that the theorem is due to relative changes and its derivation is nearly equivalent to computing a line integral. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the In Mathematics, a line integral (sometimes called a path integral or curve integral) is an Integral where the function to be integrated A simpler derivation would leave fixed and then observe that

It is doubtful that the Pythagoreans would have been able to do the above proof but they knew how to compute the area of a triangle and were familiar with figurate numbers and the gnomon, a segment added onto a geometrical figure. "Pythagoras of Samos" redirects here For the Samian statuary of the same name see Pythagoras (sculptor. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line A figurate number is a number that can be represented as a regular and discrete geometric pattern (e In Geometry, a gnomon is a plane figure formed by removing a similar Parallelogram from a corner of a larger parallelogram All of these ideas predate calculus and are an alternative for the integral.

The proportional relation between the changes and their sides is at best an approximation, so how can one justify its use? The answer is the approximation gets better for smaller changes since the arc of the circle which cuts off c more closely approaches the tangent to the circle. For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. As for the sides and triangles, no matter how many segments they are divided into the sum of these segments is always the same. The Pythagoreans were trying to understand change and motion and this led them to realize that the number line was infinitely divisible. Could they have discovered the approximation for the changes in the sides? One only has to observe that the motion of the shadow of a sundial produces the hypotenuses of the triangles to derive the figure shown. A sundial is a device that measures time by the position of the Sun.

Rational trigonometry

For a proof by the methods of rational trigonometry, see Pythagorean theorem proof (rational trigonometry). Divine Proportions Rational Trigonometry to Universal Geometry is a book by Norman Wildberger, presenting his reformulation of Trigonometry.

Converse

The converse 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. Conversion is a concept in Traditional logic referring to a "type of immediate Inference in which from a given Proposition another proposition is inferred

This converse also appears in Euclid's Elements. It can be proven using the law of cosines (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. In Trigonometry, the law of cosines (also known as Al-Kashi law or the cosine formula or cosine rule) is a statement about a general 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 congruent, and so they must have the same angles. In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i Therefore, the angle between the side of lengths a and b in our original triangle is a right angle.

A corollary of the Pythagorean theorem's converse is a simple means of determining whether a triangle is right, obtuse, or acute, as follows. A corollary is a statement which follows readily from a previously proven statement 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

Main article: Pythagorean triple

A Pythagorean triple has 3 positive numbers a, b, and c, such that a² + b² = c². A Pythagorean triple consists of three positive Integers a, b, and c, such that a 2 +  b 2 =  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 numbers, such as the square root of two, can be constructed. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction A right triangle with legs both equal to one unit has hypotenuse length square root of two. The Pythagoreans proved that the square root of two is irrational, and this proof has come down to us even though it flew in the face of their cherished belief that everything was rational. Pythagoreanism is a term used for the Esoteric and metaphysical beliefs held by Pythagoras and his followers the Pythagoreans who were much influenced In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction According to the legend, Hippasus, who first proved the irrationality of the square root of two, was drowned at sea as a consequence. Hippasus of Metapontum (Ίππασος b c 500 BC in Magna Graecia, was a Greek Philosopher. ^[10]

Distance in Cartesian coordinates

The distance formula in Cartesian coordinates is derived from the Pythagorean theorem. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane If (x₀, y₀) and (x₁, y₁) are points in the plane, then the distance between them, also called the Euclidean distance, is given by

More generally, in Euclidean n-space, the Euclidean distance between two points, and , is defined, using the Pythagorean theorem, as:

Generalizations

The Pythagorean theorem was generalized by Euclid in his Elements:

If one erects similar figures (see Euclidean geometry) 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. In Mathematics, the Euclidean distance or Euclidean metric is the "ordinary" Distance between two points that one would measure with a ruler Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria.

The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines:

where θ is the angle between sides a and b. In Trigonometry, the law of cosines (also known as Al-Kashi law or the cosine formula or cosine rule) is a statement about a general

When θ is 90 degrees, then cos(θ) = 0, so the formula reduces to the usual Pythagorean theorem.

Given two vectors v and w in a complex inner product space, 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. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i

Using mathematical induction, the previous result can be extended to any finite number of pairwise orthogonal vectors. Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i 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 real inner product spaces is known as Parseval's identity. In Mathematics, the real numbers may be described informally in several different ways In Mathematical analysis, Parseval's identity is a fundamental result on the Summability of the Fourier series of a function

When the theorem above about vectors is rewritten in terms of solid geometry, it becomes the following theorem. In Mathematics, solid geometry was the traditional name for the Geometry of three-dimensional Euclidean space &mdash for practical purposes the kind of 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 theorem, named for Jean Paul de Gua de Malves: If a tetrahedron 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. De Gua's theorem is a spatial analog of the Pythagorean theorem and named for Jean Paul de Gua de Malves. Jean Paul de Gua de Malves ( Carcassonne, 1713 &ndash June 2, 1785 Paris) was a French Mathematician who published in 1740 A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex.

There are also analogs of these theorems in dimensions four and higher.

In a triangle with three acute angles, α + β > γ holds. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called Therefore, a² + b² > c² holds.

In a triangle with an obtuse angle, α + β < γ holds. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called Therefore, a² + b² < c² holds.

Edsger Dijkstra has stated this proposition about acute, right, and obtuse triangles in this language:

sgn(α + β − γ) = sgn(a² + b² − c²)

where α is the angle opposite to side a, β is the angle opposite to side b and γ is the angle opposite to side c. Edsger Wybe Dijkstra ( May 11, 1930 &ndash August 6, 2002; ˈɛtsxər ˈwibə ˈdɛɪkstra was a Dutch computer scientist ^[11]

The Pythagorean theorem in non-Euclidean geometry

The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, the Euclidean form of the Pythagorean theorem given above does not hold in non-Euclidean geometry. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry (It has been shown in fact to be equivalent to Euclid's Parallel (Fifth) Postulate. ) For example, in spherical geometry, 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 . Spherical geometry is the Geometry of the two- Dimensional surface of a Sphere.

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 geometry and hyperbolic plane geometry; 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. Spherical geometry is the Geometry of the two- Dimensional surface of a Sphere. In In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives

For any triangle in the hyperbolic plane (with Gaussian curvature −1), the Pythagorean theorem takes the form

where cosh is the hyperbolic cosine. In In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry 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 geometry, for a right triangle one can also write,

where is the angle of parallelism of the line segment AB that where μ is the multiplicative distance function (see Hilbert's arithmetic of ends). In In Hyperbolic geometry, the angle of parallelism Φ is the Angle at one vertex of a right Hyperbolic triangle that has two asymptotic parallel sides In Algebraic geometry, \mu is said to be a multiplicative Distance function over a field if it satisfies \mu(AB>1 Hilbert's arithmetic of ends is an algebraic approach introduced by German mathematician David Hilbert for Poincaré disk model of Hyperbolic geometry

In hyperbolic trigonometry, the sine of the angle of parallelism satisfies

Thus, the equation takes the form

where a, b, and c are multiplicative distances of the sides of the right triangle (Hartshorne, 2000). In Mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular functions In Hyperbolic geometry, the angle of parallelism Φ is the Angle at one vertex of a right Hyperbolic triangle that has two asymptotic parallel sides

Cultural references to the Pythagorean theorem

• In The Wizard of Oz, when the Scarecrow receives his diploma from the Wizard, he immediately exhibits his "knowledge" by reciting a mangled and incorrect version of the theorem: "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side. The Wizard of Oz is a 1939 American musical - Fantasy film mainly directed by Victor Fleming and based on the 1900 children’s The Scarecrow is a character in the fictional Land of Oz created by American author L The Wizard of Oz (or simply The Wizard) is a Fictional character in the Land of Oz created by American author L Oh, joy, oh, rapture. I've got a brain!" The "knowledge" exhibited by the Scarecrow is incorrect. The accurate statement would have been "The sum of the squares of the legs of a right triangle is equal to the square of the remaining side. "^[12]
• In an episode of The Simpsons, Homer quotes the Oz Scarecrow's quote, thus turning the theorem into a cultural reference to a cultural reference. Homer Jay Simpson is a main fictional character in the animated television series The Simpsons and father of the eponymous family. After finding a pair of Henry Kissinger's glasses in a toilet at the Springfield Nuclear Power Plant, Homer puts them on and quotes the scarecrow's mangled formula. Henry Alfred Kissinger (born Heinz Alfred Kissinger on May 27, 1923) is a German -born American bureaucrat diplomat and 1973 Profile The plant the core of which is a Fissionator 1952 Slow-Fission Reactor is poor and badly maintained largely due to owner Montgomery Burns' negligence especially considering Homer Jay Simpson is a main fictional character in the animated television series The Simpsons and father of the eponymous family. A man in a nearby toilet stall then yells out "That's a right triangle, you idiot!" (The comment about square roots remained uncorrected. )
• Similarly, the Speech software on an Apple MacBook references the Scarecrow's incorrect statement. Apple Inc, ( formerly Apple Computer Inc, is an American Multinational corporation with a focus on designing and manufacturing Consumer electronics The MacBook is a Macintosh Notebook computer by Apple Inc that replaced the iBook G4 series It is the sample speech when the voice setting 'Ralph' is selected.
• In the English version of the seventeenth Asterix book "The Mansions of the Gods", Julius Caesar uses the services of an architect named "Squaronthehypotenus" to develop the estate near the village, through which he hopes to absorb the Gaulish village into the Roman culture. The Adventures of Asterix ( French: Astérix or Astérix le Gaulois) is a series of French
• In 2000, Uganda released a coin with the shape of a right triangle. The Republic of Uganda is a Landlocked country in East Africa. The tail has an image of Pythagoras and the Pythagorean theorem, accompanied with the mention "Pythagoras Millennium". ^[13]
• In the Major-General's Song, "About binomial theorem I'm teeming with a lot o' news, With many cheerful facts about the square of the hypotenuse. The Major-General's Song is a Patter song from Gilbert and Sullivan 's 1879 Comic opera The Pirates of Penzance. "
• Greece, Japan, San Marino, Sierra Leone, and Surinam have issued postage stamps depicting Pythagoras and the Pythagorean theorem. Greece (Ελλάδα transliterated: Elláda, historically, Ellás,) officially the Hellenic Republic (Ελληνική Δημοκρατία For a topic outline on this subject see List of basic Japan topics. The Most Serene Republic of San Marino (Serenissima Repubblica di San Marino is a country in the Apennine Mountains. Sierra Leone, officially the Republic of Sierra Leone, is a country in West Africa. Suriname ( Dutch: Suriname; Sranan Tongo: Sranan) officially the Republic of Suriname (traditionally spelled Surinam by A postage stamp is an adhesive paper evidence of pre-paying a fee for postal services ^[14]
• In South Park, Eric Cartman shows the Pythagorean theorem during a slide show at show and tell. South Park is an animated American television comedy series created and written by Trey Parker and Matt Stone for Comedy Central
• In Smart Guy, the teacher is asking the class what the formula for Pythagorean Theorem is. Smart Guy was an American Sitcom that aired on The WB for three seasons from 1997 to 1999. T. J. was supposed to be stupid but replied, "A squared plus B squared equals C squared. . . ".

See also

Notes

References

External links

• Pythagorean Theorem (more than 70 proofs from cut-the-knot)
• The Pythagorean Proposition - An e-book (in PDF format) with 367 different proofs of The Pythagorean Theorem
• Interactive links:
  • Interactive proof in Java of The Pythagorean Theorem
  • Another interactive proof in Java of The Pythagorean Theorem
  • Pythagorean theorem with interactive animation
  • Animated, Non-Algebraic, and User-Paced Pythagorean Theorem
• Eric W. Weisstein, Pythagorean theorem at MathWorld. Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in Mathematics. An e-book (for electronic book: also ebook) is the Digital media equivalent of a conventional printed Book. Java (Jawa is an Island of Indonesia and the site of its Capital city Jakarta. Java (Jawa is an Island of Indonesia and the site of its Capital city Jakarta. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
• FORTRAN code for solving with the Pythagorean theorem
• Interactive Mind Map of the Pythagorean theorem