In mathematics, the Pythagorean theorem (AmE) or Pythagoras' theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. 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. The first recorded use is in China, known as the "Gougu theorem" (勾股定理). For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... For other uses, see American English (disambiguation). ... British English (BrE, BE, en-GB) is the broad term used to distinguish the forms of the English language used in the United Kingdom from forms used elsewhere in the Anglophone world. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... A triangle. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Pythagoras of Samos (Greek: ; between 580 and 572 BC–between 500 and 490 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. ...

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: Image File history File links Pythagorean. ... Image File history File links Pythagorean. ...

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). For other uses, see Square. ... A right triangle and its hypotenuse, h, along with catheti, c1 and c2. ...

This is usually summarized as:

The square on the hypotenuse 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 For other uses of this word, see Length (disambiguation). ...

or, solved for c:

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 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. If the angle between the sides is a right angle it reduces to the Pythagorean theorem. Fig. ...

Visual proof for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.

History

The history of the theorem can be divided into three parts: knowledge of Pythagorean triples, knowledge of the relationship between the sides of a right triangle, and proofs of the theorem. The Pythagorean theorem: a2 + b2 = c2 A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

Megalithic monuments from circa 2500 BC in Egypt, and in the British Isles, incorporate right triangles with integer sides.^[2] Bartel Leendert van der Waerden conjectures that these Pythagorean triples were discovered algebraically.^[3] Megalithic tomb, Mane Braz, Brittany A megalith is a large stone which has been used to construct a structure or monument either alone or with other stones. ... This article describes the archipelago in north-western Europe. ... Bartel Leendert van der Waerden (February 2, 1903, Amsterdam, Netherlands – January 12, 1996, Zürich, Switzerland) was a Dutch mathematician. ... This article is about the branch of mathematics. ...

Written between 2000–1786 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, roughly between 2030 BC and 1640 BC. The period comprises two phases, the 11th Dynasty, which ruled from Thebes and the 12th Dynasty... The Berlin papyrus is an ancient Egyptian papyrus document that was created circa 1800 BCE. This papyrus was found at the Saqqara ancient Egyptian burial ground in the early 19th Century. ... The Pythagorean theorem: a2 + b2 = c2 A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). ...

During the reign of Hammurabi, the Mesopotamian tablet Plimpton 322, written between 1790 and 1750 BC, contains many entries closely related to Pythagorean triples. For the computer game, see Hamurabi. ... Mesopotamia was a cradle of civilization geographically located between the Tigris and Euphrates rivers, largely corresponding to modern-day Iraq. ... Of the approximately half million clay tablets excavated at the beginning of the 19th century, about 400 are of a mathematical nature. ... // Events 1787 - 1784 BC -- Amorite conquests of Uruk and Isin 1786 BC -- Egypt: Queen Sobekneferu died. ...

The Baudhayana Sulba Sutra, the dates of which are given variously as between the 8th century BCE and the 2nd century BCE, 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. ... The Sulba Sutras or Sulva Sutras are a text of Vedic mathematics. ... (9th century BC - 8th century BC - 7th century BC - other centuries) (800s BC - 790s BC - 780s BC - 770s BC - 760s BC - 750s BC - 740s BC - 730s BC - 720s BC - 710s BC - 700s BC - other decades) (2nd millennium BC - 1st millennium BC - 1st millennium AD) Events Golden age in Armenia Assyria... (3rd century BC - 2nd century BC - 1st century BC - other centuries) (2nd millennium BC - 1st millennium BC - 1st millennium AD) Events BC 168 Battle of Pydna -- Macedonian phalanx defeated by Romans BC 148 Rome conquers Macedonia BC 146 Rome destroys Carthage in the Third Punic War BC 146 Rome conquers... The Pythagorean theorem: a2 + b2 = c2 A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. ... For other uses, see Geometry (disambiguation). ... A triangle. ...

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". According to Albert Bŭrk, this is the original proof of the theorem; he further theorizes that Pythagoras visited Arakonam, India, and copied it. Apastamba (c. ... Bartel Leendert van der Waerden (February 2, 1903-January 12, 1996) was a Dutch mathematician who born in Amsterdam, Netherlands and died in Zürich, Switzerland. ... Arakkonam is a mid-sized town in the Indian state of Tamil Nadu with a population of about 77,000. ...

Pythagoras, whose dates are commonly given as 569–475 BC, used algebraic methods to construct Pythagorean triples, according to Proklos's commentary on Euclid. 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. 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. Pythagoras of Samos (Greek: ; between 580 and 572 BC–between 500 and 490 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. ... This article is about Proclus Diadochus, the Neoplatonist philosopher. ... For other uses, see Euclid (disambiguation). ... Mestrius Plutarchus (Greek: Πλούταρχος; 46 - 127), better known in English as Plutarch, was a Greek historian, biographer, essayist, and Middle Platonist. ... For other uses, see Cicero (disambiguation). ...

Around 400 BC, according to Proklos, Plato gave a method for finding Pythagorean triples that combined algebra and geometry. Circa 300 BC, in Euclid's Elements, the oldest extant axiomatic proof of the theorem is presented. For other uses, see Plato (disambiguation). ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...

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. 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.^[4] Wikipedia does not have an article with this exact name. ... Han Dynasty in 87 BC Capital Changan (202 BC–9 AD) Luoyang (25 AD–190 AD) Language(s) Chinese Religion Taoism, Confucianism Government Monarchy History  - Establishment 206 BC  - Battle of Gaixia; Han rule of China begins 202 BC  - Interruption of Han rule 9 - 24  - Abdication to Cao Wei 220... The Nine Chapters on the Mathematical Art (九章算術) is a Chinese mathematics book, probably composed in the 1st century AD, but perhaps as early as 200 BC. This book is the earliest surviving mathematical text from China that has come down to us by being copied by scribes and (centuries later...

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.^[5]

Proofs

This theorem 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. In number theory, the law of quadratic reciprocity connects the solvability of two related quadratic equations in modular arithmetic. ...

Some arguments based on trigonometric identities (such as Taylor series for sine 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 question.) In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ... Series expansion redirects here. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In logic, begging the question describes a type of logical fallacy, petitio principii, in which the conclusion of an argument is implicitly or explicitly assumed in one of the premises. ...

Proof using similar triangles

Proof using similar triangles

Like many of the proofs of the Pythagorean theorem, this one is based on the proportionality of the sides of two similar triangles. Image File history File links Proof-Pythagorean-Theorem. ... Image File history File links Proof-Pythagorean-Theorem. ... This article is about proportionality, the mathematical relation. ... Several equivalence relations in mathematics are called similarity. ...

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. 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. By a similar reasoning, the triangle CBH is also similar to ABC. The similarities lead to the two ratios..: In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to (i. ... // Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. ...

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. 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. Image File history File links Illustration_to_Euclid's_proof_of_the_Pythagorean_theorem. ... Image File history File links Illustration_to_Euclid's_proof_of_the_Pythagorean_theorem. ... For other uses, see Euclid (disambiguation). ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems...

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.
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. A parallelogram. ...

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.
5. Angles CAB and BAG are both right angles; therefore C, A, and G are colinear. 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 colinear with K and L, rectangle BDLK must be twice in area to triangle ABD.
9. Since C is colinear 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.^[6]

Garfield's proof

James A. Garfield (later President of the United States) is credited with a novel algebraic proof using a trapezoid containing two examples of the triangle[[1]], 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 a major general in the United States Army, member of the U.S. House of Representatives, and the twentieth President of the United States. ... This article is about the geometric figure. ...

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". 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. // Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. ...

Proof using area subtraction

Image File history File links Pythagorean_proof. ... Image File history File links Pythagorean_proof. ...

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. Look up QED in Wiktionary, the free dictionary. ...

Elegant animation showing a 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. 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). 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). Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ... The Banach–Tarski paradox: A ball can be decomposed and reassembled into two balls the same size as the original. ...

Proof using rearrangement

A second 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.^[7] Image File history File links Pythagorean_graphic. ... Image File history File links Pythagorean_graphic. ...

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. The area of this square is C². Thus the area of everything together is given by: Image File history File links Pythagproof. ... Image File history File links Pythagproof. ... A pair of complementary angles, because they add up to 90 degrees. ...

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.^[8] For other uses, see Calculus (disambiguation). ...

Proof using differential equations

As a result of a change in side a, Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

by similar triangles and for differential changes. So

upon separation of variables. A more general result is 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 equation so that each of two variables occurs on a different side of the equation. ...

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

Integrating gives This article is about the concept of integrals in calculus. ...

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. 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 simpler derivation would leave fixed and then observe that This article is about proportionality, the mathematical relation. ... Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ... This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ...

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. All of these ideas predate calculus and are an alternative for the integral. Pythagoras of Samos (Greek: ; between 580 and 572 BC–between 500 and 490 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. ... A triangle. ... 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 parallelogram from a corner of a larger parallelogram. ...

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. 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. For other uses, see tangent (disambiguation). ... For other uses, see Sundial (disambiguation). ...

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 Dr. Norman Wildberger of The University of New South Wales, presenting the authors reformulation of trigonometry. ... The Pythagorean theorem, expressed as a relation between the quadrances of the sides of a right triangle, is one of the five basic laws of the rational trigonometry system devised in the early 2000s by Dr. Norman Wildberger. ...

Converse

The converse of the theorem is also true: In traditional logic Conversion is a form of immediate inference in which from a given categorical proposition another proposition is inferred which has as its subject the predicate of the original proposition, and has as its predicate the subject of the original proposition, with the quality of the proposition remaining...

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 cosines (see below under Generalizations), or by the following proof: Fig. ...

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 congruent, 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. An example of congruence. ...

A corollary 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: A theorem is a statement which can be proven true within some logical framework. ...

• 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 consists of three positive integers a, b, and c, such that a² + b² = c². 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 British Isles 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). The Pythagorean theorem: a2 + b2 = c2 A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). ... The integers are commonly denoted by the above symbol. ...

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. 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. According to the legend, Hippasus, who first proved the irrationality of the square root of two, was drowned at sea as a consequence.^[9] In mathematics, an irrational number is any real number that is not a rational number, i. ... The Pythagoreans were an Hellenic organization of astronomers, musicians, mathematicians, and philosophers; who believed that all things are, essentially, numeric. ... 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 m/n, where m and n are integers, with n non-zero. ... Hippasus of Metapontum, born circa 500 B.C. in Magna Graecia, was a Greek philosopher. ...

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 distance, is given by Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... In mathematics, the Euclidean distance or Euclidean metric is the ordinary distance between two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ...

More generally, in Euclidean n-space, the Euclidean distance between two points, and , is defined, using the Pythagorean theorem, as: Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

Generalizations

The Pythagorean theorem was generalised by Euclid in his Elements: For other uses, see Euclid (disambiguation). ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems...

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. Euclid 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: Fig. ...

where θ is the angle between sides a and b.

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 mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

In particular, ||v + w||² = ||v||² + ||w||² if and only if v and w are orthogonal. In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...

Using mathematical induction, the previous result can be extended to any finite 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 Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...

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 as numbers that can be given by an infinite decimal representation, such as 2. ... In functional analysis, Parsevals identity, also known as Parsevals equality, is the Pythagorean theorem for inner-product spaces. ...

When the theorem above about vectors is rewritten in terms of solid geometry, 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. In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for practical purposes the kind of space we live in. ...

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 Guas theorem is a generalization of the Pythagorean theorem to three dimensions and named for Jean Paul de Gua de Malves. ... Jean Paul de Gua de Malves (1713 Carcassonne - 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 algebraical curve. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, 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. Therefore, a² + b² > c² holds. This article is about angles in geometry. ...

In a triangle with an obtuse angle, α + β < γ holds. Therefore, a² + b² < c² holds. This article is about angles in geometry. ...

Edsger Dijkstra has stated this proposition about acute, right, and obtuse triangles in this language: Edsger Dijkstra Edsger Wybe Dijkstra (Rotterdam, May 11, 1930 – Nuenen, August 6, 2002; IPA: ) was a Dutch computer scientist. ...

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.^[10] Signum function In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real number. ... Signum function In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real number. ...

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. (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 . Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ... 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: Spherical geometry is the geometry of the two-dimensional surface of a sphere. ... Lines through a given point P and asymptotic to line l. ...

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. As the degree of the taylor series rises, it approaches the correct function. ...

For any triangle in the hyperbolic plane (with Gaussian curvature −1), the Pythagorean theorem takes the form Lines through a given point P and asymptotic to line l. ... In mathematics, curvature refers to a number of loosely related concepts in different areas of geometry. ...

where cosh is the hyperbolic cosine. A ray through the origin intercepts the hyperbola in the point , where is the area between the ray, its mirror image with respect to the -axis, and the hyperbola (see animated version with comparison with the trigonometric (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, Lines through a given point P and asymptotic to line l. ...

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 hyperbolic geometry, the angle of parallelism &#934; is the angle at one vertex of an right hyperbolic triangle that has two parallel sides. ... is said to be a multiplicative distance function over a field if it satisfies, a. ... An algebraic approach introduced by German mathematician David Hilbert for Poincaré disk model of hyperbolic geometry [1]. One can also set up a hyperbolic analytic geometry and hyperbolic trigonometry, whereby any geometric problem can be translated into an algebraic problem in the field. ...

In hyperbolic trigonometry, the sine of the angle of parallelism satisfies A ray through the origin intercepts the hyperbola in the point , where is the area between the ray, its mirror image with respect to the -axis, and the hyperbola (see animated version with comparison with the trigonometric (circular) functions). ...

Thus, the equation takes the form

where a, b, and c are multiplicative distances of the sides of the right triangle (Hartshorne, 2000).

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. 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."^[11]
• 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. 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. A man in a nearby toilet stall then yells out "That's a right triangle, you idiot!" (The comment about square roots remained uncorrected.)
• 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.
• In 2000, Uganda released a coin with the shape of a right triangle. The tail has an image of Pythagoras and the Pythagorean theorem, accompanied with the mention "Pythagoras Millennium".^[12]
• 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."
• Greece, Japan, San Marino, Sierra Leone, and Surinam have issued postage stamps depicting Pythagoras and the Pythagorean theorem.^[13]
• In South Park, Eric Cartman shows the Pythagorean theorem during a slide show at show and tell.

The Wizard of Oz (film) redirects here. ... The Scarecrow is a character in the fictional Land of Oz created by American author L. Frank Baum and illustrator William Wallace Denslow. ... The Wizard, on the cover of Dorothy and the Wizard in Oz The Wizard of Oz (or simply The Wizard) is a fictional character in the Land of Oz created by American author L. Frank Baum and further popularized by the classic 1939 movie. ... Simpsons redirects here. ... Homer Simpson is also a character in the book and film The Day of the Locust. ... Henry Alfred Kissinger (born Heinz Alfred Kissinger on May 27, 1923) is a German-born American politician, and 1973 Nobel Peace Prize laureate. ... Springfield Nuclear Power Plant Springfield Nuclear Power Plant is a fictional nuclear power plant in the television animated cartoon series The Simpsons. ... Homer Simpson is also a character in the book and film The Day of the Locust. ... This article is about the comic book series. ... Henry Lytton as the Major-General. ... This 1974 stamp from Japan depicts a Class 8620 steam locomotive. ... This article is about the TV series. ...

See also

Baudhāyana, (fl. ... Katyayana was probably a priest who lived in India around 200 BC. Like Baudhayana, he composed Sulba Sutra, or sacred mathematical texts. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... In mathematics, the simplest form of the parallelogram law belongs to elementary geometry. ... Synthetic geometry is a descriptive term that identifies a methodology of geometry which makes use of theorems and synthetic observations to create theorems or solve problems, as opposed to analytic geometry which uses algebra, numbers, computations to draw theorems or solve problems. ... Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ... Pythagorean expectation is a formula invented by Bill James to estimate how many games a baseball team should have won based on the number of runs they scored and allowed. ... In mathematics, a nonhypotenuse number is a natural number whose square can not be written as the sum of two distinct nonzero squares. ...

Notes

Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... is the 353rd day of the year (354th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 353rd day of the year (354th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 215th day of the year (216th in leap years) in the Gregorian calendar. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 218th day of the year (219th in leap years) in the Gregorian calendar. ...

References

The Mathematical Intelligencer is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common amongst such journals. ... Sir Thomas Little Heath (October 5, 1861 - March 16, 1940) was a mathematician, classical scholar, historian of ancient Greek mathematics, and translator. ...

External links

Wikimedia Commons has media related to:

Pythagorean theorem

• Eric W. Weisstein, Pythagorean theorem at MathWorld.
• Proof of the Pythagorean theorem
• Pythagorean theorem with interactive animation
• Pythagorean Theorem (more than 70 proofs from cut-the-knot)
• FORTRAN code for solving with the Pythagorean theorem

Image File history File links Commons-logo. ... Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. ...