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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. The theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and proof, although it is often argued that knowledge of the theory predates him. (There is much evidence that Babylonian mathematicians understood the principle, if not the mathematical significance). 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).

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.

In formulae

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:

${\displaystyle a^{2}+b^{2}=c^{2}\,}$

or, solved for c:

${\displaystyle c={\sqrt {a^{2}+b^{2}}}.\,}$

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):

${\displaystyle c^{2}-a^{2}=b^{2}\,}$

or

${\displaystyle c^{2}-b^{2}=a^{2}.\,}$

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.

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History

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The history of the theorem can be divided into four parts: knowledge of Pythagorean triples, knowledge of the relationship between the sides of a right triangle, knowledge of the relationship between adjacent angles, and proofs of the theorem.

Megalithic monuments from circa 2500 BC in Egypt, and in Northern Europe, incorporate right triangles with integer sides. Bartel Leendert van der Waerden conjectures that these Pythagorean triples were discovered algebraically.

Written between 2000 and 1786 BC, the Middle Kingdom Egyptian papyrus Berlin 6619 includes a problem whose solution is a Pythagorean triple.

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.

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.

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.

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 was 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.

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.

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.

The first recorded use is in China, known as the "Gougu theorem" (勾股定理) and in India known as the Bhaskara Theorem.

There is much debate on whether the Pythagorean theorem was discovered once or many times. Boyer (1991) thinks the elements found in the Shulba Sutras may be of Mesopotamian derivation.

Proofs

This is a theorem that may have more known proofs than any other (the law of quadratic reciprocity being also a contender for that distinction); the book Pythagorean Proposition, by Elisha Scott Loomis, contains 367 proofs.

Some arguments based on trigonometric identities (such as Taylor 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.)

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.

YAY

Converse

The converse of the theorem is also true:

For any three positive numbers a, b, and c such that a + b = c, 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:

Let ABC be a triangle with side lengths a, b, and c, with a + b = c. 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.

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:

• If a + b = c, then the triangle is right.
• If a + b > c, then the triangle is acute.
• If a + b < c, then the triangle is obtuse.

Consequences and uses of the theorem

Pythagorean triples

A Pythagorean triple has 3 positive numbers a, b, and c, such that ${\displaystyle a^{2}+b^{2}=c^{2}}$. 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 incommensurable lengths (ie. their ratio is irrational number), such as the square root of 2, can be constructed. A right triangle with legs both equal to one unit has hypotenuse length square root of 2. The Pythagoreans proved that the square root of 2 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.

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

${\displaystyle {\sqrt {(x_{1}-x_{0})^{2}+(y_{1}-y_{0})^{2}}}.}$

More generally, in Euclidean n-space, the Euclidean distance between two points, ${\displaystyle \scriptstyle A\,=\,(a_{1},a_{2},\dots ,a_{n})}$ and ${\displaystyle \scriptstyle B\,=\,(b_{1},b_{2},\dots ,b_{n})}$, is defined, using the Pythagorean theorem, as:

${\displaystyle {\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}+\cdots +(a_{n}-b_{n})^{2}}}={\sqrt {\sum _{i=1}^{n}(a_{i}-b_{i})^{2}}}.}$

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.

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

${\displaystyle a^{2}+b^{2}-2ab\cos {\theta }=c^{2},\,}$

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:

${\displaystyle \|\mathbf {v} +\mathbf {w} \|^{2}=\|\mathbf {v} \|^{2}+\|\mathbf {w} \|^{2}+2\,{\mbox{Re}}\,\langle \mathbf {v} ,\mathbf {w} \rangle .}$

In particular, ||v + w|| = ||v|| + ||w|| if v and w are orthogonal, although the converse is not necessarily true.

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

${\displaystyle \left\|\,\sum _{k=1}^{n}\mathbf {v} _{k}\,\right\|^{2}=\sum _{k=1}^{n}\|\mathbf {v} _{k}\|^{2}.}$

The generalization of this result to infinite-dimensional real inner product spaces is known as Parseval's identity.

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.

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.

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

In a triangle with three acute angles, α + β > γ holds. Therefore, a + b > c holds.

In a triangle with an obtuse angle, α + β < γ holds. Therefore, a + b < c holds.

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

sgn(α + β − γ) = sgn(a + b − c)

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.

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 ${\displaystyle \scriptstyle \pi /2}$; this violates the Euclidean Pythagorean theorem because ${\displaystyle \scriptstyle (\pi /2)^{2}+(\pi /2)^{2}\neq (\pi /2)^{2}}$.

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

${\displaystyle \cos \left({\frac {c}{R}}\right)=\cos \left({\frac {a}{R}}\right)\,\cos \left({\frac {b}{R}}\right).}$

This equation can be derived as a special case of the spherical law of cosines. By using the Maclaurin series for the cosine function, it can be shown that as the radius R approaches infinity, the spherical form of the Pythagorean theorem approaches the Euclidean form.

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

${\displaystyle \cosh c=\cosh a\,\cosh b}$

where cosh is the hyperbolic cosine.

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,

${\displaystyle \sin {\bar {a}}\sin {\bar {b}}=\sin {\bar {c}}}$

where ${\displaystyle \scriptstyle {\bar {a}}}$ is the angle of parallelism of the line segment AB that ${\displaystyle \scriptstyle \mu (AB)\,=\,a}$ where μ is the multiplicative distance function (see Hilbert's arithmetic of ends).

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

${\displaystyle \sin {\bar {a}}={\frac {2a}{1+a^{2}}}.}$

Thus, the equation takes the form

${\displaystyle {\frac {2a}{1+a^{2}}}{\frac {2b}{1+b^{2}}}={\frac {2c}{1+c^{2}}}}$

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

Cultural references to the Pythagorean theorem

The Pythagorean theorem has been referenced in a variety of mass media throughout history. A verse of the Major-General's Song in the Gilbert and Sullivan musical The Pirates of Penzance, "About binomial theorem I'm teeming with a lot o' news, With many cheerful facts about the square of the hypotenuse", with oblique reference to the theorem. The Scarecrow of The Wizard of Oz makes a more specific reference to the theorem when he 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." 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.) Similarly, the Speech software on an Apple MacBook references the Scarecrow's incorrect statement. It is the sample speech when the voice setting 'Ralph' is selected.

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". Greece, Japan, San Marino, Sierra Leone, and Suriname have issued postage stamps depicting Pythagoras and the Pythagorean theorem.

See also

• List of triangle topics
• Baudhayana
• Katyayana
• Linear algebra
• Orthogonality
• Parallelogram law
• Synthetic geometry
• Fermat's Last Theorem
• Pythagorean expectation
• Nonhypotenuse number
• Pythagorean theorem proof (rational trigonometry)

Notes

1. Heath, Vol I, p. 144.
2. The "other two sides" are also known as legs or catheti.
3. "Megalithic Monuments".
4. van der Waerden 1983.
5. Heath, Vol I, p. 144.
6. Swetz.
7. Boyer (1991). "China and India". p. 207. we find rules for the construction of right angles by means of triples of cords the lengths of which form Pythagorean triages, such as 3, 4, and 5, or 5, 12, and 13, or 8, 15, and 17, or 12, 35, and 37. However all of these triads are easily derived from the old Babylonian rule; hence, Mesopotamian influence in the Sulvasutras is not unlikely. Aspastamba knew that the square on the diagonal of a rectangle is equal to the sum of the squares on the two adjacent sides, but this form of the Pythagorean theorem also may have been derived from Mesopotamia. So conjectural are the origin and period of the Sulbasutras that we cannot tell whether or not the rules are related to early Egyptian surveying or to the later Greek problem of alter doubling. They are variously dated within an interval of almost a thousand years stretching from the eighth century B.C. to the second century of our era. {{cite book}}: Missing or empty |title= (help)
8. Heath, Vol I, pp. 65, 154; Stillwell, p. 8–9.
9. "Dijkstra's generalization" (PDF).
10. "The Scarecrow's Formula".
11. "Le Saviez-vous ?".
12. Miller, Jeff (2007-08-03). "Images of Mathematicians on Postage Stamps". Retrieved 2007-08-06. {{cite web}}: Check date values in: |date= (help)

References

• Bell, John L., The Art of the Intelligible: An Elementary Survey of Mathematics in its Conceptual Development, Kluwer, 1999. ISBN 0-7923-5972-0.
• Euclid, The Elements, Translated with an introduction and commentary by Sir Thomas L. Heath, Dover, (3 vols.), 2nd edition, 1956.
• Hardy, Michael, "Pythagoras Made Difficult". Mathematical Intelligencer, 10 (3), p. 31, 1988.
• Heath, Sir Thomas, A History of Greek Mathematics (2 Vols.), Clarendon Press, Oxford (1921), Dover Publications, Inc. (1981), ISBN 0-486-24073-8.
• Loomis, Elisha Scott, The Pythagorean proposition. 2nd edition, Washington, D.C : The National Council of Teachers of Mathematics, 1968. ISBN 978-0873530361.
• Maor, Eli, The Pythagorean Theorem: A 4,000-Year History. Princeton, New Jersey: Princeton University Press, 2007, ISBN 978-0-691-12526-8.
• Stillwell, John, Mathematics and Its History, Springer-Verlag, 1989. ISBN 0-387-96981-0 and ISBN 3-540-96981-0.
• Swetz, Frank, Kao, T. I., Was Pythagoras Chinese?: An Examination of Right Triangle Theory in Ancient China, Pennsylvania State University Press. 1977.
• van der Waerden, B.L., Geometry and Algebra in Ancient Civilizations, Springer, 1983.

External links

• Pythagorean Theorem (more than 70 proofs from cut-the-knot)
• 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
• Weisstein, Eric W. "Pythagorean theorem". MathWorld.

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