Pythagorean theorem

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The Pythagorean theorem: The sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse.

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry between the three sides of a right-angled triangle.

The theorem is as follows:

In any right triangle, the area of the square whose side is the hypotenuse (the side of the triangle opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.

Where c is the length of the hypotenuse and a and b are the lengths of the other two sides, the theorem can be expressed as the following equation:

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

This means that knowing the lengths of two sides of a right triangle is enough to calculate the length of the third.

A paraphrase of the Pythagorean theorem is :

In the diagram, the sum of the areas of the blue and red squares is equal to the area of the purple square.

This works for any right triangle laid out on a flat plane.

A visual proof

Perhaps this theorem has a greater variety of different known proofs than any other (the law of quadratic reciprocity may also be a contender for that distinction).

This illustration depicts one of them. 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.

NB: This proof is 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. For this reason, axiomatic introductions to geometry usually employ another proof based on the similarity of triangles (see proof 6 in the external link).

History

Visual proof as in the Chou Pei Suan Ching 500-200 BCE

The history of the theorem called Pythagorean 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.

• Circa 2500 BCE, Megalithic monuments on the British Isles incorporate right triangles with integer sides. B.L. van der Waerden conjectures that these Pythagorean triples were discovered algebraicly.

• Written between 2000 - 1786 BCE, the Middle Kingdom Egyptian papyrus Berlin 6619 has a problem the solution to which is a Pythagorean triple.

• Written between 1790 - 1750 BCE, during the reign of Hammurabi, the Mesopotamian tablet Plimpton 322 contains a large number of entries closely related to Pythagorean triples.

• Written sometime between 800 - 500 BCE in India, the Sulba Sutras contain a statement of the Pythagorean theorem and a list of Pythagorean triples discovered algebraicly. The Apastamba Sulba Sutra (c. 600 BC) also contains a numerical proof of the theorem, using an area computation. (Numerical proof is a proof that uses specific numbers but in such a way that it can be generalized.) Van der Waerden believes that "it was certainly based on earlier traditions". According to Albert Bŭrk, this is the original proof of the theorem, and Pythagoras copied it. Many scholars find Bŭrk's claim unsubstantiated however.

• Pythagoras, whose dates are commonly given as 569 - 475 BCE, used algebraic methods to construct Pythagorean triples, according to Proklos' commentary on Euclid. Proklos, however, wrote between 410 - 485 CE. 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 did attribute the theorem to Pythagoras, they did so in such a way as to suggests that the attribution was widely known and undoubted.

• Circa 400 BCE, according to Proklos, Plato gave a method for finding Pythagorean triples that combined algebra and geometry.

• Circa 300 BCE, in Euclid's "Elements" we find the oldest extant abstract proof of the theorem (that is, a proof that does not use specific numbers).

• Written during the Han dynasty, 200 BCE - 200 CE the Chinese text "Chou Pei Suan Ching" gives a numerical proof of the Pythagorean theorem, using the (3, 4, 5) right triangle. From the same period, Pythagorean triples appear in "Nine Chapters on the Mathematical Art", together with a mention of right triangles. (Some authorities place the "Chou Pei Suan Ching" as much as three hundred years earlier than the Han dynasty.)

There has been much debate on whether the Pythagorean theorem was discovered once or many times. B.L. van der Waerden asserts a single discovery, by someone in Neolithic Britain, knowledge of which then spread to Mesopotamia and Egypt circa 2000 BCE, and from there to India, China, and Greece circa 600 BCE. Most authorities, however, favor independent discovery.

In the West, the theorem is named after and commonly attributed to the 6th century BC Greek philosopher and mathematician Pythagoras. In China, the theorem goes by the name "Gougu Theorem" (勾股定理), based on the numerical proof in the Chou Pei Suan Ching (周髀算经) (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven, variously dated between 500 BCE -200 CE, see image above).

James Garfield, who later became a President of the United States, devised an original proof of the Pythagorean theorem in 1876. (See the external links below for a sampling of the many different proofs of the Pythagorean theorem.)

In heraldry, the Pythagorean theorem appears as a charge in the arms of Seissenegger.

Other facts

The converse of the Pythagorean 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.

This converse also appears in Euclid's Elements. This can be proven using the law of cosines (see below under Generalizations).

It can also be proven by reductio ad absurdum. Suppose there exists a triangle for which a² + b² = c² but the angle between sides a and b is not a right angle. Then we can construct another triangle with a right angle between sides of lengths a and b. It follows that the hypotenuse of this triangle also has length c. Thus we have two triangles with the side lengths a, b and c but different angles between the a and b sides. But triangles with the same side lengths are congruent, and so we have a contradiction.

The theorem is referenced in an episode of The Simpsons. After finding a pair of glasses at the Nuclear Power Plant, Homer puts them on and in an attempt to sound smart, comments "the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side." A man in a nearby toilet stall then yells out "That's a right triangle, you idiot!" (This was a homage to The Wizard of Oz. When the Scarecrow receives his diploma from the Wizard, he recites the Pythagorean theorem incorrectly).

Generalizations

• The Pythagorean theorem was generalised 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 law of cosines (or cosine rule) is a version of the Pythagorean theorem that applies to all (Euclidean) triangles, not just right-angled ones. It states that:

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

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.

• The Pythagorean theorem stated in Cartesian coordinates is the formula for the distance between points in the plane -- if (x₀, y₀) and (x₁, y₁) are points in the plane, then the distance between them is given by

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

• More generally, given two vectors v and w in a complex inner product space, the Pythagorean theorem takes the following form:

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

In particular, ||v + w||² = ||v||² + ||w||² if v and w are orthogonal, and these two statements are equivalent in any real inner product space.

• 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

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

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

• The Pythagorean theorem also generalises to higher-dimensional simplexes. If a tetrahedron has a right angle corner (a corner of a cube), then the square of the area of the face opposite the right angle corner is the sum of the squares of the areas of the other three faces. This is called de Gua's theorem.

• Edsger Dijkstra discovered this related proposition, not actually a generalisation:

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.

This formula holds in all triangles, not just the right triangles. If γ is a right angle (γ equals $\pi /2$ radians or 90°), then sgn(α + β − γ) = 0 since the sum of the angles of a triangle is $\pi$ radians (or 180°). Thus, a² + b² − c² = 0.

In a triangle with three acute angles, α + β > γ holds. Therefore, a² + b² > c² holds.

In a triangle with an obtuse angle, α + β < γ holds. Therefore, a² + b² < c² holds.

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. For example, in spherical geometry, all three sides of the right triangle bounding an octant of the unit sphere have length equal to $\pi /2$; this violates the Euclidean Pythagorean theorem because $(\pi /2)^{2}+(\pi /2)^{2}\neq (\pi /2)^{2}$.

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

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

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

$\cosh c=\cosh a\,\cosh b$

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.

See also

• Baudhayana
• Euclid
• Fermat's last theorem
• Katyayana
• linear algebra
• orthogonality
• parallelogram law
• Pythagoras
• Pythagorean triple
• synthetic geometry

References

Euclid, The Elements, Translated with an introduction and commentary by Sir Thomas L. Heath, Dover, (3 vols.), 2nd edition, 1956.

B. L. van der Waerden, Geometry and Algebra in Ancient Civilizations, Springer, 1983.

External links

• Over 50 proofs of the Pythagorean theorem
• Using the Pythagorean theorem and a Microsoft Windows calculator to calculate the sides of an octagon
• Dijkstra's generalization
• The Pythagorean Theorem is Equivalent to the Parallel Postulate.
• The true origin
• Some animated proofs of the Pythagorean theorem
• Vedic Mathematics
• Pythagorean Theorem -- From MathWorld