Encyclopedia
In
mathematics, the
Pythagorean theorem or
Pythagoras' theorem 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.
The theorem is as follows:
In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of areas of the squares whose sides are the two legs .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
-
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.
This theorem may have more known proofs than any other.
The Pythagorean Proposition, a book published in 1940, contains 370 proofs of Pythagoras' theorem, including one by American President
James Garfield.
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.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.
Megalithic monuments from 4000 BC in
Egypt, and in the
British Isles from circa 2500 BC, incorporate right triangles with integer sides but the builders did not necessarily understand the theorem.
Bartel Leendert van der Waerden conjectures that these Pythagorean triples were discovered
algebraically.
Written between 2000–1786 BC, the Middle Kingdom
Egyptian papyrus
Berlin 6619 includes a problem, whose solution is a
Pythagorean triple.
During the reign of
Hammurabi, the
Mesopotamian tablet
Plimpton 322, written between 1790 and 1750 BC, contains many entries closely related to Pythagorean triples.
The
Baudhayana Sulba Sutra, written in the 8th 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 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 Burk, this is the original proof of the theorem, and Pythagoras copied it on his visit to India. Many scholars find van der Waerden and Burk's claims unsubstantiated.
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.
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
abstract proof of the theorem is presented.
Written sometime between 500 BC and 200 BC, the
Chinese text
Chou Pei Suan Ching gives a visual proof of the Pythagorean theorem for the triangle. During the
Han Dynasty, from 200 BC to 200 AD, Pythagorean triples appear in
The Nine Chapters on the Mathematical Art is a Chinese mathematics [i] book, probably composed by th ...
, together with a mention of right triangles.
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 circa 2000 BC, and from there to India, China, and Greece by 600 BC. Most scholars disagree however, and 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" , based on the numerical proof in the Chou Pei Suan Ching , presented in
The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven, variously dated between 500 BC–200 AD.
Proofs
This theorem may have more known proofs than any other ; the book
Pythagorean Proposition, by Elisha Scott Loomis, contains over 350 proofs.
James Garfield, who later became
President of the United States, devised an original proof of the Pythagorean theorem in 1876. The external links below provide a sampling of the many proofs of the Pythagorean theorem.
Some arguments based on
trigonometric identities 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.
For similar reasons, no proof can be based on analytic geometry or
calculus.
More recently, Shri Bharati Krishna Tirthaji in his book claimed ancient Indian Hindu Vedic proofs for the Pythagoras Theorem.
Geometrical proof
Like many of the proofs of the Pythagorean theorem, this one is based on the proportionality of the sides of two similar triangles.
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 , 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:
These can be written as:
and
Summing these two equalities, we obtain:
In other words, the Pythagorean theorem:
Visual proofs
A visual proof is given by this illustration. The area of each large square is ². In both, the area of four identical triangles is removed. The remaining areas,
a² +
b² and
c², are equal. Q.E.D.
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. For this reason, axiomatic introductions to geometry usually employ another proof based on the similarity of triangles .
A second graphic illustration of the Pythagorean theorem 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.
Algebraic proof
A more algebraic variant of this proof is provided by the following reasoning. Looking at the illustration, the area of each of the four red, yellow, green and pink right-angled triangles is given by:
The blue square in the middle has side length
c, so its area 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 ². This can be shown by considering the angles.
Therefore:
-
Simplifying this equation:
-
-
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.
As a result of a change in side
a,
by similar triangles and for differential changes. So
upon separation of variables. 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. 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
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.
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 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.
Rational trigonometry
For a proof by the methods of
rational trigonometry, see Pythagoras' theorem proof .
Proof of the converse
only if you are smart will you get this. try this prblem.... one of the legs is 6578.95768 and the other is 95786.36457 what is the hpotenuse = to. use a2+b2=c2
- 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 , or by the following proof:
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.
Consequences and uses of the theorem
Pythagorean triples
A Pythagorean triple consists of three positive integers
a,
b, and
c, such that . 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 , and a well-known example is .
A Pythagorean triple is primitive if
a,
b, and
c have no common divisor other than 1. There are infinitely many primitive triples, and all Pythagorean triples can be explicitly generated using the following formula: choose two integers
m and
n with
m >
n, and let
, , . Then we have . Also, any multiple of a Pythagorean triple is again a Pythagorean triple.
Pythagorean triples allow the construction of right angles. The fact that the lengths of the sides are integers means that, for example, tying knots at equal intervals along a string allows the string to be stretched into a triangle with sides of length three, four, and five, in which case the largest angle will be a right angle. This methods was used to step masts at sea and by the Egyptians in construction work.
A generalization of the concept of Pythagorean triples is a triple of positive integers
a,
b, and
c, such that
an +
bn =
cn, for some
n strictly greater than 2.
Pierre de Fermat in 1637 claimed that no such triple exists, a claim that came to be known as
Fermat's last theorem. The first proof was given by
Andrew Wiles in 1994.
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.
Distance in Cartesian coordinates
The distance formula in
Cartesian coordinates is derived from the Pythagorean theorem. If and 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 generalised by
Euclid in his
Elements:- If one erects similar figures 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:
- where ? is the angle between sides a and b.
- When ? is 90 degrees, then cos = 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 and only if
v and
w are orthogonal.
Using
mathematical induction, the previous result can be extended to any finite number of pairwise orthogonal vectors. Let
v1,
v2,...,
vn be vectors in an inner product space such that <
vi,
vj> = 0 for 1 =
i <
j =
n. Then
The generalisation 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: If a
tetrahedron has a right angle corner , 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,
a +
ß >
? holds. Therefore,
a2 +
b2 >
c2 holds.
In a triangle with an
obtuse angle,
a +
ß <
? holds. Therefore,
a2 +
b2 <
c2 holds.
Edsger Dijkstra's has stated this proposition about acute, right, and obstuse triangles in this language:
sgn =
sgnwhere
a is the angle opposite to side
a,
ß is the angle opposite to side
b and
? 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. 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 . However, in
hyperbolic geometry, the Pythagorean theorem does hold in the limit of small distances.
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.
For any triangle in the
hyperbolic plane , the Pythagorean theorem takes the form
- 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 , the hyperbolic form of the Pythagorean theorem approaches the Euclidean form.
Pythagoras's theorem and complex numbers
This proof is only valid if
a and
b are real. If
a and/or
b have imaginary parts, Pythagoras's theorem breaks down because the concept of areas loses its meaning because in the
complex plane loci of the type
y =
f cannot separate an inside from an outside because there they are 2-dimensional in a 4-dimensional space .
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!"
- In an episode of The Simpsons is an Emmy [i] and Peabody [i]-winning American [i] animated [i] ...
, homage is paid to 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 at the Springfield Nuclear Power Plant, Homer puts them on and quotes the scarecrow verbatim. A man in a nearby toilet stall then yells out "That's a right triangle, you idiot!"
- 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".
See also
- Baudhayana
- Katyayana
- linear algebra
- orthogonality
- parallelogram law
- synthetic geometry
- Fermat's Last Theorem
Notes
References
External links
