In
mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
between the three sides of a
right triangle. It states that 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 on the other two sides. This
theorem can be written as an
equation relating the lengths of the sides ''a'', ''b'' and the hypotenuse ''c'', often called the Pythagorean equation:
:
The theorem is named for the
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
philosopher
Pythagoras
Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samian, or simply ; in Ionian Greek; ) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His politic ...
, born around 570 BC. The theorem has been
proven
Proven is a rural village in the Belgian province of West Flanders, and a "deelgemeente" of the municipality Poperinge. The village has about 1400 inhabitants.
The church and parish of Proven are named after Saint Victor. The Saint Victor ...
numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.
When
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
is represented by a
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured ...
in
analytic geometry,
Euclidean distance satisfies the Pythagorean relation: the squared distance between two points equals the sum of squares of the difference in each coordinate between the points.
The theorem can be
generalized in various ways: to
higher-dimensional spaces, to
spaces that are not Euclidean, to objects that are not right triangles, and to objects that are not triangles at all but
''n''-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical
abstruseness,
mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps, and cartoons abound.
Other forms of the theorem
If ''c'' denotes the
length of the hypotenuse and ''a'' and ''b'' denote the two lengths of the legs of a right triangle, then the Pythagorean theorem can be expressed as the Pythagorean equation:
:
If only the lengths of the legs of the right triangle are known but not the hypotenuse, then the length of the hypotenuse can be calculated with the equation
:
If the length of the hypotenuse and of one leg is known, then the length of the other leg can be calculated as
:
or
:
A generalization of this theorem is the
law of cosines, which allows the
computation of the length of any side of any triangle, given the lengths of the other two sides and the angle between them. If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation.
Proofs using constructed squares
Rearrangement proofs
In one rearrangement proof, two squares are used whose sides have a measure of
and which contain four right triangles whose sides are ''a'', ''b'' and ''c'', with the hypotenuse being ''c''. In the square on the right side, the triangles are placed such that the corners of the square correspond to the corners of the right angle in the triangles, forming a square in the center whose sides are length ''c''. Each outer square has an area of
as well as
, with
representing the total area of the four triangles. Within the big square on the left side, the four triangles are moved to form two similar rectangles with sides of length ''a'' and ''b''. These rectangles in their new position have now delineated two new squares, one having side length ''a'' is formed in the bottom-left corner, and another square of side length ''b'' formed in the top-right corner. In this new position, this left side now has a square of area
as well as
. Since both squares have the area of
it follows that the other measure of the square area also equal each other such that
=
. With the area of the four triangles removed from both side of the equation what remains is
In another proof rectangles in the second box can also be placed such that both have one corner that correspond to consecutive corners of the square. In this way they also form two boxes, this time in consecutive corners, with areas
and
which will again lead to a second square of with the area
.
English mathematician
Sir Thomas Heath gives this proof in his commentary on Proposition I.47 in
Euclid's ''
Elements
Element or elements may refer to:
Science
* Chemical element, a pure substance of one type of atom
* Heating element, a device that generates heat by electrical resistance
* Orbital elements, parameters required to identify a specific orbit of ...
'', and mentions the proposals of German mathematicians
Carl Anton Bretschneider and
Hermann Hankel that Pythagoras may have known this proof. Heath himself favors a different proposal for a Pythagorean proof, but acknowledges from the outset of his discussion "that the Greek literature which we possess belonging to the first five centuries after Pythagoras contains no statement specifying this or any other particular great geometric discovery to him." Recent scholarship has cast increasing doubt on any sort of role for Pythagoras as a creator of mathematics, although debate about this continues.
Algebraic proofs
The theorem can be proved algebraically using four copies of the same triangle arranged symmetrically around a square with side ''c'', as shown in the lower part of the diagram. This results in a larger square, with side and area . The four triangles and the square side ''c'' must have the same area as the larger square,
:
giving
:
A similar proof uses four copies of a right triangle with sides ''a'', ''b'' and ''c'', arranged inside a square with side ''c'' as in the top half of the diagram.
[
] The triangles are similar with area
, while the small square has side and area . The area of the large square is therefore
:
But this is a square with side ''c'' and area ''c''
2, so
:
:
Other proofs of the theorem
This theorem may have more known proofs than any other (the
law of
quadratic reciprocity being another contender for that distinction); the book ''The Pythagorean Proposition'' contains 370 proofs.
Proof using similar triangles
This proof is based on the
proportionality of the sides of three
similar triangles, that is, upon the fact that the
ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles.
Let ''ABC'' represent a right triangle, with the
right angle located at ''C'', as shown on the figure. Draw the
altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
from point ''C'', and call ''H'' its intersection with the side ''AB''. Point ''H'' divides the length of the hypotenuse ''c'' into parts ''d'' and ''e''. The new triangle, ''ACH,'' is
similar to 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, marked as ''θ'' in the figure. By a similar reasoning, the triangle ''CBH'' is also similar to ''ABC''. The proof of similarity of the triangles requires the
triangle postulate: The sum of the angles in a triangle is two right angles, and is equivalent to the
parallel postulate. Similarity of the triangles leads to the equality of ratios of corresponding sides:
:
The first result equates the
cosines of the angles ''θ'', whereas the second result equates their
sines.
These ratios can be written as
:
Summing these two equalities results in
:
which, after simplification, demonstrates the Pythagorean theorem:
:
The role of this proof in history is the subject of much speculation. The underlying question is why Euclid did not use this proof, but invented another. One
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...
is that the proof by similar triangles involved a theory of proportions, a topic not discussed until later in the ''Elements'', and that the theory of proportions needed further development at that time.
[
This proof first appeared after a computer program was set to check Euclidean proofs.]
Euclid's proof
In outline, here is how the proof in
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ...
's ''
Elements
Element or elements may refer to:
Science
* Chemical element, a pure substance of one type of atom
* Heating element, a device that generates heat by electrical resistance
* Orbital elements, parameters required to identify a specific orbit of ...
'' proceeds. The large square is divided into a left and right rectangle. A triangle is constructed that has half the area of the left rectangle. Then another triangle is constructed that has half the area of the square on the left-most side. These two triangles are shown to be
congruent, proving this square has the same area as the left rectangle. This argument is followed by a similar version for the right rectangle and the remaining square. Putting the two rectangles together to reform the square on the hypotenuse, its area is the same as the sum of the area of the other two squares. The details follow.
Let ''A'', ''B'', ''C'' be the
vertices of a right triangle, with a right angle at ''A''. Drop a perpendicular from ''A'' to the side opposite the hypotenuse in the square on the hypotenuse. That line divides the square on the hypotenuse into two rectangles, each having the same area as one of the two squares on the legs.
For the formal proof, we require four elementary
lemmata:
# If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles included by those sides equal, then the triangles are congruent (
side-angle-side Solution of triangles ( la, solutio triangulorum) is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere. Appl ...
).
# The area of a triangle is half the area of any parallelogram on the same base and having the same altitude.
# The area of a rectangle is equal to the product of two adjacent sides.
# The area of a square is equal to the product of two of its sides (follows from 3).
Next, each top square is related to a triangle congruent with another triangle related in turn to one of two rectangles making up the lower square.
The proof is as follows:
#Let ACB be a right-angled triangle with right angle CAB.
#On each of the sides BC, AB, and CA, squares are drawn, CBDE, BAGF, and ACIH, in that order. The construction of squares requires the immediately preceding theorems in Euclid, and depends upon the parallel postulate.
[
]
#From A, draw a line parallel to BD and CE. It will perpendicularly intersect BC and DE at K and L, respectively.
#Join CF and AD, to form the triangles BCF and BDA.
#Angles CAB and BAG are both right angles; therefore C, A, and G are
collinear
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
.
#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.
#Since AB is equal to FB, BD is equal to BC and angle ABD equals angle FBC, triangle ABD must be congruent to triangle FBC.
#Since A-K-L is a straight line, parallel to BD, then rectangle BDLK has twice the area of triangle ABD because they share the base BD and have the same altitude BK, i.e., a line normal to their common base, connecting the parallel lines BD and AL. (lemma 2)
#Since C is collinear with A and G, and this line is parallel to FB, then square BAGF must be twice in area to triangle FBC.
#Therefore, rectangle BDLK must have the same area as square BAGF = AB
2.
#By applying steps 3 to 10 to the other side of the figure, it can be similarly shown that rectangle CKLE must have the same area as square ACIH = AC
2.
#Adding these two results, AB
2 + AC
2 = BD × BK + KL × KC
#Since BD = KL, BD × BK + KL × KC = BD(BK + KC) = BD × BC
#Therefore, AB
2 + AC
2 = BC
2, since CBDE is a square.
This proof, which appears in Euclid's ''Elements'' as that of Proposition 47 in Book 1, demonstrates that the area of the square on the hypotenuse is the sum of the areas of the other two squares.
This is quite distinct from the proof by similarity of triangles, which is conjectured to be the proof that Pythagoras used.
[The proof by Pythagoras probably was not a general one, as the theory of proportions was developed only two centuries after Pythagoras; see ]
Proofs by dissection and rearrangement
Another by rearrangement is given by the middle animation. A large square is formed with area ''c''2, from four identical right triangles with sides ''a'', ''b'' and ''c'', fitted around a small central square. Then two rectangles are formed with sides ''a'' and ''b'' by moving the triangles. Combining the smaller square with these rectangles produces two squares of areas ''a''2 and ''b''2, which must have the same area as the initial large square.
The third, rightmost image also gives a proof. The upper two squares are divided as shown by the blue and green shading, into pieces that when rearranged can be made to fit in the lower square on the hypotenuse – or conversely the large square can be divided as shown into pieces that fill the other two. This way of cutting one figure into pieces and rearranging them to get another figure is called dissection
Dissection (from Latin ' "to cut to pieces"; also called anatomization) is the dismembering of the body of a deceased animal or plant to study its anatomical structure. Autopsy is used in pathology and forensic medicine to determine the cause ...
. This shows the area of the large square equals that of the two smaller ones.
Einstein's proof by dissection without rearrangement
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
gave a proof by dissection in which the pieces do not need to be moved. Instead of using a square on the hypotenuse and two squares on the legs, one can use any other shape that includes the hypotenuse, and two similar shapes that each include one of two legs instead of the hypotenuse (see Similar figures on the three sides). In Einstein's proof, the shape that includes the hypotenuse is the right triangle itself. The dissection consists of dropping a perpendicular from the vertex of the right angle of the triangle to the hypotenuse, thus splitting the whole triangle into two parts. Those two parts have the same shape as the original right triangle, and have the legs of the original triangle as their hypotenuses, and the sum of their areas is that of the original triangle. Because the ratio of the area of a right triangle to the square of its hypotenuse is the same for similar triangles, the relationship between the areas of the three triangles holds for the squares of the sides of the large triangle as well.
Proof by area-preserving shearing
As shown in the accompanying animation, area-preserving shear mappings and translations can transform the squares on the sides adjacent to the right-angle onto the square on the hypotenuse, together covering it exactly. Each shear leaves the base and height unchanged, thus leaving the area unchanged too. The translations also leave the area unchanged, as they do not alter the shapes at all. Each square is first sheared into a parallelogram, and then into a rectangle which can be translated onto one section of the square on the hypotenuse.
Algebraic proofs
A related proof was published by future U.S. President James A. Garfield (then a U.S. Representative) (see diagram).[
Published in a weekly mathematics column: as noted in and i]
A calendar of mathematical dates: April 1, 1876
by V. Frederick Rickey
Instead of a square it uses a trapezoid, which can be constructed from the square in the second of the above proofs by bisecting along a diagonal of the inner square, to give the trapezoid as shown in the diagram. The area of the trapezoid can be calculated to be half the area of the square, that is
:
The inner square is similarly halved, and there are only two triangles so the proof proceeds as above except for a factor of , which is removed by multiplying by two to give the result.
Proof using differentials
One can arrive at the Pythagorean theorem by studying how changes in a side produce a change in the hypotenuse and employing calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
.[
][
]
The triangle ''ABC'' is a right triangle, as shown in the upper part of the diagram, with ''BC'' the hypotenuse. At the same time the triangle lengths are measured as shown, with the hypotenuse of length ''y'', the side ''AC'' of length ''x'' and the side ''AB'' of length ''a'', as seen in the lower diagram part.
If ''x'' is increased by a small amount ''dx'' by extending the side ''AC'' slightly to ''D'', then ''y'' also increases by ''dy''. These form two sides of a triangle, ''CDE'', which (with ''E'' chosen so ''CE'' is perpendicular to the hypotenuse) is a right triangle approximately similar to ''ABC''. Therefore, the ratios of their sides must be the same, that is:
:
This can be rewritten as , which is a differential equation that can be solved by direct integration:
:
giving
:
The constant can be deduced from ''x'' = 0, ''y'' = ''a'' to give the equation
:
This is more of an intuitive proof than a formal one: it can be made more rigorous if proper limits are used in place of ''dx'' and ''dy''.
Converse
The converse of the theorem is also true:[
]
Given a triangle with sides of length ''a'', ''b'', and ''c'', if then the angle between sides ''a'' and ''b'' is a right angle.
For any three positive real numbers ''a'', ''b'', and ''c'' such that , there exists a triangle with sides ''a'', ''b'' and ''c'' as a consequence of the converse of the triangle inequality.
This converse appears in Euclid's ''Elements'' (Book I, Proposition 48): "If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right."
It can be proven using the law of cosines or as follows:
Let ''ABC'' be a triangle with side lengths ''a'', ''b'', and ''c'', with Construct a second triangle with sides of length ''a'' and ''b'' containing a right angle. By the Pythagorean theorem, it follows that the hypotenuse of this triangle has length ''c'' = , the same as the hypotenuse of the first triangle. Since both triangles' sides are the same lengths ''a'', ''b'' and ''c'', the triangles are congruent and must have the same angles. Therefore, the angle between the side of lengths ''a'' and ''b'' in the original triangle is a right angle.
The above proof of the converse makes use of the Pythagorean theorem itself. The converse can also be proven without assuming the Pythagorean theorem.
A corollary of the Pythagorean theorem's converse is a simple means of determining whether a triangle is right, obtuse, or acute, as follows. Let ''c'' be chosen to be the longest of the three sides and (otherwise there is no triangle according to the triangle inequality). The following statements apply:[
]
* If then the triangle is right.
* If then the triangle is acute.
* If then the triangle is obtuse.
Edsger W. Dijkstra
Edsger Wybe Dijkstra ( ; ; 11 May 1930 – 6 August 2002) was a Dutch computer scientist, programmer, software engineer, systems scientist, and science essayist. He received the 1972 Turing Award for fundamental contributions to developing progra ...
has stated this proposition about acute, right, and obtuse triangles in this language:
:
where ''α'' is the angle opposite to side ''a'', ''β'' is the angle opposite to side ''b'', ''γ'' is the angle opposite to side ''c'', and sgn is the sign function.
Consequences and uses of the theorem
Pythagorean triples
A Pythagorean triple has 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.[ Such a triple is commonly written Some well-known examples are and
A primitive Pythagorean triple is one in which ''a'', ''b'' and ''c'' are coprime (the greatest common divisor of ''a'', ''b'' and ''c'' is 1).
The following is a list of primitive Pythagorean triples with values less than 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)
]
Inverse Pythagorean theorem
Given a right triangle with sides and altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
(a line from the right angle and perpendicular to the hypotenuse ). The Pythagorean theorem has,
:
while the inverse Pythagorean theorem relates the two legs to the altitude ,
:
The equation can be transformed to,
:
where for any non-zero real . If the are to be integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s, the smallest solution is then
:
using the smallest Pythagorean triple . The reciprocal Pythagorean theorem is a special case of the optic equation
In number theory, the optic equation is an equation that requires the sum of the reciprocals of two positive integers ''a'' and ''b'' to equal the reciprocal of a third positive integer ''c'':Dickson, L. E., ''History of the Theory of Numbers, V ...
:
where the denominators are squares and also for a heptagonal triangle whose sides are square numbers.
Incommensurable lengths
One of the consequences of the Pythagorean theorem is that line segments whose lengths are incommensurable (so the ratio of which is not a rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
) can be constructed using a straightedge and compass
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an Idealiz ...
. Pythagoras' theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by the square root operation.
The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer. Each triangle has a side (labeled "1") that is the chosen unit for measurement. In each right triangle, Pythagoras' theorem establishes the length of the hypotenuse in terms of this unit. If a hypotenuse is related to the unit by the square root of a positive integer that is not a perfect square, it is a realization of a length incommensurable with the unit, such as , , . For more detail, see Quadratic irrational.
Incommensurable lengths conflicted with the Pythagorean school's concept of numbers as only whole numbers. The Pythagorean school dealt with proportions by comparison of integer multiples of a common subunit.[
] According to one legend, Hippasus of Metapontum (''ca.'' 470 B.C.) was drowned at sea for making known the existence of the irrational or incommensurable.[
; Hippasus was on a voyage at the time, and his fellows cast him overboard. See
][
A careful discussion of Hippasus's contributions is found in
]
Complex numbers
For any complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
:
the absolute value or modulus is given by
:
So the three quantities, ''r'', ''x'' and ''y'' are related by the Pythagorean equation,
:
Note that ''r'' is defined to be a positive number or zero but ''x'' and ''y'' can be negative as well as positive. Geometrically ''r'' is the distance of the ''z'' from zero or the origin ''O'' in the complex plane.
This can be generalised to find the distance between two points, ''z''1 and ''z''2 say. The required distance is given by
:
so again they are related by a version of the Pythagorean equation,
:
Euclidean distance
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, by generalization of the Pythagorean theorem, as:
:
If instead of Euclidean distance, the square of this value (the squared Euclidean distance, or SED) is used, the resulting equation avoids square roots and is simply a sum of the SED of the coordinates:
:
The squared form is a smooth, convex function of both points, and is widely used in optimization theory and statistics, forming the basis of least squares
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the r ...
.
Euclidean distance in other coordinate systems
If Cartesian coordinates are not used, for example, if polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to t ...
are used in two dimensions or, in more general terms, if curvilinear coordinates are used, the formulas expressing the Euclidean distance are more complicated than the Pythagorean theorem, but can be derived from it. A typical example where the straight-line distance between two points is converted to curvilinear coordinates can be found in the applications of Legendre polynomials in physics. The formulas can be discovered by using Pythagoras' theorem with the equations relating the curvilinear coordinates to Cartesian coordinates. For example, the polar coordinates can be introduced as:
:
Then two points with locations and are separated by a distance ''s'':
:
Performing the squares and combining terms, the Pythagorean formula for distance in Cartesian coordinates produces the separation in polar coordinates as:
:
using the trigonometric product-to-sum formulas. This formula is the law of cosines, sometimes called the generalized Pythagorean theorem. From this result, for the case where the radii to the two locations are at right angles, the enclosed angle and the form corresponding to Pythagoras' theorem is regained: The Pythagorean theorem, valid for right triangles, therefore is a special case of the more general law of cosines, valid for arbitrary triangles.
Pythagorean trigonometric identity
In a right triangle with sides ''a'', ''b'' and hypotenuse ''c'', trigonometry determines the sine and cosine of the angle ''θ'' between side ''a'' and the hypotenuse as:
:
From that it follows:
:
where the last step applies Pythagoras' theorem. This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity.[
] In similar triangles, the ratios of the sides are the same regardless of the size of the triangles, and depend upon the angles. Consequently, in the figure, the triangle with hypotenuse of unit size has opposite side of size sin ''θ'' and adjacent side of size cos ''θ'' in units of the hypotenuse.
Relation to the cross product
The Pythagorean theorem relates the cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and i ...
and dot product in a similar way:[
]
:
This can be seen from the definitions of the cross product and dot product, as
:
with n a unit vector normal to both a and b. The relationship follows from these definitions and the Pythagorean trigonometric identity.
This can also be used to define the cross product. By rearranging the following equation is obtained
:
This can be considered as a condition on the cross product and so part of its definition, for example in seven dimensions.[
][
]
Generalizations
Similar figures on the three sides
The Pythagorean theorem generalizes beyond the areas of squares on the three sides to any similar figures. This was known by Hippocrates of Chios in the 5th century BC, and was included by Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ...
in his ''Elements
Element or elements may refer to:
Science
* Chemical element, a pure substance of one type of atom
* Heating element, a device that generates heat by electrical resistance
* Orbital elements, parameters required to identify a specific orbit of ...
'':[Euclid's ''Elements'': Book VI, Proposition VI 31: "In right-angled triangles the figure on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle."]
If one erects similar figures (see Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
) with corresponding sides on the sides of a right triangle, then the sum of the areas of the ones on the two smaller sides equals the area of the one on the larger side.
This extension assumes that the sides of the original triangle are the corresponding sides of the three congruent figures (so the common ratios of sides between the similar figures are ''a:b:c'').[Putz, John F. and Sipka, Timothy A. "On generalizing the Pythagorean theorem", ''The College Mathematics Journal'' 34 (4), September 2003, pp. 291–295.] While Euclid's proof only applied to convex polygons, the theorem also applies to concave polygons and even to similar figures that have curved boundaries (but still with part of a figure's boundary being the side of the original triangle).[
The basic idea behind this generalization is that the area of a plane figure is ]proportional
Proportionality, proportion or proportional may refer to:
Mathematics
* Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant
* Ratio, of one quantity to another, especially of a part compare ...
to the square of any linear dimension, and in particular is proportional to the square of the length of any side. Thus, if similar figures with areas ''A'', ''B'' and ''C'' are erected on sides with corresponding lengths ''a'', ''b'' and ''c'' then:
:
:
But, by the Pythagorean theorem, ''a''2 + ''b''2 = ''c''2, so ''A'' + ''B'' = ''C''.
Conversely, if we can prove that ''A'' + ''B'' = ''C'' for three similar figures without using the Pythagorean theorem, then we can work backwards to construct a proof of the theorem. For example, the starting center triangle can be replicated and used as a triangle ''C'' on its hypotenuse, and two similar right triangles (''A'' and ''B'' ) constructed on the other two sides, formed by dividing the central triangle by its altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
. The sum of the areas of the two smaller triangles therefore is that of the third, thus ''A'' + ''B'' = ''C'' and reversing the above logic leads to the Pythagorean theorem ''a''2 + ''b''2 = ''c''2. (''See also Einstein's proof by dissection without rearrangement'')
Law of cosines
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 and .
When is radians or 90°, then , and the formula reduces to the usual Pythagorean theorem.
Arbitrary triangle
At any selected angle of a general triangle of sides ''a, b, c'', inscribe an isosceles triangle such that the equal angles at its base θ are the same as the selected angle. Suppose the selected angle θ is opposite the side labeled ''c''. Inscribing the isosceles triangle forms triangle ''CAD'' with angle θ opposite side ''b'' and with side ''r'' along ''c''. A second triangle is formed with angle θ opposite side ''a'' and a side with length ''s'' along ''c'', as shown in the figure. Thābit ibn Qurra
Thābit ibn Qurra (full name: , ar, أبو الحسن ثابت بن قرة بن زهرون الحراني الصابئ, la, Thebit/Thebith/Tebit); 826 or 836 – February 19, 901, was a mathematician, physician, astronomer, and translator ...
stated that the sides of the three triangles were related as:[
][
]
:
As the angle θ approaches /2, the base of the isosceles triangle narrows, and lengths ''r'' and ''s'' overlap less and less. When θ = /2, ''ADB'' becomes a right triangle, ''r'' + ''s'' = ''c'', and the original Pythagorean theorem is regained.
One proof observes that triangle ''ABC'' has the same angles as triangle ''CAD'', but in opposite order. (The two triangles share the angle at vertex A, both contain the angle θ, and so also have the same third angle by the triangle postulate.) Consequently, ''ABC'' is similar to the reflection of ''CAD'', the triangle ''DAC'' in the lower panel. Taking the ratio of sides opposite and adjacent to θ,
:
Likewise, for the reflection of the other triangle,
:
Clearing fractions and adding these two relations:
:
the required result.
The theorem remains valid if the angle is obtuse so the lengths ''r'' and ''s'' are non-overlapping.
General triangles using parallelograms
Pappus's area theorem is a further generalization, that applies to triangles that are not right triangles, using parallelograms on the three sides in place of squares (squares are a special case, of course). The upper figure shows that for a scalene triangle, the area of the parallelogram on the longest side is the sum of the areas of the parallelograms on the other two sides, provided the parallelogram on the long side is constructed as indicated (the dimensions labeled with arrows are the same, and determine the sides of the bottom parallelogram). This replacement of squares with parallelograms bears a clear resemblance to the original Pythagoras' theorem, and was considered a generalization by Pappus of Alexandria in 4 AD[For the details of such a construction, see ]
The lower figure shows the elements of the proof. Focus on the left side of the figure. The left green parallelogram has the same area as the left, blue portion of the bottom parallelogram because both have the same base ''b'' and height ''h''. However, the left green parallelogram also has the same area as the left green parallelogram of the upper figure, because they have the same base (the upper left side of the triangle) and the same height normal to that side of the triangle. Repeating the argument for the right side of the figure, the bottom parallelogram has the same area as the sum of the two green parallelograms.
Solid geometry
In terms of solid geometry
In mathematics, solid geometry or stereometry is the traditional name for the geometry of three-dimensional, Euclidean spaces (i.e., 3D geometry).
Stereometry deals with the measurements of volumes of various solid figures (or 3D figures), inc ...
, Pythagoras' theorem can be applied to three dimensions as follows. Consider a rectangular solid as shown in the figure. The length of diagonal ''BD'' is found from Pythagoras' theorem as:
:
where these three sides form a right triangle. Using horizontal diagonal ''BD'' and the vertical edge ''AB'', the length of diagonal ''AD'' then is found by a second application of Pythagoras' theorem as:
:
or, doing it all in one step:
:
This result is the three-dimensional expression for the magnitude of a vector v (the diagonal AD) in terms of its orthogonal components (the three mutually perpendicular sides):
:
This one-step formulation may be viewed as a generalization of Pythagoras' theorem to higher dimensions. However, this result is really just the repeated application of the original Pythagoras' theorem to a succession of right triangles in a sequence of orthogonal planes.
A substantial generalization of the Pythagorean theorem to three dimensions is de Gua's theorem
__NOTOC__
In mathematics, De Gua's theorem is a three-dimensional analog of the Pythagorean theorem named after Jean Paul de Gua de Malves. It states that if a tetrahedron has a right-angle corner (like the corner of a cube), then the square of th ...
, named for Jean Paul de Gua de Malves: If a tetrahedron has a right angle corner (like a corner of a cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the on ...
), 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 result can be generalized as in the "''n''-dimensional Pythagorean theorem":[
]
This statement is illustrated in three dimensions by the tetrahedron in the figure. The "hypotenuse" is the base of the tetrahedron at the back of the figure, and the "legs" are the three sides emanating from the vertex in the foreground. As the depth of the base from the vertex increases, the area of the "legs" increases, while that of the base is fixed. The theorem suggests that when this depth is at the value creating a right vertex, the generalization of Pythagoras' theorem applies. In a different wording:[
For an extended discussion of this generalization, see, for example]
Willie W. Wong
2002, ''A generalized n-dimensional Pythagorean theorem''.
Inner product spaces
The Pythagorean theorem can be generalized to inner product spaces,[
] which are generalizations of the familiar 2-dimensional and 3-dimensional Euclidean spaces. For example, a function may be considered as a vector with infinitely many components in an inner product space, as in functional analysis.[
]
In an inner product space, the concept of perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
ity is replaced by the concept of orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
ity: two vectors v and w are orthogonal if their inner product is zero. The inner product is a generalization of the dot product of vectors. The dot product is called the ''standard'' inner product or the ''Euclidean'' inner product. However, other inner products are possible.[
]
The concept of length is replaced by the concept of the norm , , v, , of a vector v, defined as:[
]
:
In an inner-product space, the Pythagorean theorem states that for any two orthogonal vectors v and w we have
:
Here the vectors v and w are akin to the sides of a right triangle with hypotenuse given by the vector sum v + w. This form of the Pythagorean theorem is a consequence of the properties of the inner product:
:
where because of orthogonality.
A further generalization of the Pythagorean theorem in an inner product space to non-orthogonal vectors is the '' parallelogram law'' :[
:
which says that twice the sum of the squares of the lengths of the sides of a parallelogram is the sum of the squares of the lengths of the diagonals. Any norm that satisfies this equality is '' ipso facto'' a norm corresponding to an inner product.][
The Pythagorean identity can be extended to sums of more than two orthogonal vectors. If v''1'', v''2'', ..., v''n'' are pairwise-orthogonal vectors in an inner-product space, then application of the Pythagorean theorem to successive pairs of these vectors (as described for 3-dimensions in the section on ]solid geometry
In mathematics, solid geometry or stereometry is the traditional name for the geometry of three-dimensional, Euclidean spaces (i.e., 3D geometry).
Stereometry deals with the measurements of volumes of various solid figures (or 3D figures), inc ...
) results in the equation
:
Sets of ''m''-dimensional objects in ''n''-dimensional space
Another generalization of the Pythagorean theorem applies to Lebesgue-measurable sets of objects in any number of dimensions. Specifically, the square of the measure of an ''m''-dimensional set of objects in one or more parallel ''m''-dimensional flats in ''n''-dimensional Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
is equal to the sum of the squares of the measures of the orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
projections of the object(s) onto all ''m''-dimensional coordinate subspaces.[
]
In mathematical terms:
:
where:
* is a measure in ''m''-dimensions (a length in one dimension, an area in two dimensions, a volume in three dimensions, etc.).
* is a set of one or more non-overlapping ''m''-dimensional objects in one or more parallel ''m''-dimensional flats in ''n''-dimensional Euclidean space.
* is the total measure (sum) of the set of ''m''-dimensional objects.
* represents an ''m''-dimensional projection of the original set onto an orthogonal coordinate subspace.
* is the measure of the ''m''-dimensional set projection onto ''m''-dimensional coordinate subspace . Because object projections can overlap on a coordinate subspace, the measure of each object projection in the set must be calculated individually, then measures of all projections added together to provide the total measure for the set of projections on the given coordinate subspace.
* is the number of orthogonal, ''m''-dimensional coordinate subspaces in ''n''-dimensional space () onto which the ''m''-dimensional objects are projected (''m'' ≤ ''n''):
Non-Euclidean geometry
The Pythagorean theorem is derived from the axioms of Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, and in fact, were the Pythagorean theorem to fail for some right triangle, then the plane in which this triangle is contained cannot be Euclidean. More precisely, the Pythagorean theorem implies, and is implied by, Euclid's Parallel (Fifth) Postulate.[
][
] Thus, right triangles in a non-Euclidean geometry[
]
do not satisfy the Pythagorean theorem. For example, in spherical geometry
300px, A sphere with a spherical triangle on it.
Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...
, all three sides of the right triangle (say ''a'', ''b'', and ''c'') bounding an octant of the unit sphere have length equal to /2, and all its angles are right angles, which violates the Pythagorean theorem because
.
Here two cases of non-Euclidean geometry are considered—spherical geometry
300px, A sphere with a spherical triangle on it.
Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...
and hyperbolic plane geometry; in each case, as in the Euclidean case for non-right triangles, the result replacing the Pythagorean theorem follows from the appropriate law of cosines.
However, the Pythagorean theorem remains true in hyperbolic geometry and elliptic geometry if the condition that the triangle be right is replaced with the condition that two of the angles sum to the third, say ''A''+''B'' = ''C''. The sides are then related as follows: the sum of the areas of the circles with diameters ''a'' and ''b'' equals the area of the circle with diameter ''c''.
Spherical geometry
For any right triangle on a sphere of radius ''R'' (for example, if γ in the figure is a right angle), with sides ''a'', ''b'', ''c'', the relation between the sides takes the form:[
]
:
This equation can be derived as a special case of the spherical law of cosines that applies to all spherical triangles:
:
By expressing the Maclaurin series for the cosine function as an asymptotic expansion with the remainder term in big O notation,
:
it can be shown that as the radius ''R'' approaches infinity and the arguments ''a/R'', ''b/R'', and ''c/R'' tend to zero, the spherical relation between the sides of a right triangle approaches the Euclidean form of the Pythagorean theorem. Substituting the asymptotic expansion for each of the cosines into the spherical relation for a right triangle yields
:
The constants ''a''4, ''b''4, and ''c''4 have been absorbed into the big ''O'' remainder terms since they are independent of the radius ''R''. This asymptotic relationship can be further simplified by multiplying out the bracketed quantities, cancelling the ones, multiplying through by −2, and collecting all the error terms together:
:
After multiplying through by ''R''2, the Euclidean Pythagorean relationship ''c''2 = ''a''2 + ''b''2 is recovered in the limit as the radius ''R'' approaches infinity (since the remainder term tends to zero):
:
For small right triangles (''a'', ''b'' << ''R''), the cosines can be eliminated to avoid loss of significance, giving
:
Hyperbolic geometry
In a hyperbolic space with uniform Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...
−1/''R''2, for a right triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colli ...
with legs ''a'', ''b'', and hypotenuse ''c'', the relation between the sides takes the form:[
]
:
where cosh is the hyperbolic cosine. This formula is a special form of the hyperbolic law of cosines that applies to all hyperbolic triangles:[
]
:
with γ the angle at the vertex opposite the side ''c''.
By using the Maclaurin series for the hyperbolic cosine, , it can be shown that as a hyperbolic triangle becomes very small (that is, as ''a'', ''b'', and ''c'' all approach zero), the hyperbolic relation for a right triangle approaches the form of Pythagoras' theorem.
For small right triangles (''a'', ''b'' << ''R''), the hyperbolic cosines can be eliminated to avoid loss of significance, giving
:
Very small triangles
For any uniform curvature ''K'' (positive, zero, or negative), in very small right triangles (, ''K'', ''a''2, , ''K'', ''b''2 << 1) with hypotenuse ''c'', it can be shown that
:
Differential geometry
The Pythagorean theorem applies to infinitesimal triangles seen in differential geometry. In three dimensional space, the distance between two infinitesimally separated points satisfies
:
with ''ds'' the element of distance and (''dx'', ''dy'', ''dz'') the components of the vector separating the two points. Such a space is called a Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
. However, in Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to po ...
, a generalization of this expression useful for general coordinates (not just Cartesian) and general spaces (not just Euclidean) takes the form:[
]
:
which is called the metric tensor. (Sometimes, by abuse of language, the same term is applied to the set of coefficients .) It may be a function of position, and often describes curved space
Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. ...
. A simple example is Euclidean (flat) space expressed in curvilinear coordinates. For example, in polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to t ...
:
:
History
There is debate whether the Pythagorean theorem was discovered once, or many times in many places, and the date of first discovery is uncertain, as is the date of the first proof. Historians of Mesopotamian mathematics have concluded that the Pythagorean rule was in widespread use during the Old Babylonian period (20th to 16th centuries BC), over a thousand years before Pythagoras was born. The history of the theorem can be divided into four parts: knowledge of Pythagorean triples, knowledge of the relationship among the sides of a right triangle, knowledge of the relationships among adjacent angles, and proofs of the theorem within some deductive system.
Written between 2000 and 1786 BC, the Egypt
Egypt ( ar, مصر , ), officially the Arab Republic of Egypt, is a transcontinental country spanning the northeast corner of Africa and southwest corner of Asia via a land bridge formed by the Sinai Peninsula. It is bordered by the Med ...
ian Middle Kingdom '' Berlin Papyrus 6619'' includes a problem whose solution is the Pythagorean triple 6:8:10, but the problem does not mention a triangle. The Mesopotamian tablet ''Plimpton 322
Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G.A. Plimpton Collection at Columbia University. This tablet, believed to have been written about 1800 BC, has a tab ...
'', written between 1790 and 1750 BC during the reign of King Hammurabi the Great, contains many entries closely related to Pythagorean triples.
In India
India, officially the Republic of India ( Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the ...
, the '' Baudhayana Shulba Sutra'', the dates of which are given variously as between the 8th and 5th century BC, contains a list of Pythagorean triples and a statement of the Pythagorean theorem, both in the special case of the isosceles right triangle and in the general case, as does the '' Apastamba Shulba Sutra'' (c. 600 BC).
Byzantine
The Byzantine Empire, also referred to as the Eastern Roman Empire or Byzantium, was the continuation of the Roman Empire primarily in its eastern provinces during Late Antiquity and the Middle Ages, when its capital city was Constantin ...
Neoplatonic philosopher and mathematician Proclus
Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophe ...
, writing in the fifth century AD, states two arithmetic rules, "one of them attributed to Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institutio ...
, the other to Pythagoras", for generating special Pythagorean triples. The rule attributed to Pythagoras () starts from an odd number and produces a triple with leg and hypotenuse differing by one unit; the rule attributed to Plato (428/427 or 424/423 – 348/347 BC) starts from an even number and produces a triple with leg and hypotenuse differing by two units. According to Thomas L. Heath
Sir Thomas Little Heath (; 5 October 1861 – 16 March 1940) was a British civil servant, mathematician, classical scholar, historian of ancient Greek mathematics, translator, and mountaineer. He was educated at Clifton College. Heath translat ...
(1861–1940), no specific attribution of the theorem to Pythagoras exists in the surviving Greek literature from the five centuries after Pythagoras lived.[page 351]
/ref> However, when authors such as Plutarch
Plutarch (; grc-gre, Πλούταρχος, ''Ploútarchos''; ; – after AD 119) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo in Delphi. He is known primarily for his ...
and Cicero
Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Roman statesman, lawyer, scholar, philosopher, and academic skeptic, who tried to uphold optimate principles during the political crises that led to the est ...
attributed the theorem to Pythagoras, they did so in a way which suggests that the attribution was widely known and undoubted.[: "Though this is the proposition universally associated by tradition with the name of Pythagoras, no really trustworthy evidence exists that it was actually discovered by him. The comparatively late writers who attribute it to him add the story that he sacrificed an ox to celebrate his discovery."][
An extensive discussion of the historical evidence is provided in]
page=351
/ref> Classicist Kurt von Fritz wrote, "Whether this formula is rightly attributed to Pythagoras personally, but one can safely assume that it belongs to the very oldest period of Pythagorean mathematics
Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the ancient Greek colony of Kroton ...
."[ Around 300 BC, in Euclid's ''Elements'', the oldest extant axiomatic proof of the theorem is presented.][
]
With contents known much earlier, but in surviving texts dating from roughly the 1st century BC, the Chinese text ''Zhoubi Suanjing
The ''Zhoubi Suanjing'' () is one of the oldest Chinese mathematical texts. "Zhou" refers to the ancient Zhou dynasty (1046–256 BCE); "Bì" literally means " thigh", but in the book refers to the gnomon of a sundial. The book is dedicated to ...
'' (周髀算经), (''The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven'') gives a reasoning for the Pythagorean theorem for the (3, 4, 5) triangle — in China it is called the "Gougu theorem" (勾股定理).[
][
A rather extensive discussion of the origins of the various texts in the Zhou Bi is provided by
] During the Han Dynasty
The Han dynasty (, ; ) was an Dynasties in Chinese history, imperial dynasty of China (202 BC – 9 AD, 25–220 AD), established by Emperor Gaozu of Han, Liu Bang (Emperor Gao) and ruled by the House of Liu. The dynasty was preceded by th ...
(202 BC to 220 AD), Pythagorean triples appear in '' The Nine Chapters on the Mathematical Art'',[
This work is a compilation of 246 problems, some of which survived the book burning of 213 BC, and was put in final form before 100 AD. It was extensively commented upon by Liu Hui in 263 AD. See particularly §3: ''Nine chapters on the mathematical art'', pp. 71 ''ff''.
] together with a mention of right triangles.[
] Some believe the theorem arose first in China,[
In particular, Li Jimin; see
] where it is alternatively known as the "Shang Gao theorem" (商高定理),[
] named after the Duke of Zhou's astronomer and mathematician, whose reasoning composed most of what was in the ''Zhoubi Suanjing''.[
]
See also
* Addition in quadrature
* At Dulcarnon
* British flag theorem
*Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have bee ...
* Inverse Pythagorean theorem
* Kepler triangle
*Linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matric ...
* List of triangle topics
* L''p'' space
* Nonhypotenuse number
* Parallelogram law
* Parseval's identity
* Ptolemy's theorem
* Pythagorean expectation
* Pythagorean tiling
* Rational trigonometry in Pythagoras' theorem
*Thales theorem
In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved ...
Notes and references
Notes
References
Works cited
*
* On-line text a
archive.org
*
* This high-school geometry text covers many of the topics in this WP article.
* For full text of 2nd edition of 1940, see Originally published in 1940 and reprinted in 1968 by National Council of Teachers of Mathematics, .
*
*
* Robson, Eleanor and Jacqueline Stedall, eds., The Oxford Handbook of the History of Mathematics, Oxford: Oxford University Press, 2009. pp. vii + 918. .
* Also .
*
*
External links
* In HTML with Java-based interactive figures.
*
* Interactive links:
*
in Java
Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's mo ...
of the Pythagorean theorem
*
Another interactive proof
in Java
Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's mo ...
of the Pythagorean theorem
*
Pythagorean theorem
with interactive animation
*
Pythagorean theorem
Pythagorean theorem water demo
on YouTube
Pythagorean theorem
(more than 70 proofs from cut-the-knot)
*
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