Pythagoras Thorem

8/7/2019 Pythagoras Thorem

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Pythagoras' Theorem

Years ago, a man namedPythagoras found an amazing factabout triangles:

If the triangle had a right angle(90) ...

... and you made a square on eachof the three sides, then ...

... the biggest square had the exact same

area as the other two squares put together!

The longest side of the triangle is called the "hypotenuse",so the formal definition is:

In a right angled triangle the square of thehypotenuse is equal to the sum of the squares of

the other two sides.

So, the square of a (a) plus the square of b (b) is

equal to the square of c (c):

a2 + b2 = c2

Life


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Pythagoras was born on Samos, a Greek island in the

eastern Aegean, off the coast ofAsia Minor. He was born to

Pythais (his mother, a native of Samos) and Mnesarchus (his

father, a Phoenician merchant fromTyre). As a young man,

he left his native city for Croton, Calabria, in Southern Italy,to escape the tyrannical government ofPolycrates.

According to Iamblichus,Thales, impressed with his abilities,

advised Pythagoras to head to Memphis in Egypt and study

with the priests there who were renowned for their wisdom.

He was also discipled in the temples of Tyre and Byblos in

Phoenicia. It may have been in Egypt where he learned some

geometric principles which eventually inspired his

formulation of the theorem that is now called by his name.

This possible inspiration is presented as an extraordinaire

problem in the Berlin Papyrus. Upon his migration from

Samos to Croton, Calabria, Italy, Pythagoras established a

secret religious society very similar to (and possibly

influenced by) the earlier Orphic cult.

Bust of Pythagoras, Vatican

Pythagoras undertook a reform of the cultural life of Croton,

urging the citizens to follow virtue and form an elite circle of

followers around himself called Pythagoreans. Very strict

rules of conduct governed this cultural center. He opened his

school to both male and female students uniformly. Those

who joined the inner circle of Pythagoras's society called

themselves the Mathematikoi. They lived at the school,

owned no personal possessions and were required to

assume a mainly vegetarian diet (meat that could besacrificed was allowed to be eaten). Other students who

lived in neighboring areas were also permitted to attend

Pythagoras's school. Known asAkousmatikoi, these students

were permitted to eat meat and own personal belongings.

Richard Blackmore, in his book The Lay Monastery(1714),
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saw in the religious observances of the Pythagoreans, "the

first instance recorded in history of a monastic life."

According to Iamblichus, the Pythagoreans followed a

structured life of religious teaching, common meals,

exercise, reading and philosophical study. Music featured as

an essential organizing factor of this life: the disciples would

sing hymns to Apollo together regularly; they used

the lyre to cure illness of the soul or body; poetry recitations

occurred before and after sleep to aid the memory.

Flavius Josephus, in his polemicalAgainst Apion, in defence

ofJudaism against Greek philosophy, mentions that

according to Hermippus of Smyrna, Pythagoras was familiar

with Jewish beliefs, incorporating some of them in his ownphilosophy.

Towards the end of his life he fled to Metapontum because of

a plot against him and his followers by a noble of Croton

named Cylon. He died in Metapontum around 90 years old

from unknown causes.

Influence on Plato

Pythagoras or in a broader sense, the Pythagoreans,

allegedly exercised an important influence on the work

ofPlato. According to R. M. Hare, his influence consists of

three points: a) the platonic Republic might be related to the

idea of "a tightly organized community of like-minded

thinkers", like the one established by Pythagoras in Croton.

b) there is evidence that Plato possibly took from Pythagoras

the idea that mathematics and, generally speaking, abstract

thinking is a secure basis for philosophical thinking as well as"for substantial theses in scienceand morals". c) Plato and

Pythagoras shared a "mystical approach to the soul and its

place in the material world". It is probable that both have

been influenced by Orphism.[11]
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Plato's harmonics were clearly influenced by the work

ofArchytas, a genuine Pythagorean of the third generation,

who made important contributions to geometry, reflected in

Book VIII ofEuclid's Elements.

Roman influence

In the legends ofancient Rome, Numa Pompilius, the second

King of Rome, is said to have studied under Pythagoras. This

is unlikely, since the commonly accepted dates for the two

lives do not overlap.

Influence on esoteric groups

Pythagoras started a secret society called the Pythagorean

brotherhood devoted to the study of mathematics. This hada great effect on future esoteric traditions, such

as Rosicrucianism and Freemasonry, both of which were

occult groups dedicated to the study of mathematics and

both of which claimed to have evolved out of the

Pythagorean brotherhood. The mystical and occult qualities

of Pythagorean mathematics are discussed in a chapter of

Manly P. Hall's The Secret Teachings of All Ages entitled

"Pythagorean Mathematics".

Pythagorean theory was tremendously influential on

later numerology, which was extremely popular throughout

the Middle East in the ancient world. The 8th-

century MuslimalchemistJabir ibn Hayyan grounded his

work in an elaborate numerology greatly influenced by

Pythagorean theory.

THEORUM

In mathematics, the Pythagorean theorem (American

English) or Pythagoras' theorem (British English) is a

relation in Euclidean geometry among the three sides of

a right triangle. The theorem is named after
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the GreekmathematicianPythagoras, who by tradition is

credited with its discovery and proof,[1] although it is often

argued that knowledge of the theory predates him. (There is

much evidence that Babylonian mathematicians understood

the principle, if not the mathematical significance). Thetheorem is as follows:

In any right triangle, the area of the square whose side isthe hypotenuse (the side opposite the right angle) is equal tothe sum of the areas of the squares whose sides are the twolegs (the two sides that meet at a right angle).

This is usually summarized as follows:

The square of the hypotenuse of a right triangle is equal to

the sum of the squares on the other two sides.[2]

In formulae

If we let c be the length of the hypotenuse and a and b be the length

the other two sides, the theorem can be expressed as the equation:

or, solved for c:

Ifc is already given, and the length of one of the legs must be

found, the following equations can be used (The following

equations are simply the converse of the original equation):

or

This equation provides a simple relation among the thresides of a right triangle so that if the lengths of any two

sides are known, the length of the third side can be fou

generalization of this theorem is the law of cosines, wh

allows the computation of the length of the third side o

triangle, given the lengths of two sides and the size of t
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angle between them. If the angle between the sides is a

right angle it reduces to the Pythagorean theorem.

Visual proof for the (3, 4, 5) triangle as in the Chou Pei Ching 500200 BC.

[edit]History

The history of the theorem can be divided into four part

knowledge ofPythagorean triples, knowledge of the

relationship between the sides of a right triangle,

knowledge of the relationship between adjacent anglesproofs of the theorem.

Megalithic monuments from circa 2500 BC in Egypt, an

in Northern Europe, incorporate right triangles with inte

sides.[3]Bartel Leendert van der Waerden conjectures t

these Pythagorean triples were discovered algebraically

Written between 2000 and 1786 BC, the Middle

KingdomEgyptian papyrus Berlin 6619 includes a prob

whose solution is a Pythagorean triple.During the reign ofHammurabi the Great,

the Mesopotamian tablet Plimpton 322, written

between 1790and 1750 BC, contains many entries clos

related to Pythagorean triples.

The BaudhayanaSulba Sutra, the dates of which are gi

variously as between the 8th century BC and the 2nd

century BC, in India, contains a list ofPythagorean

triples discovered algebraically, a statement of thePythagorean theorem, and a geometrical proof of the

Pythagorean theorem for an isosceles right triangle.

TheApastamba Sulba Sutra (circa 600 BC) contains a

numerical proof of the general Pythagorean theorem, u

an area computation. Van der Waerden believes that "i
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certainly based on earlier traditions". According to Albe

Brk, this is the original proof of the theorem; he furthe

theorizes that Pythagoras visited Arakonam, India, and

copied it.

Pythagoras, whose dates are commonly given as 5694

BC, used algebraic methods to construct Pythagorean

triples, according to Proklos's commentary on Euclid.

Proklos, however, wrote between 410 and 485 AD.

According to Sir Thomas L. Heath, there was no attribut

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 thattribution was widely known and undoubted.[5]

Around 400 BC, according to Proklos, Plato gave a meth

for finding Pythagorean triples that combined algebra a

geometry. Circa 300 BC, in Euclid's Elements, the oldes

extant axiomatic proofof the theorem is presented.

Written sometime between 500 BC and 200 AD,

the Chinese text Chou Pei Suan Ching (), (The

Arithmetical Classic of the Gnomon and the Circular Patof Heaven) gives a visual proof of the Pythagorean theo

in China it is called the "Gougu Theorem" ()

the (3, 4, 5) triangle. During theHan Dynasty, from 202 BC to

AD, Pythagorean triples appear inThe Nine Chapters on

Mathematical Art, together with a mention of right trian[6]

The first recorded use is in China, known as the "Gougu

theorem" () and inIndia known as the BhaskaraTheorem.

There is much debate on whether the Pythagorean theo

was discovered once or many times. Boyer (1991) think

the elements found in the Shulba Sutras may be of

Mesopotamian derivation.[7]
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Proof using similar triangles

Proof using similar triangles.

Like most of the proofs of the Pythagorean theorem, this one is based on

theproportionality of the sides of two similar triangles.

LetABC represent a right triangle, with the right angle located at C, as shown on the

figure. We draw thealtitude from point C, and call H its intersection with the sideAB.

The new triangleACH is similar to our triangleABC, because they both have a right

angle (by definition of the altitude), and they share the angle atA, meaning that the

third angle will be the same in both triangles as well. By a similar reasoning, the

triangle CBH is also similar toABC. The similarities lead to the two ratios..: As

so

These can be written as

Summing these two equalities, we obtain

In other words, the Pythagorean theorem:

Consequences and uses of thetheorem
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[edit]Pythagorean triples

Main article: Pythagorean triple

A Pythagorean triple has 3 positive numbers a, b, and c, such that a2 + b2 = c2. In

other words, a Pythagorean triple represents the lengths of the sides of a right

triangle where all three sides have integer lengths. Evidence from megalithic

monuments on the Northern Europe shows that such triples were known before

the discovery of writing. Such a triple is commonly written (a, b, c). Some well-

known examples are (3, 4, 5) and (5, 12, 13).

[edit]List of primitive Pythagorean triples up to 100

(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37),

(13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85),

(39, 80, 89), (48, 55, 73), (65, 72, 97)

[edit]The existence of irrational numbers

One of the consequences of the Pythagorean theorem is

that incommensurablelengths (ie. their ratio is irrational number), such as the

square root of 2, can be constructed. A right triangle with legs both equal to one

unit has hypotenuse length square root of 2. The Pythagoreansproved that the

square root of 2 is irrational, and this proof has come down to us even though it

flew in the face of their cherished belief that everything was rational. According

to the legend, Hippasus, who first proved the irrationality of the square root of

two, was drowned at sea as a consequence.[12]

[edit]Distance in Cartesian coordinates

The distance formula inCartesian coordinatesis derived from the Pythagorean

theorem. If (x0,y0) and (x1,y1) are points in the plane, then the distance between

them, also called the Euclidean distance, is given by

More generally, inEuclidean n-space, the Euclidean distance between two

points, and , is defined, using the

Pythagorean theorem, as:
http://en.wikipedia.org/w/index.php?title=Pythagorean_theorem&action=edit&section=13http://en.wikipedia.org/wiki/Pythagorean_triplehttp://en.wikipedia.org/w/index.php?title=Pythagorean_theorem&action=edit&section=14http://en.wikipedia.org/w/index.php?title=Pythagorean_theorem&action=edit&section=15http://en.wikipedia.org/wiki/Commensurability_(mathematics)http://en.wikipedia.org/wiki/Commensurability_(mathematics)http://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Pythagoreanismhttp://en.wikipedia.org/wiki/Irrational_number#The_square_root_of_2http://en.wikipedia.org/wiki/Irrational_number#The_square_root_of_2http://en.wikipedia.org/wiki/Hippasushttp://en.wikipedia.org/wiki/Hippasushttp://en.wikipedia.org/wiki/Pythagoras_Theorem#cite_note-11%23cite_note-11http://en.wikipedia.org/w/index.php?title=Pythagorean_theorem&action=edit&section=16http://en.wikipedia.org/wiki/Cartesian_coordinateshttp://en.wikipedia.org/wiki/Cartesian_coordinateshttp://en.wikipedia.org/wiki/Cartesian_coordinateshttp://en.wikipedia.org/wiki/Euclidean_distancehttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/w/index.php?title=Pythagorean_theorem&action=edit&section=13http://en.wikipedia.org/wiki/Pythagorean_triplehttp://en.wikipedia.org/w/index.php?title=Pythagorean_theorem&action=edit&section=14http://en.wikipedia.org/w/index.php?title=Pythagorean_theorem&action=edit&section=15http://en.wikipedia.org/wiki/Commensurability_(mathematics)http://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Pythagoreanismhttp://en.wikipedia.org/wiki/Irrational_number#The_square_root_of_2http://en.wikipedia.org/wiki/Irrational_number#The_square_root_of_2http://en.wikipedia.org/wiki/Hippasushttp://en.wikipedia.org/wiki/Pythagoras_Theorem#cite_note-11%23cite_note-11http://en.wikipedia.org/w/index.php?title=Pythagorean_theorem&action=edit&section=16http://en.wikipedia.org/wiki/Cartesian_coordinateshttp://en.wikipedia.org/wiki/Euclidean_distancehttp://en.wikipedia.org/wiki/Euclidean_space

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Pythagoras Thorem