PYTHAGORAS THEOREM IN PYTHAGORAS THEOREM IN BABYLONIAN MATHEMATICSBABYLONIAN MATHEMATICS

ARITHMETIC AND GEOMETRYARITHMETIC AND GEOMETRY

PYTHAGOREAN TRIPLESPYTHAGOREAN TRIPLES

RATIONAL POINTS ON RATIONAL POINTS ON CIRCLE OF CONSTRUCTIONCIRCLE OF CONSTRUCTION

DEFINITIONDEFINITION

WHAT IS PYTHAGOREAN THEOREMWHAT IS PYTHAGOREAN THEOREM

BIOGRAPHYCAL NOTES: BIOGRAPHYCAL NOTES: PYTHAGORASPYTHAGORAS

THE USE OF THEOREMTHE USE OF THEOREM

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Born on island SamosLearned mathematics from Thales (624 - 547 BCE) (Miletus)Croton (around 540 BCE)Founded a school (Pythagoreans)

“All is number”strict code of conduct (secrecy, vegetarianism, taboo on eating beans etc.)explanation of musical harmony in terms of whole-number ratiosPythagoras

(580 BCE – 497 BCE)

Bhiographical Notes : Pythagoras

1. Pythagoras was a Greek mathematician and philosopher. 2. He was born on the island of Samos off the Greek coast.(582 BC -

496 BC).3. At a very early age he travelled to Mesopotamia and Egypt where he

undertook his basic studies and eventually founded his first school.4. He was known best for the Pythagorean Theorem. 5. He is also called "the father of numbers" .6. Pythagoras believed that mathematics could exist without music or

astronomy but mathematical principles were universal and implicit in all things; thus nothing could exist without numbers.

7. His teachings encompassed not only the investigation into the self but into the whole of the known universe of his time.

8. Pythagoras is widely regarded as the founder of modern mathematics, musical theory, philosophy and the science of health (hygiene).

What is the Pythagorean Theorem?

We have a page that talks all about the Pythagorean Theorem, but here is a quick summary: The Pythagorean Theorem states that, in

a right triangle, the square of a (a²) plus the square of b (b²) is equal to the square of c (c²):

a2 + b2 = c2

In a right angled triangle the square of the long side (the "hypotenuse") is equal to the sum of the squares of the other two sides. It is stated in this formula: a2 + b2 = c2

a

b

c c2c2

b2b2

a2a2

+ =

Years ago, a man named Pythagoras found an amazing fact about triangles:

If the triangle had a right angle (90°) ...... and you made a square on each of 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 the hypotenuse is equal to the sum of the squares of the other two sides.

Formal Definition…

So, the square of a (a²) plus the square of b (b²) is equal to the square of c (c²):

a2 + b2 = c2

a

b

c

If we know the lengths of two sides of a right angled triangle, then Pythagoras' Theorem allows us to find the length of the third side. (But

remember it only works on right angled triangles!)

a2 + b2 = c2 52 + 122 = c2 25 + 144 = 169 c2 = 169 c = √169 c = 13

a2 + b2 = c2 92 + b2 = 152 81 + b2 = 225 Take 81 from both sides b2 = 144 b = √144 b = 12

The Use of Theorem…

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HISTORY OF BABYLONIANHISTORY OF BABYLONIAN

MAP OF THE REGIONMAP OF THE REGION

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THE BABYLONIAN TABLETTHE BABYLONIAN TABLET

In this article we examine four Babylonian tablets which all have some connection with Pythagoras's theorem. Certainly the Babylonians were familiar with Pythagoras's theorem. A translation of a Babylonian tablet which is preserved in the British museum goes as follows:-

4 is the length and 5 the diagonal. What is the breadth ?Its size is not known.

4 times 4 is 16. 5 times 5 is 25.

You take 16 from 25 and there remains 9. What times what shall I take in order to get 9 ?

3 times 3 is 9. 3 is the breadth.

All the tablets we wish to consider in detail come from roughly the same period, namely that of the Old Babylonian Empire which flourished in Mesopotamia between 1900 BC and 1600 BC.

History of Babylonian…

Here is a map of the region where the Babylonian Civilisation flourished between 1900 BC and 1600 BC

Pythagoras Theorem in Babylonian

1. There are four types of babylonian tablets called Yale tablet YBC 7289, Plimpton 322 (shown), the Susa tablet, and the Tell Dhibayi tablet.

2. The Yale tablet YBC 7289 is one of a large collection of tablets held in the Yale Babylonian collection of Yale University.

3. It consists of a tablet on which a diagram appears. The diagram is a square of side 30 with the diagonals drawn in.

The Babylonian Tablet

Plimpton 322 is the tablet numbered 322 in the collection of G A Plimpton housed in Columbia University

Its date is not known accurately but it is put as between 1800 BC and 1650 BC. It is thought to be only part of a larger tablet, the remainder of which has been destroyed.

Plimpton 322 Tablet

1. The Susa tablet was discovered at the present town of Shush in the Khuzistan region of Iran. The town is about 350 km from the ancient city of Babylon.

2. W K Loftus identified this as an important archaeological site as early as 1850 but excavations were not carried out until much later.

3. The particular tablet which interests us here investigates how to calculate the radius of a circle through the vertices of an isosceles triangle.

The Susa Tablet

1. The Tell Dhibayi tablet was one of about 500 tablets found near Baghdad by archaeologists in 1962.

2. Most relate to the administration of an ancient city which flourished in the time of Ibalpiel II of Eshunna and date from around 1750.

3. The particular tablet which concerns us is not one relating to administration but one which presents a geometrical problem which asks for the dimensions of a rectangle whose area and diagonal are known.

The Tell Dhibayi Tablet

1. It has on it a diagram of a square with 30 on one side, the diagonals are drawn in and near the centre is written 1,24,51,10 and 42,25,35.

2. Of course these numbers are written in Babylonian numerals to base 60. See our article on Babylonian numerals.

3. Now the Babylonian numbers are always ambiguous and no indication occurs as to where the integer part ends and the fractional part begins.

The Yale Tablet

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DEFINITION IN ARITHMETIC AND DEFINITION IN ARITHMETIC AND GEOMETRYGEOMETRY

CONTRIBUTIONS BY THE CONTRIBUTIONS BY THE PYTHAGOREANPYTHAGOREAN

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CONVERSE STATEMENTCONVERSE STATEMENT

The Theorem of Pythagoras

If c is the hypotenuse of a right-angled triangleand a, b are two other sides then

a2+b2=c2

a

b

c

c2b2

a2

“Let no one unversed in geometry enter here”was written over the door of Plato’s Academy (≈ 387 BCE)

Arithmetic and Geometry...

1. If a,b and c satisfy a2+b2=c2 then there exists a right-angled triangle with corresponding sides.

2. One can consider a2+b2=c2 as an equation3. It has some simple solutions: (3,4,5), (5,12,13) etc.4. Practical use - construction of right angles5. Deep relationship between arithmetic and geometry6. Discovery of irrational numbers

Converse Statement...

VISUAL PROOF OF THE PYTHAGOREAN THEOREM FOR THE

(3, 4, 5) TRIANGLE AS IN THE CHOU PEI

SUAN CHING 500–200 BC.

Contributions by the Pythagorean is include :-1. Various theorems about triangles, parallel

lines, polygons, circles, spheres and regular polyhedra.

2. Work on a class of problems in the applications of areas. (e.g. to construct a polygon of given area and similar to another polygon.

The Pythagoreans had four branches of Number: 1. Arithmetic consisted of solely Number,2. Geometry was Number combined with space,3. Music was Number in time, and4. Astronomy was the mixture of Number,

space, and time.

Contributions by the Pythagorean

Pythagoreans liked to express numbers as geometrical figures;

the foremost of these is the Tetraktys

Starting at Unity (one), the Tetraktys proceeds through the ordered numbers two, three, and four, which add up to a second Unity, ten. These four numbers also correspond to the ideas of a point,

line, plane, and three-dimensional surface

Pythagorean number theory make abundantly clear how the early Greeks

achieved the insights necessary to develop geometry. They devised a

new number system which expressed numbers in a highly visual and

physical form, through the arrangement

of dots.

1. The figulate numbers were found by the Pythagorean.

2. These numbers, considered as the number of dots in

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DEFINITIONDEFINITION

LIST OF PYTHOGOREAN LIST OF PYTHOGOREAN TRIPLES UP TO 100TRIPLES UP TO 100

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GENERAL FORMULAGENERAL FORMULA

1. Definition Integer triples (a,b,c) satisfying a2+b2=c2 are called Pythagorean triples

2. Examples: (3,4,5), (5,12,13), (8,15,17) etc.3. Pythagoras: around 500 BCE4. Babylonia 1800 BCE: clay tablet

“Plimpton 322” lists integer pairs (a,c) such that there is an integer b satisfying a2+b2=c2

5. China (200 BCE -220 CE), India (500-200 BCE)6. Greeks: between Euclid (300 BCE) and Diophantus (250 CE)

7. Diophantine equation (after Diophantus, 300 CE) - polynomial equation with integer coefficients to which integer solutions are sought

8. It was shown that there is no algorithm for deciding which polynomial equations have integer solutions.

Pythagorean Triples

•Theorem Any Pythagorean triple can be obtained as follows:a = (p2-q2)r, b = 2qpr, c = (p2+q2)r

where p, q and r are arbitrary integers.

•Special case: a = p2-q2, b = 2qp, c = p2+q2

•Proof of general formula:Euclid’s “Elements” Book X (around 300 BCE)

General Formula…

1. A Pythagorean triple has three positive integers a, b, and c, such that a2 + b2 = c2.

2. In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths.

3. Evidence from megalithic monuments on the Northern Europe shows that such triples were known before the discovery of writing.

4. Such a triple is commonly written (a, b, c). Some well-known examples are (3, 4, 5) and (5, 12, 13).

How To Know The Pythagorean Triples?

• (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, 80) • (48, 55, 73)• (65, 72, 97)

List of Pythagorean Triples Up To 100

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RATIONAL POINTS OF CIRCLERATIONAL POINTS OF CIRCLE

CONSTRUCTION ON CIRCLECONSTRUCTION ON CIRCLE

• Pythagorean triple (a,b,c)• Triangle with rational sidesx = a/c, y = b/c and hypotenuse c = 1

• x2 + y2 = 1 → P (x,y) is a rational point on the unit circle.Y

XO x

y1

P

Rational Point On Circle

1. Base point (trivial solution) Q(x,y) = (-1,0)2. Line through Q with rational slope t

y = t(x+1)intersects the circle at a second rational point R

3. As t varies we obtain all rational points on the circle which have the formx = (1-t2) / (1+t2), y = 2t / (1+t2)where t = p/q

Y

X

O 1

Q

-1

R

Construction Of The Rational Point On Circle