Definition Pythagoras Theorem In Babylonian Mathematics Arithmetic And Geometry Pythagorean

DEFINITION PYTHAGORAS THEOREM IN BABYLONIAN MATHEMATICS ARITHMETIC AND GEOMETRY PYTHAGOREAN TRIPLES RATIONAL POINTS ON CIRCLE OF CONSTRUCTION

BIOGRAPHYCAL NOTES: PYTHAGORAS WHAT IS PYTHAGOREAN THEOREM THE USE OF THEOREM VIDEO SEGMENT

Bhiographical Notes : Born on island Samos Pythagoras

Pythagoras (580 BCE – 497 BCE)

Learned 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 wholenumber ratios

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Pythagoras was a Greek mathematician and philosopher. He was born on the island of Samos off the Greek coast.(582 BC - 496 BC). At a very early age he travelled to Mesopotamia and Egypt where he undertook his basic studies and eventually founded his first school. He was known best for the Pythagorean Theorem . He is also called "the father of numbers" . 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. His teachings encompassed not only the investigation into the self but into the whole of the known universe of his time. Pythagoras is widely regarded as the founder of modern mathematics, musical theory, philosophy and the science of health (hygiene).

What is the Pythagorean Theorem?

c2

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 2 a a c (c²): 2 a2 + btriangle = c2 the In a right angled bb2 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 + b 2 = c 2

Formal Definition…  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 c is equal to the sum of the squares of the other  So,two the sides. square of a (a²) plus the a square of b (b²) is equal to the square of c (c²): b a2 + b 2 = c 2

The of Use Theorem… If we know the lengths twoof 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!)

a 2 + b 2 = c2 52 + 122 = c2 25 + 144 = 169 c2 = 169 c = √169 c = 13

a2 + b 2 = c 2 92 + b2 = 152 81 + b2 = 225 Take 81 from both sides b2 = 144 b = √144 b = 12

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HISTORY OF BABYLONIAN MAP OF THE REGION THE BABYLONIAN TABLET VIDEO SEGMENT

In this article we examine four Babylonian tablets which all have some connection with Pythagoras's theorem. Certainly of Babylonian… the BabyloniansHistory 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.

Pythagoras Theorem in Babylonian

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

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

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

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

Plimpton 322 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.

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

Tablet

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

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

The Tell Dhibayi Tablet •

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

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

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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

1. It has on it a diagram of a square with 30 on one Yale Tablet side, the diagonals are drawn in and near the centre is written 1,24,51,10 and 42,25,35. 3. Of course these numbers are written in Babylonian numerals to base 60. See our article on Babylonian numerals. 5. Now the Babylonian numbers are always ambiguous and no indication occurs as to where the integer part ends and the fractional part begins.

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DEFINITION IN ARITHMETIC AND GEOMETRY CONVERSE STATEMENT CONTRIBUTIONS BY THE PYTHAGOREAN VIDEO SEGMENT

Arithmetic and Geometry... The Theorem of Pythagoras If c is the hypotenuse of a right-angled triangle and a, b area2two other 2 sides then +b2=c

c2

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

a2 a b

b2

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Converse Statement...

If a,b and c satisfy a2+b2=c2 then there exists a right-angled triangle with corresponding sides. One can consider a2+b2=c2 as an equation It has some simple solutions: (3,4,5), (5,12,13) etc. Practical use - construction of right angles Deep relationship between arithmetic and geometry Discovery of irrational numbers

Contributions by the Pythagorean is include :2. Various theorems about Contributions by the triangles, parallel lines, polygons, Pythagorean circles, spheres and regular polyhedra. 3. 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: VISUAL PROOF 5. Arithmetic consisted of solely OF THE PYTHAGOREAN Number, THEOREM FOR 6. Geometry was Number combined THE (3, 4, 5) TRIANGLE AS IN with space, THE CHOU PEI 7. Music was Number in time, and SUAN CHING 8. Astronomy was the mixture of 500–200 BC. Number, space, and time.

Starting at Unity (one), the Pythagoreans liked to Tetraktys proceeds through express numbers as the ordered numbers two, geometrical figures; thethree, and four, which add up foremost of these is the to a second Unity, ten. These Tetraktys 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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DEFINITION GENERAL FORMULA LIST OF PYTHOGOREAN VIDEO TRIPLES SEGMENT UP TO 100

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Pythagorean Triples Definition Integer triples (a,b,c) satisfying a2+b2=c2 are called Pythagorean triples Examples: (3,4,5), (5,12,13), (8,15,17) etc. Pythagoras: around 500 BCE Babylonia 1800 BCE: clay tablet “Plimpton 322” lists integer pairs (a,c) such that there is an integer b satisfying a2+b2=c2 China (200 BCE -220 CE), India (500-200 BCE) Greeks: between Euclid (300 BCE) and Diophantus (250 CE) Diophantine equation (after Diophantus, 300 CE) - polynomial equation with integer coefficients to which integer solutions are sought It was shown that there is no algorithm for deciding which polynomial equations have integer solutions.

General Formula… •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)

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How To Know The Pythagorean Triples? A Pythagorean triple has three positive integers a, b, and c, such that a2 + b2 = c2.

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

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

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

List of 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, (20, (28, (33, (36, (39, (48, (65,

63, 21, 45, 56, 77, 80, 55, 72,

65) 29) 53) 65) 85) 80) 73) 97)

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RATIONAL POINTS OF CIRCLE CONSTRUCTION ON CIRCLE

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Point On(a,b,c) Circle • Rational Pythagorean triple • Triangle with rational sides x = a/c, y = b/c and hypotenuse c = 1 x2 + y2 = 1 → P (x,y) is a rational point on the unit circle. Y P 1 O

x

y

X

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Construction Of The Rational Point On Circle Base point (trivial solution) Q(x,y) = (-1,0) Line through Q with rational slope t y = t(x+1) intersects the circle at a second rational point R Y As t varies we obtain all rational points on the circle which have the form R x = (1-t2) / (1+t2), y = 2t / (1+t2) where t = p/q Q X -1

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