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Diapositiva 1 We Should Not Call It the Pythagorean Theorem! Presented by Eric Hutchinson and Aminul KM The Pythagorean Theorem • The sum of the squares of each leg of a right angled triangle is equal to the square of the hypotenuse a² + b² = c² The Pythagorean Theorem • The Pythagorean Theorem takes its name from the ancient Greek Mathematician Pythagoras • Purportedly first to offer a proof of the theorem • Long before Pythagoras people noticed right triangle special relationships Pythagoras • Pythagoras was a Greek mathematician and a philosopher, but was best known for his Pythagorean Theorem. • He was born around 572 B.C. on the island of Samos, in Greece. • For about 22 years, Pythagoras spent time traveling though Egypt and Babylonia to educate himself. Pythagoras • At about 530 B.C., Pythagoras settled in a Greek town in southern Italy called Crotona. • Pythagoras formed a brotherhood that was an exclusive society devoted to moral, political and social life. This society was known as Pythagoreans. Pythagoras • The Pythagorean School excelled in many subjects, such as music, medicine and mathematics • In the society, members were known as mathematikoi, which is Greek for mathematicians • History tells us that this theorem has been introduced through drawings, texts, legends, and stories from Babylon, Egypt and China, dating back to 1800- 1500 BC Proof (Pythagoras) • Start with this figure: • The area of this figure is: Proof (Pythagoras) • Rearrange the triangles while keeping the area the the same: • The area of this figure is: Proof (Pythagoras) The Babylonians • Ancient clay tablets from Babylonia reveal that the Babylonians, 1000 years before Pythagoras, knew the Pythagorean relationships. • Tablets reveal Babylonians used the Pythagorean Theorem to approximate the square root of 2. Babylonian Tablets (1800 BC) Babylonian Tablets (1800 BC) • Last two columns: • Column heading is translated loosely as “number” • Contains row numbers • Green numbers represent missing data Babylonian Tablets (1800 BC) • Second column: • Column heading translated as “width” • Error corrections in red • Sexagesimal numbers For example 1,59 is 1*60+59=159 and 5,19 = 5*60+19=319 Babylonian Tablets (1800 BC) • Third column • Column heading translated “diagonal” • Sexagesimal #s • The number 3,12,1 represents 3*3600 + 12*60 + 1 = 11521 Babylonian Tablets (1800 BC) • This tablet contains the expression d2 / l2 where l2 = d2 - w2 Babylonian Tablets (1800 BC) Width Length Diagonal • Translating the tablet 119 120 169 numbers, we see that 3367 11018 11521 for most #s the result 4601 4800 6649 12709 13500 18541 is Pythagorean triples 65 72 97 • The Babylonians did 319 360 481 2291 2700 3541 understand the 799 960 1249 Pythagorean Thm, 541 546 769 but they did not prove 4961 6480 8161 45 60 75 the theorem 1679 2400 2929 1771 7290000 3229 Babylonian Tablets (1800 BC) • We know something about how the tablet was constructed • We do not know exactly why it was created • Ordering of rows indicate it may have been used in an early form of trigonometry in order to make the arithmetic easier Babylonian Tablets (1800 BC) • Another Babylonian tablet states the following which indicates an understanding of Pythagorean Theorem: “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.” Sulbasutras (800BC – 200BC) • The Vedic people entered India about 1500BC from the region that today is Iran • The word Vedic describes the religion of these people • The name comes from their collections of sacred texts known as the Vedas. • The Sulbasutras are appendices to the Vedas Sulbasutras (800BC – 200BC) • The Sulbasutras are Ancient Indian manuals of geometrical constructions used by Vedic priests • Geometry of Shulba Sutras was used in the construction of altars required for sacrificial ritual Sulbasutras (800BC – 200BC) • The Baudhayana Sulbasutra gives one case of the theorem explicitly: “The rope which is stretched across the diagonal of a square produces an area double the size of the original square” Sulbasutras (800BC – 200BC) • The Katyayana Sulbasutra gives a more general case: “The rope which is stretched along the length of the diagonal of a rectangle produces an area which the vertical and horizontal sides make together” Sulbasutras (800BC – 200BC) • Results are stated in terms of “ropes”. A rope is an instrument used for measuring an altar • No formal proofs are shown because this text is meant to be a construction manual • However, there are many examples of Pythagorean triples found in the Sulbasutras • (5,12,13), (12,16,20), (8,15,17), and more all occur in the text. Sulbasutras (800BC – 200BC) • The following construction occurs in most of the different Sulbasutras. • ABCD and PQRS are given squares • Mark X such that PX is equal to AB • Square on SX has an area equal to the sum of the areas of ABCD and ABCD. Then Zhou Bi Suan Jing • One of the oldest Chinese mathematical books • Dedicated to astronomical observation and calculation • From Zhoa Dynasty (1046 BC to 256 BC) • At the end of each Chinese dynasty, libraries were burned, however the Zhoa Dynasty works were saved. Zhou Bi Suan Jing • Anonymous collection of 245 problems encountered by the Duke of Zhou & Shang Goa (astronomer, mathematician) • Contains one of the first recorded proofs of the Pythagorean Theorem Proof (Zhou Bi Suan Jing) • Here is the Chinese original drawing Proof (Zhou Bi Suan Jing) • Here is a redrawing of the original Proof (Zhou Bi Suan Jing) President Garfield • 20th President of the United States • Served from March 4, 1881 to September 19, 1881 (assassinated) • Before politics, Garfield wanted to become a mathematics professor • While in the House of Representatives, he came up with a proof of the Pythagorean Theorem (1876) Proof (Garfield) • The two key facts that are needed for Garfield’s proof are: 1.) The sum of the angles of any triangle is 180 degrees. 2.) The area of a trapezoid formula: Proof (Garfield) • Start with the following diagram made up of three right triangles. These form a trapezoid with bases a and b with a height of a + b. The area of the 3 triangles equals the area of the trapezoid. Proof (Garfield) • Trapezoidal Area = Area of 3 triangles Thank You! • Eric Hutchinson, College of Southern Nevada, Las Vegas [email protected] • Aminul KM, College of Southern Nevada, Las Vegas [email protected].
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