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π Consider the value of π below we recall that two of the more accurate fractional approximations of π are: 22 3.142857142857 7 355 3.141592920353982300884655722124 113 The 7th, 22nd, 113th, and 355th positions in the decimal value of π are all “2”. Is this coincidental, or does it have some mysterious meaning? 1 2 – Babylonian & Egyptian Mathematics The student will learn about Numeral systems from the Babylonian and Egyptian cultures. 2 Cultural Connection The Agricultural Revolution The Cradles of Civilization – ca. 3,000 – 525 B.C. Student led discussion. 3 Cultural Connection The Agricultural Revolution The Cradles of Civilization – ca. 3,000 – 525 B.C. Ends at 525 B.C. when Persia conquered Babylonia. Climatic changes caused the savannahs to change into forest or desserts. Population density prohibited hunter/gathers (about 40 people per square mile) so man turned to agriculture. continued 4 Cultural Connection The Agricultural Revolution The Cradles of Civilization – ca. 3,000 – 525 B.C. Civilization centered about rivers – Africa Nile River Mid-East Tigrus and Euphrates Rivers (Mesopotamia) with city of Ur about 24,000 people. India Indus River China Yellow River continued 5 Cultural Connection The Agricultural Revolution The Cradles of Civilization – ca. 3,000 – 525 B.C. Civilization needed and developed – A written language Engineering skills Commercial skills Astronomical skills Geodetic skills continued 6 Cultural Connection The Agricultural Revolution The Cradles of Civilization – ca. 3,000 – 525 B.C. Governments were developed – Oligarchy – small clique of privileged citizens. Monarchies – king or queen. Theocracies – rule by religious leaders. Republics – broad citizen participation 7 §2-1 The Ancient Orient Student Discussion. 8 §2-1 The Ancient Orient Calendars. Weights and measures to harvest, store and apportion food. Surveying for canals and reservoirs and to parcel land. Financial and commercial practices – raising and collecting taxes and trade. 9 §2-2 Babylonian Sources of Information Student Discussion. 10 §2-2 Babylonian Sources of Information About 500,000 clay tablets found in Mesopotamia. Many were deciphered by Sir Henry Creswicke Rawlinson in the mid 1800’s. Tablets were small. Several inches on a side. 11 §2-3 Babylonian Commercial and Agrarian Math Student Discussion. 12 §2-3 Babylonian Commercial and Agrarian Math Commercial examples – bills, receipts, promissory notes, interest, etc. Agrarian examples – field measurement, crop calculation, sales of crops, etc. Many tablets were math tables – reciprocals, squares, cubes, exponents, etc. continued 13 §2-3 Babylonian Commercial and Agrarian Math Remember they worked in base 60 with only two symbols and for 1 and 10 respectively. meant 11 or 11 · 60 or 11· 60 2 or …. 765 was 12 · 60 + 45 or . A fraction was also in base 60 where ½ = 30/60 = continued 14 §2-3 Babylonian Commercial and Agrarian Math There is a modern notation for base 60 which is quite helpful. 1, 02, 34; 15 means 1 · 60 2 + 2 · 60 + 34 + 15/60 = 3600 + 120 + 34 + 0.25 = 3754.25 ten 15 §2-4 Babylonian Geometry Student Discussion. 16 §2-4 Babylonian Geometry Area of rectangles, right triangles, isosceles triangles, and trapezoids was known. Volume of rectangular parallelepipeds, and right prisms was known. π was assumed to be 3 1/8. Proportions between similar triangles were known. The Pythagorean theorem was known. 17 §2-5 Babylonian Algebra Student Discussion. 18 §2-5 Babylonian Algebra Solved some quadratics by substitution and completing the square. Solved some cubic, biquadratic and a few of higher degree. 19 2 by Babalonian Methods The ancients knew that if 2 < x then 2/x < 2 . Show why. This implied: 2/x < 2 < x First iteration: Let x = 1.5 For a better approximation average x and 2/x: x 2/x Average 3/2 4/3 17/12 17/12 24/17 577/408 continued 20 2 by Babalonian Methods With basically two iterations we arrive at 577 / 408 In decimal form this is 1.414212963 In base sixty notation this is 1 ; 24, 51, 10, 35, . . . To three decimal places 1 ; 24, 51, 10 is what the Babylonians used for 2 ! Accuracy to - 0.0000006 or about the equivalency of about 1 foot over the distance to Boston! This calculation was on Tablet No. 7289 from the Yale Collection. 21 YBC 7289 On the Yale Babylonian Collection Tablet 7289 there are three numbers: a = 30 b = 1, 24, 51, 10 and c = 42, 25, 35 Note that c = a ∙ b = 30 ∙ (1, 24, 51, 10) Instead of multiplying b by 30 the Babylonians no doubt divided it by 2. Why? Do it! b = 1, 24, 51, 10 OR 84, 50, 70 ÷ 2 = 42, 25, 35 Just like in our base ten system multiplying by 5 and dividing by 2 yield the same numeric results less decimal 22 point placement. §2-6 Babylonian Plimpton 322 Student Discussion. 23 §2-6 Babylonian Plimpton 322 (c/a) 2 1.9326 1.8696 1.8107 … 1.3611 a 120 3456 480 … 90 b 119 3367 4601 … 56 c A B 90 b a c 169 4825 6649 … 106 1 2 3 … 15 (c/a) 2 is the secant 2 of 44°, 43 °, 42 °, … , 31 °. Accuracy is from 0.02 to 0.08. We will see the significance of secant later in the course. Column a is regular sexagesimal numbers. Columns b and c are generated parametrically from regular sexagesimal numbers. 24 Egyptian 25 §2-7 Egyptian Sources of Information Student Discussion. 26 §2-7 Egyptian Sources of Information Egypt was more seclude and naturally protected. Their society was a theocracy with slaves doing manual labor. The dry climate preserved many of their documents. It has been felt recently that they were not as sophisticated as the Babylonians. continued 27 §2-7 Egyptian Sources of Information 3100 B.C. 2600 B.C. Numbers to millions Great Pyramid – 13 acres, 2,000,000 stones from 2.5 to 54 tons granite blocks from 600 miles away. Square to 1/14,000, and right angles to 1/27,000. 100,000 laborers for 30 years 1850 B.C. 1650 B.C. Moscow papyrus – 25 problems Rhine papyrus – 85 problems continued 28 §2-7 Egyptian Sources of Information 1500 B.C. 1350 B.C. Sundial Papyrus with bread accounts. 1167 B.C. Harris papyrus – Rameses III 196 B.C. Rosetta Stone – Egyptian hieroglyphics, Egyptian Demotic, and Greek. 29 by MIKE PETERS 30 §2-8 Egyptian Arithmetic and Algebra Student Discussion. 31 §2-8 Egyptian Arithmetic and Algebra Duplation and Mediation for multiplication. 26 · 33. 1 33 Pick the numbers in the left 2 66 column that add to 26. Cross 4 8 16 132 264 528 out the remaining rows. The sum of the right column is the answer. 858 continued 32 §2-8 Egyptian Arithmetic and Algebra Duplation and Mediation – Why It Works! 26 · 33. (26) x (33) 1 33 = (2 + 8 + 16) x (33) 2 66 4 8 16 132 264 528 = (2)(33) + (8)(33) + (16)(33) = (66) + (264) + (528) = 858 858 continued 33 §2-8 Egyptian Arithmetic and Algebra Duplation and Mediation for division. 753 26. Pick the numbers in the right 1 26 column that add to 753 or less. 2 52 Cross out the remaining rows. The 4 8 16 104 208 416 28 728 + 25 = 753 sum in the left column is the quotient and the difference between the right column and 753 is the remainder. continued Quotient remainder 34 §2-8 Egyptian Arithmetic and Algebra Duplation and Mediation for division. Why it works! 753 ÷ 26 753 26. 1 2 26 52 753 = 28 x 26 + 25 4 8 16 104 208 416 753 = (104 + 208 + 416) + 25 28 728 + 25 = 753 753 = (4 + 8 + 16) x 26 + 25 753 = (728) + 25 continued Quotient remainder 35 36 §2-8 Egyptian Arithmetic and Algebra Unit fractions to avoid fractional difficulties. 2 1 1 7 4 28 3 1 1 5 2 10 5 ? 18 continued 37 §2-8 Egyptian Arithmetic and Algebra Rule of False Positioning. x – x/3 = 8 Pick a number to try. A good choice would be a number divisible by three, Why? Try 6. 6 – 6/3 = 4 Notice 4 is one-half the correct answer hence the correct answer must be double 6 (6 was your guess) or 12. 38 §2-9 Egyptian Geometry Student Discussion. 39 40 §2-9 Egyptian Geometry They knew the area of a circle as (8/9 d)2, area of a triangle as ½ ab, area of a quadrilateral as (a + c) (b + d) / 4 which is incorrect. Knew the volume of a right circular cylinder as bh, = (16/9) 2 which is off by 0.0189. 3 1/8 is more accurate. No Pythagorean Theorem. 41 §2-10 Egyptian Rhind Papyrus Student Discussion. 42 §2-10 Egyptian Rhind Papyrus Curious Problem. Knew regular sexagesimal numbers – that is a number divisible by factors of 60. This made work with fractions easier since they produced reciprocals which were terminating fractions.. 43 Assignment 1. Read Chapter 3. 2. Calculate the cost of building a pyramid at 100,000 laborers, six days a week at twelve hours a day for 30 years at $7.15 an hour. 3. By Duplation and Mediation (346)(53) 4. By Duplation and Mediation (7634) (24) 5. Handouts. 44