1 π Consider the value of π below we recall that two of the more accurate fractional approximations of π are: The 7 th, 22 nd, 113 th, and 355 th positions in the decimal value of π are all “2”. Is this coincidental, or does it have some mysterious meaning?

2 2 – Babylonian & Egyptian Mathematics The student will learn about Numeral systems from the Babylonian and Egyptian cultures.

3 Cultural Connection The Agricultural Revolution The Cradles of Civilization – ca. 3,000 – 525 B.C. Student led discussion.

4 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

5 Cultural Connection The Agricultural Revolution The Cradles of Civilization – ca. 3,000 – 525 B.C. Civilization centered about rivers – AfricaNile River Mid-EastTigrus and Euphrates Rivers (Mesopotamia) with city of Ur about 24,000 people. IndiaIndus River ChinaYellow River continued

6 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

7 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

8 §2-1 The Ancient Orient Student Discussion.

9 §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.

10 §2-2 Babylonian Sources of Information Student Discussion.

11 §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.

12 §2-3 Babylonian Commercial and Agrarian Math Student Discussion.

13 §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

14 §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 · or. A fraction was also in base 60 where ½ = 30/60 = continued

15 §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 = 1, 02, 34; 15 means 1 · · /60 = = ten

16 §2-4 Babylonian Geometry Student Discussion.

17 §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.

18 §2-5 Babylonian Algebra Student Discussion.

19 §2-5 Babylonian Algebra Solved some quadratics by substitution and completing the square. Solved some cubic, biquadratic and a few of higher degree.

20  2 by Babalonian Methods The ancients knew that if  2 < x then 2/x <  2. First iteration: Let x = 1.5 For a better approximation average x and 2/x: x2/xAverage 3/24/317/12 continued Show why. This implied: 2/x <  2 < x x2/xAverage 3/24/317/12 24/17577/408

21  2 by Babalonian Methods With basically two iterations we arrive at 577 / 408 In decimal form this is 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 This calculation was on Tablet No from the Yale Collection. Accuracy to or about the equivalency of about 1 foot over the distance to Boston!

YBC Do it! Just like in our base ten system multiplying by 5 and dividing by 2 yield the same numeric results less decimal point placement. Do it! b = 1, 24, 51, 10 OR 84, 50, 70Do it! b = 1, 24, 51, 10 OR 84, 50, 70 ÷ 2 = 42, 25, 35 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? 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)

23 §2-6 Babylonian Plimpton 322 Student Discussion.

24 §2-6 Babylonian Plimpton 322 bc ……… b c a B A 90 abc ………… Column a is regular sexagesimal numbers. Columns b and c are generated parametrically from regular sexagesimal numbers. (c/a) 2 abc …………… (c/a) 2 is the secant 2 of 44°, 43 °, 42 °, …, 31 °. Accuracy is from 0.02 to We will see the significance of secant later in the course.

25 Egyptian

26 §2-7 Egyptian Sources of Information Student Discussion.

27 §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

28 §2-7 Egyptian Sources of Information 3100 B.C.Numbers to millions 2600 B.C.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 laborers for 30 years 1850 B.C.Moscow papyrus – 25 problems 1650 B.C.Rhine papyrus – 85 problems continued

29 §2-7 Egyptian Sources of Information 1500 B.C.Sundial 1350 B.C.Papyrus with bread accounts B.C.Harris papyrus – Rameses III 196 B.C.Rosetta Stone – Egyptian hieroglyphics, Egyptian Demotic, and Greek.

30 by MIKE PETERS

31 §2-8 Egyptian Arithmetic and Algebra Student Discussion.

32 §2-8 Egyptian Arithmetic and Algebra Duplation and Mediation for multiplication. 26 · continued Pick the numbers in the left column that add to 26. Cross out the remaining rows. 858 Pick the numbers in the left column that add to 26. Cross out the remaining rows. The sum of the right column is the answer.

33 §2-8 Egyptian Arithmetic and Algebra Duplation and Mediation – Why It Works! 26 · continued (26) x (33) = ( ) x (33) = (2)(33) + (8)(33) + (16)(33) = (66) + (264) + (528) = 858

34 §2-8 Egyptian Arithmetic and Algebra continued = 753 Duplation and Mediation for division. 753  26. Quotient remainder Pick the numbers in the right column that add to 753 or less. Cross out the remaining rows. Pick the numbers in the right column that add to 753 or less. Cross out the remaining rows. The sum in the left column is the quotient and the difference between the right column and 753 is the remainder.

35 §2-8 Egyptian Arithmetic and Algebra continued = 753 Duplation and Mediation for division. Why it works! 753  26. Quotient remainder 753 ÷ = 28 x = ( ) x = ( ) = (728) + 25

36

37 §2-8 Egyptian Arithmetic and Algebra Unit fractions to avoid fractional difficulties. continued

38 §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? 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. Pick a number to try. A good choice would be a number divisible by three, Why? Try 6.

39 §2-9 Egyptian Geometry Student Discussion.

40

41 §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 /8 is more accurate. No Pythagorean Theorem.

42 §2-10 Egyptian Rhind Papyrus Student Discussion.

43 §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..

44 Assignment 1. Read Chapter 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.