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

$Page 1: 1 π Consider the value of π below we recall that two of the more accurate fractional approximations of π are: The 7 th, 22 nd, 113 th, and 355 th positions$
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πConsider the value of π below we recall that two of the more accurate fractional approximations of π are:

142857142857.37

22

465572212453982300881415929203.3113

355

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?

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2 – Babylonian & Egyptian Mathematics

The student will learn about

Numeral systems from the Babylonian and Egyptian cultures.

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Cultural ConnectionThe Agricultural Revolution

The Cradles of Civilization – ca. 3,000 – 525 B.C.

Student led discussion.

4



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



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

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Civilization needed and developed –

A written languageEngineering skillsCommercial skillsAstronomical skills

Geodetic skills

continued

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Governments were developed –

Oligarchy – small clique of privileged citizens.Monarchies – king or queen.Theocracies – rule by religious leaders.Republics – broad citizen participation

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§2-1 The Ancient Orient

Student Discussion.

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§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.

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§2-2 Babylonian Sources of Information

Student Discussion.

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§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.

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§2-3 BabylonianCommercial and Agrarian Math

Student Discussion.

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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

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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

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There is a modern notation for base 60 which is quite helpful.

1, 02, 34; 15 means 1, 02, 34; 15 means

1 · 60 2 + 2 · 60 + 34 + 15/60 =

1, 02, 34; 15 means

1 · 60 2 + 2 · 60 + 34 + 15/60 =

3600 + 120 + 34 + 0.25 = 3754.25 ten

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§2-4 BabylonianGeometry

Student Discussion.

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§2-4 BabylonianGeometry

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.

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§2-5 BabylonianAlgebra

Student Discussion.

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§2-5 BabylonianAlgebra

Solved some quadratics by substitution and completing the square.

Solved some cubic, biquadratic and a few of higher degree.

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2 by Babalonian MethodsThe ancients knew that if 2 < x then 2/x < 2 .

First iteration: Let x = 1.5For a better approximation average x and 2/x:

x 2/x Average

3/2 4/3 17/12

continued

Show why.

This implied: 2/x < 2 < x

x 2/x Average

3/2 4/3 17/12

17/12 24/17 577/408

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2 by Babalonian MethodsWith 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

This calculation was on Tablet No. 7289 from the Yale Collection.

Accuracy to - 0.0000006 or about the equivalency of about 1 foot over the distance to Boston!

YBC 7289

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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 = 30b = 1, 24, 51, 10 andc = 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 = 30b = 1, 24, 51, 10 andc = 42, 25, 35

Note that c = a ∙ b = 30 ∙ (1, 24, 51, 10)

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§2-6 BabylonianPlimpton 322

Student Discussion.

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§2-6 BabylonianPlimpton 322

b c

119 169 1

3367 4825 2

4601 6649 3

… … …

56 106 15

bc

aB

A

90

a b c

120 119 169 1

3456 3367 4825 2

480 4601 6649 3

… … … …

90 56 106 15

Column a is regular sexagesimal numbers. Columns b and c are generated parametrically from regular sexagesimal numbers.

(c/a) 2 a b c

1.9326 120 119 169 1

1.8696 3456 3367 4825 2

1.8107 480 4601 6649 3

… … … … …

1.3611 90 56 106 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.

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Egyptian

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§2-7 EgyptianSources of Information

Student Discussion.

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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

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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. 100,000 laborers for 30 years

1850 B.C. Moscow papyrus – 25 problems

1650 B.C. Rhine papyrus – 85 problems

continued

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1500 B.C. Sundial

1350 B.C. Papyrus with bread accounts.1167 B.C. Harris papyrus – Rameses III

196 B.C. Rosetta Stone – Egyptian hieroglyphics, Egyptian Demotic, and Greek.

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by MIKE PETERS

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§2-8 EgyptianArithmetic and Algebra

Student Discussion.

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Duplation and Mediation for multiplication. 26 · 33.

1 332 664 132

continued

8 26416 528

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.

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Duplation and Mediation – Why It Works! 26 · 33.

1 332 664 132

continued

8 26416 528

858

(26) x (33)

= (2 + 8 + 16) x (33)= (2)(33) + (8)(33) + (16)(33)

= (66) + (264) + (528)

= 858

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1 262 524 104

continued

8 20816 416

28 728 + 25 = 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.

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1 262 524 104

continued

8 20816 416

28 728 + 25 = 753

Duplation and Mediation for division. Why it works! 753 26.

Quotient remainder

753 ÷ 26

753 = 28 x 26 + 25

753 = (4 + 8 + 16) x 26 + 25

753 = (104 + 208 + 416) + 25

753 = (728) + 25

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37


Unit fractions to avoid fractional difficulties.

continued

2 1 1

7 4 28

3 1 1

5 2 10

5?

18

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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 EgyptianGeometry

Student Discussion.

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§2-9 EgyptianGeometry

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.

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§2-10 EgyptianRhind Papyrus

Student Discussion.

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§2-10 EgyptianRhind 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..

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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.

Documents

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