AMA1D01C Egypt and Mesopotamia

Transcription

1 Hong Kong Polytechnic University 2017

2 Outline Cultures we will cover: Ancient Egypt Ancient Mesopotamia (Babylon) Ancient Greece Ancient India Medieval Islamic World Europe since Renaissance

3 References These notes follow the following book: Katz, V. A History of Mathematics: an Introduction. Addison-Wesley, 1998.

4 Introduction Origin of mathematics: Earliest motivation: tax collection, measurement, building, trade, calendar making, ritual practices

5 Babylon Mesopotamian civilization Emerged in the Tigris and Euphrates river valleys around 3500 BC Many kingdoms rose over the next 3000 years The one based in the city of Babylon conquered the entire area around 1700 BC Writing was done by styli on clay tablets Thousands of such tablets have been excavated and well documented by archeologists many of these tablets contain mathematical problems, solutions and tables

6 Egypt First dynasty to rule over both Upper Egypt and Lower Egypt dated from around 3100 BC Much of what we know about ancient Egyptian mathematics comes from the two following sources

7 Egypt Rhind Mathematical Papyrus Named for Scotsman A. H. Rhind ( ) Bought by Rhind at a market in Luxor, Egypt Has remained in the British Museum since 1864 Moscow Mathematical Papyrus Purchased by V. S. Golenishchev in 1893 and later sold to the Moscow Museum of Fine Arts

8 Arithmetic ARITHMETIC

9 Counting Egyptian numerals (hieroglyphic system, used for writing on temple walls or carving on columns): Each power of ten has a symbol Numbers are represented by corresponding repetitions of such symbols

10 Counting Egyptian numerals (hieratic system, used for writing on papyrus (something like paper made from the pith of the papyrus plant)): Each number from 1 to 9, each multiple of 10 from 10 to 90, and each multiple of 100 from 100 to 900, has its own symbol Example: 37 is represented by the symbol for 7 next to the symbol for 30 Example: 243 is represented by the symbol for 3, 40 and 200.

11 Hieroglyph Figure: Hieroglyphic numerals. Source: st-and.ac.uk/histtopics/egyptian_numerals.html

12 Hieroglyph Figure: Hieroglyphic numerals-examples. Source: mcs.st-and.ac.uk/histtopics/egyptian_numerals.html

13 Hieratic Figure: Hieratic numerals. Source: ac.uk/histtopics/egyptian_numerals.html

14 Hieratic Figure: Hieratic numerals - Example. Source: mcs.st-and.ac.uk/histtopics/egyptian_numerals.html

15 Counting Babylonian numerals (Base 60): Numbers smaller than 60 are written in base 10 In these notes (following Katz), we write, for example, a, b, c; d, e, f for the number a b 60 + c + d 1 60, +e f, where a, b, c, are numbers 0 and 59.

16 Babylonian numerals Figure: Babylonian numerals. Source: st-and.ac.uk/histtopics/babylonian_numerals.html

17 Babylonian numerals Figure: Babylonian numerals. Source: st-and.ac.uk/histtopics/babylonian_numerals.html

18 Arithmetic Egyptian algorithms for addition and multiplication in the hieroglyphic system Addition: Combine the symbols, then convert Subtraction: Convert (i.e., borrowing, if necessary), then subtract Multiplication: continuous doubling There is no evidence, however, to show how the doubling was done Depends on the fact that every number can be written as a sum of powers of 2. We are not sure, however, if the Egyptians knew this fact. Maybe they just observed by experimentation Division is the inverse of multiplication, therefore the number a/b would be phrased as muliply by b so that we get a

19 Arithmetic Example: x 12 x Keep doubling. Doubling the last row will give x = 16 and 16 > 13 Find numbers in the first column so that the sum is 13: 13 = , therefore = = 156

20 Arithmetic Algorithms for the hieratic system no evidence to show how addition was done addition tables probably existed

21 Arithmetic Egyptian fractions The Egyptians used only unit fractions, with the single exception of 2/3 To write a reciporcal, in hieroglyphics they put a flat circle above the number and in hieratics they put a dot over the number Problem 3 of the Rhind Papyrus: How to divide 6 loaves among 10 men? Each man gets To use a notation more similar to the hieroglyphic system, we write

22 Arithmetic Figure: Reciprocals in hieroglyphics. Source: st-and.ac.uk/histtopics/babylonian_numerals.html

23 Arithmetic Check: How was the doubling done? The first section of the Rhind Papyrus is a table which contains numbers of the form 2 times n where n is an odd integer between 3 and 101 To check the correctness of the answer, need to know adding and 1 5 gives 6 It is conjectured that addition tables existed

24 Modern proof Modern proof that every fraction can be written as a sum of unit fractions Given a b, let c be the smallest integer such that 1 c a b < 1 c 1 Consider a b 1 c = ac b bc From a b < 1 c 1, we get ac a < b, so ac b < a Any time we subtract the biggest possible unit fraction from a b, the numerator becomes smaller A decreasing sequence of non-negative integers must reach 0 in finitely many steps

25 Modern proof Example: Consider Note 1 2 < Note 1 4 < < 1 3 < 1. Do = Do = Note 1 17 < 5 84 < Do = Therefore = Note, however, the expression is not unique, since is also equal to , which is easier to work with. Practical concern: cutting a pizza into 7 equal slices is easier than cutting it into 21 equal slices

26 Arithmetic Babylonian arithmetic Extensive use of multiplication tables proved by tablets preserved to this day However no addition tables have been found Since the Babylonian place-value system is similar to ours, we may assume their adding algorithm is similar to ours. Example: add 23,35 to 40,33

27 Arithmetic 23,35+40, =1,05 23,00+40,00=1,03,00 Therefore 25,35+40,33=1,04,05

28 Linear Equations and Linear Systems LINEAR EQUATIONS

29 Linear Equations and Linear Systems There is evidence of the Egyptians and the Babylonians solving interesting linear problems. Problem 64, Rhind Papyrus: Arithmetic progression Babylonian text VAT8389: Linear system

30 Linear Equations and Linear Systems Problem 64 of the Rhind Papyrus: If it is said to thee, divide 10 hekats of barley among 10 men so that the difference of each man and his neighbour in hekats of barley is 1 8, what is each man s share? (Gillings, Mathematics in the Times of the Pharaohs) Arithmetic progression Average is 1 hekat per man Add half the common difference = 1 16 nine times to get the largest share (or ) Subtract 1 8 from the largest share nine times to get the size of each share

31 Linear Equations and Linear Systems Problem from VAT8389: One of two fields yields 2 3 sila per sar, the second yields 1 2 sila per sar. The yield of the first field was 500 sila more than that of the second; the areas of the two fields were together 1800 sar. How large is each field? (Katz) We have the system { 2 3 x 1 2 y = 500 x + y = 1800 Assume x and y are both = = 350, and each unit increase in x (with a corresponding unit decrease in y) increases 2 3 x 1 2 y by = 7 6 Number of such increments needed is 350 divided by 7 6, which is 300 Add 300 to 900 to get x = 1200 and subtract 300 from 900 to get y = 600

32 Elementary Geometry ELEMENTARY GEOMETRY

33 Elementary Geometry Problem 50 of the Rhind Papyrus: Example of a round field of diameter 9. What is the area? Take away 1/9 of the diameter; the remainder is 8. Multiply 8 times 8; it makes 64. Therefore, the area is 64. Area is given by ( 8d 9 )2 = 64d2 81 = 256r = How did they come up with this number?

34 Elementary Geometry Hint: Problem 48 Using the octagon to approximate the area of the circle, we get 7d 2 /9 = 63d 2 /81 The Egyptians may be interested in squaring the circle, i.e., finding a number x such that the area of the circle is x 2. 64/81 is close to 63/81.

35 Rhind 48 Figure: Rhind Papyrus Problem 48. Source: edu/~don.allen/history/egypt_old/egypt.html

36 Elementary Geometry The Babylonians used the formula (C/2)(d/2) C is the circumference while d is the diameter They also used A = C 2 /12, obtained by taking d = C/3 Possible explanation: they divide the circles into sectors and rearranged

37 Elementary Geometry Volume There are problems in the Rhind Papyrus where the formula V = Bh was used One would expect the Egyptians knew how to calculate the volumes of pyramids No such formula has been found. However, the Moscow Papyrus contained a problem on the volume of a truncated pyramid

38 Elementary Geometry If it is said to thee, a truncated pyramid of 6 cubits in height, of 4 cubits of the base by 2 of the top; reckon thou with this 4, squaring. Result 16. Double thou this 4. Result 8. Reckon thou with this 2, squaring. Result 4. Add together this 16 with this 8 and with this 4. Result 28. Calculate thou 1/3 of 6. Result 2. Calculate thou with 28 twice. Result 56. Lo! It is 56. Thou has found rightly. (Gillings, Mathematics in the Times of the Pharaohs) 4 2 = 16, 4 2 = 8, 2 2 = 4, = 28, 6/3 = 2, 28 2 = 56 If base width is a, top width is b, truncated height is h, then the method follows the correct formula V = (h/3)(a 2 + ab + b 2 ).

39 Calendar CALENDAR

40 Calendar Egyptian calendar 12 months of 30 days with 5 additional days The priests were aware that the beginning of the year would move through the seasons in 1460-year cycles 1460 = 4 365, since the length of a year is approximately days

41 Calendar Babylonian calendar Months alternate between 29 and 30 days Closer to the actual lunar cycle, which averages at about days 12 months give us 354 days (( ) 6) 7 leap years (with 13 months) occur every 19 years Lengths of the months were adjusted once in a while to ensure there are 6940 days in each 19-year cycle (which contains = 235 months Note 6940/ and 6940/ The current Jewish calendar is similar to the Babylonian calendar, with minor modifications

42 Square Roots SQUARE ROOTS

43 Square Roots The Babylonians had extensive square, square root, cube, and cube root tables Very often problems are set up so that the square root is one of the numbers in the square root table. There are problems, however, where the square root of 2 is needed The square root of 2 is given by 1; 25 =

44 Square Roots How was the value found? Let N = 2, write N = a 2 + b = (a + c) 2 = a 2 + 2ac + c 2 Pick a so that it is very close to N, so c is small, which makes c 2 small relative to 2ac Therefore b 2ac, or c b 2a = N a2 2a. N (a + N a2 2a )2 For N = 2, let a = 1; 20 = 4/3, then a 2 = 1; 46, 40, b = 0; 13, 20, 1/a = 0; 45 Therefore 2 = 1; 40, ; 13, 20 1; 20 + (0; 30)(0; 13, 20)(0; 45) = 1; ; 5 = 1; 25 Note 1; 25 = 17/12 and (17/12) 2 = 289/144 =

45 Pythagoren Theorem PYTHAGOREAN THEOREM

46 Pythagorean Theorem A table with four columns was found on the Babylonian Table Plimpton 322 y(reconstructed) (x/y) 2 x d

47 Pythagorean Theorem y(reconstructed) (x/y) 2 x d

48 Pythagorean Theorem A guess on how to generate Pythagorean Triples x 2 + y 2 = d 2 (x/y) = (d/y) 2 Let u = x/y, v = d/y We have v 2 u 2 = 1, or (v + u)(v u) = 1

49 Pythagorean Theorem Example (Katz) (v + u) = 2; 15, (v u) = 0; 26, 40 Solving for v and u we get v = 1; 20, 50 = /72 and u = 0; 54, 10 = 65/72 Multiply each value by 1, 12 = 72 gives x = 65 and d = 97 Note: The value v + u for every line form a decreasing sequence of regular sexagesimal numbers of no more than four places

50 Quadratic Equations QUADRATIC EQUATIONS

51 Quadratic Equations The Babylonians conisdered the following system { x + y = b xy = c Find length and width given perimeter and area There was no general formula. Problems were presented with concrete numbers In tablet YBC4663 they considered x + y = 6 1 2, xy = First half to get 3 1 4, then square to get From they subtracted to get 3 16, then take the square root to get Length is = 5 and width is = 1 1 2

52 Quadratic Equations In modern notation (assume x > y): ( x+y 2 )2 = xy + ( x y Therefore x y 2 = 2 )2 ( x+y 2 )2 xy = ( b 2 )2 c x = b 2 + x y 2 and y = b 2 x y 2 x = b 2 ( + x+y 2 )2 xy = ( b 2 )2 c and y = b 2 ( x+y 2 )2 xy = ( b 2 )2 c

53 Quadratic Equations

54 Quadratic Equations Another problem considered by the Babylonians { x y = b x 2 + y 2 = c x 2 + y 2 = 2( x+y 2 )2 + 2( x y Therefore x+y 2 = x = x+y 2 + x y 2 = y = x+y 2 x y 2 = c 2 )2 2 ( b 2 )2 c 2 ( b 2 )2 + b 2, c 2 ( b 2 )2 b 2

55 Quadratic Equations

56 Quadratic Equations Another problem considered by the Babylonians (from BM13901) x x = The problem was asked with concrete numbers However, generalizing the method and presenting it in modern notation, we get x = ( b 2 )2 + c b 2 (b is the coefficient of x on the left and c is the constant on the right Most likely obtained by a geometric method: if x 2 + bx = c, then (x + b 2 )2 = ( b 2 ) + c

57 Quadratic Equations

58 Quadratic Equations A solution to x 2 bx = c (b > 0) may be found in a similar way In modern notation, x = ( b 2 )2 + c + b 2 Geometric method: if x 2 bx = c, then (x b 2 )2 = ( b 2 )2 + c Cannot be thought of as the same type of problems as x 2 + bx = c The geometric meaning is different, so the scribes gave a different procedure for finding a solutions We may guess that they did not have abstract algebra

59 Quadratic Equations

60 Quadratic Equations Problems of the form x 2 + c = bx were not considered Problems of the same essence were solved, i.e., the system x + y = b, xy = c However equations of the form x 2 + c = bx did not appear on the tablets Guess: the scribes were not comfortable with equations having more than one solutions, so they set up their problems with two variables instead.

61 References Katz, V. A History of Mathematics: an Introduction. Addison-Wesley, 1998.