Hong Kong Polytechnic University 2017
Outline Cultures we will cover: Ancient Egypt Ancient Mesopotamia (Babylon) Ancient Greece Ancient India Medieval Islamic World Europe since Renaissance
References These notes follow the following book: Katz, V. A History of Mathematics: an Introduction. Addison-Wesley, 1998.
Introduction Origin of mathematics: Earliest motivation: tax collection, measurement, building, trade, calendar making, ritual practices
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
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
Egypt Rhind Mathematical Papyrus Named for Scotsman A. H. Rhind (1833-1863) 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
Arithmetic ARITHMETIC
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
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.
Hieroglyph Figure: Hieroglyphic numerals. Source:http://www-history.mcs. st-and.ac.uk/histtopics/egyptian_numerals.html
Hieroglyph Figure: Hieroglyphic numerals-examples. Source:http://www-history. mcs.st-and.ac.uk/histtopics/egyptian_numerals.html
Hieratic Figure: Hieratic numerals. Source:http://www-history.mcs.st-and. ac.uk/histtopics/egyptian_numerals.html
Hieratic Figure: Hieratic numerals - Example. Source:http://www-history. mcs.st-and.ac.uk/histtopics/egyptian_numerals.html
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 60 2 + b 60 + c + d 1 60, +e 1 1 + f, where a, b, c,... 60 2 60 3 are numbers 0 and 59.
Babylonian numerals Figure: Babylonian numerals. Source:http://www-history.mcs. st-and.ac.uk/histtopics/babylonian_numerals.html
Babylonian numerals Figure: Babylonian numerals. Source:http://www-history.mcs. st-and.ac.uk/histtopics/babylonian_numerals.html
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
Arithmetic Example: 12 13 x 12 x 1 12 2 24 4 48 8 96 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 = 1 + 4 + 8, therefore 12 13 = 12 + 48 + 96 = 156
Arithmetic Algorithms for the hieratic system no evidence to show how addition was done addition tables probably existed
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 1 2 + 1 10. To use a notation more similar to the hieroglyphic system, we write 2 + 10
Arithmetic Figure: Reciprocals in hieroglyphics. Source:http://www-history.mcs. st-and.ac.uk/histtopics/babylonian_numerals.html
Arithmetic Check: 1 2 10 2 1 5 4 2 3 15 8 4 3 10 30 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 4 3 10 30 and 1 5 gives 6 It is conjectured that addition tables existed
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
Modern proof Example: Consider 17 21 Note 1 2 < 17 21 Note 1 4 < 13 42 < 1 3 < 1. Do 17 21 1 2 = 13 42.. Do 13 42 1 4 = 5 84. Note 1 17 < 5 84 < 1 16. Do 5 84 1 17 = 1 1428. Therefore 17 21 = 1 2 + 1 4 + 1 17 + 1 1428 Note, however, the expression is not unique, since 17 21 is also equal to 1 2 + 1 6 + 1 7, which is easier to work with. Practical concern: cutting a pizza into 7 equal slices is easier than cutting it into 21 equal slices
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
Arithmetic 23,35+40,33 35+33=1,05 23,00+40,00=1,03,00 Therefore 25,35+40,33=1,04,05
Linear Equations and Linear Systems LINEAR EQUATIONS
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
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 2 1 8 = 1 16 nine times to get the largest share 1 + 9 16 (or 1 2 16) Subtract 1 8 from the largest share nine times to get the size of each share
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 900. 2 3 900 1 2 900 = 150 500 150 = 350, and each unit increase in x (with a corresponding unit decrease in y) increases 2 3 x 1 2 y by 2 3 + 1 2 = 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
Elementary Geometry ELEMENTARY GEOMETRY
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 2 81 256 81 = 3.16049... How did they come up with this number?
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.
Rhind 48 Figure: Rhind Papyrus Problem 48. Source:http://www.math.tamu. edu/~don.allen/history/egypt_old/egypt.html
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
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
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, 16 + 8 + 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 ).
Calendar CALENDAR
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 365 1 4 days
Calendar Babylonian calendar Months alternate between 29 and 30 days Closer to the actual lunar cycle, which averages at about 29.5306 days 12 months give us 354 days ((29 + 30) 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 12 19 + 7 = 235 months Note 6940/235 29.5319 and 6940/19 365.26 The current Jewish calendar is similar to the Babylonian calendar, with minor modifications
Square Roots SQUARE ROOTS
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 = 1 + 25 60
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, 40 + 0; 13, 20 1; 20 + (0; 30)(0; 13, 20)(0; 45) = 1; 20 + 0; 5 = 1; 25 Note 1; 25 = 17/12 and (17/12) 2 = 289/144 = 2 + 1 144.
Pythagoren Theorem PYTHAGOREAN THEOREM
Pythagorean Theorem A table with four columns was found on the Babylonian Table Plimpton 322 y(reconstructed) (x/y) 2 x d 120 0.9834028 119 169 1 3456 0.9491586 3367 4825 2 4800 0.9188021 4601 6649 3 13500 0.8862479 12709 18541 4 72 0.8150077 65 97 5 360 0.7851929 319 481 6 2700 0.7199837 2291 3541 7
Pythagorean Theorem y(reconstructed) (x/y) 2 x d 960 0.6845877 799 1249 8 600 0.6426694 481 769 9 6480 0.5861226 4961 8161 10 60 0.5625 45 75 11 2400 0.4894168 1679 2929 12 240 0.4500174 161 289 13 2700 0.4302388 1771 3229 14 90 0.3871605 56 106 15
Pythagorean Theorem A guess on how to generate Pythagorean Triples x 2 + y 2 = d 2 (x/y) 2 + 1 = (d/y) 2 Let u = x/y, v = d/y We have v 2 u 2 = 1, or (v + u)(v u) = 1
Pythagorean Theorem Example (Katz) (v + u) = 2; 15, (v u) = 0; 26, 40 Solving for v and u we get v = 1; 20, 50 = 1 + 25/72 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
Quadratic Equations QUADRATIC EQUATIONS
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 = 7 1 2 First half 6 1 2 to get 3 1 4, then square 3 1 9 4 to get 10 16. From 10 9 16 they subtracted 7 1 1 2 to get 3 16, then take the square root to get 1 3 4 Length is 3 1 4 + 1 3 4 = 5 and width is 3 1 4 1 3 4 = 1 1 2
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
Quadratic Equations
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
Quadratic Equations
Quadratic Equations Another problem considered by the Babylonians (from BM13901) x 2 + 4 3 x = 11 12 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
Quadratic Equations
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
Quadratic Equations
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.
References Katz, V. A History of Mathematics: an Introduction. Addison-Wesley, 1998.