The Saga of Mathematics A Brief History Babylonians � The Babylonians lived in Mesopotamia, a fertile Mesopotamia Here We Come plain between the Tigris and Euphrates rivers. � Babylonian society replaced both the Sumerian and Akkadian civilizations. � The Sumerians built cities, developed a legal Chapter 2 system, administration, a postal system and irrigation structure. � The Akkadians invaded the area around 2300 BC and mixed with the Sumerians. 1 Lewinter & Widulski The Saga of Mathematics 2 Lewinter & Widulski The Saga of Mathematics Babylonians Babylonians � It was the use of a stylus on a clay medium � The Akkadians invented the abacus, methods for that led to the use of cuneiform symbols since addition, subtraction, multiplication and division. curved lines could not be drawn. � The Sumerians revolted against Akkadian rule � Around 1800 BC, Hammurabi, the King of the and, by 2100 BC, had once more attained control. city of Babylon, came into power over the � They developed an abstract form of writing based entire empire of Sumer and Akkad, founding on cuneiform (i.e. wedge-shaped) symbols. the first Babylonian dynasty. � Their symbols were written on wet clay tablets � While this empire was not always the center of which were baked in the hot sun and many culture associated with this time in history, the thousands of these tablets have survived to this name Babylonian is used for the region of day. Mesopotamia from 2000 BC to 600 BC. 3 4 Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski The Saga of Mathematics Babylonian Cuneiform Babylonian Cuneiform � Because the Latin word for “wedge” is cuneus , � Babylonians used a positional system with the Babylonian writing on clay tablets using a base 60 or the sexagesimal system. wedge-shaped stylus is called cuneiform . � A positional system is based on the notion � Originally, deciphered by a German of place value in which the value of a schoolteacher Georg Friedrich Grotefend symbol depends on the position it occupies (1775-1853) as a drunken wager with friends. in the numerical representation. � Later, re-deciphered by H.C. Rawlinson (1810- � For numbers in the base group (1 to 59), 1895) in 1847. they used a simple grouping system � Over 300 tablets have been found containing mathematics. 5 6 Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski 1
The Saga of Mathematics A Brief History Babylonian Cuneiform Babylonian Numerals Numbers 1 to 59 � We will use for 10 and for 1, so the number 59 is � For numbers larger than 59, a “digit” is moved to the left whose place value increases by a factor of 60. � So 60 would also be . Picture from http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_numerals.html. Lewinter & Widulski The Saga of Mathematics 7 Lewinter & Widulski The Saga of Mathematics 8 Babylonian Numerals Babylonian Numerals � Drawbacks: � Consider the following number � The lack of a sexagesimal point � Ambiguous use of symbols � The absence of zero, until about 300 BC � We will use the notation (3, 25, 4) 60 . when a separate symbol was used to act as a placeholder. � This is equivalent to � These lead to difficulties in determining the value of a number unless the context × 2 + × + = 3 60 25 60 4 12 , 304 gives an indication of what it should be. 9 10 Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski The Saga of Mathematics Babylonian Numerals Babylonian Numerals � To see this imagine that we want to � The Babylonians never achieved an determine the value of absolute positional system. � We will use 0 as a placeholder, commas to separate the “digits” and a semicolon � This could be any of the following: to indicate the fractional part. 2 × 60 + 24 = 144 � For example, (25, 0, 3; 30) 60 will × 2 + × = 2 60 24 60 8640 represent 24 2 + = 2 2 30 1 × 2 + × + + = 60 5 25 60 0 60 3 90 , 003 60 2 11 12 Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski 2
The Saga of Mathematics A Brief History More Examples More Examples � (25, 0; 3, 30) 60 represents � (5; 5, 50, 45) 60 represents 3 30 7 5 50 45 1403 × + + + 2 = 25 60 0 1500 + + + = 5 5 60 60 120 60 60 2 60 3 14400 � (10, 20; 30, 45) 60 represents � Note: Neither the comma (,) nor the semicolon (;) had any counterpart in the 30 45 41 original Babylonian cuneiform. × + + + 2 = 10 60 20 620 60 60 80 Lewinter & Widulski The Saga of Mathematics 13 Lewinter & Widulski The Saga of Mathematics 14 Babylonian Arithmetic Babylonian Arithmetic � Babylonian tablets contain evidence of � For the Babylonians, addition and subtraction are very much as it is for us today except that their highly developed mathematics carrying and borrowing center around 60 not � Some tablets contain squares of the 10. numbers from 1 to 59, cubes up to 32, � Let’s add (10, 30; 50) 60 + (30; 40, 25) 60 square roots, cube roots, sums of 10 , 30 ; 50 , 0 squares and cubes, and reciprocals. + � See Table 1 in The Saga of Mathematics 30; 40, 25 (page 29) 11, 1; 30, 25 15 16 Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski The Saga of Mathematics Babylonian Arithmetic Babylonian Multiplication � Remember to line the numbers up at the � Some tablets list the multiples of a single number, p . sexagesimal point, that is, the semicolon (;) and add zero when necessary. � Because the Mesopotamians used a sexagesimal (base 60) number system, you � Note that since 40 + 50 = 90 which is would expect that a multiplication table would greater than 60, we write 90 in list all the multiples from 1 p , 2 p , ..., up to 59 p . sexagesimal as (1, 30) 60 . � But what they did was to give all the multiples � So we put down 30 and carry the 1. from 1 p up to 20 p , and then go up in multiples of 10, thus finishing the table with 30 p , 40 p � Similarly for the 30 + 30 + 1 (that we and 50 p . carried). 17 18 Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski 3
The Saga of Mathematics A Brief History Babylonian Multiplication Babylonian Multiplication � They would then use the distributive law � Using tablets containing squares, the a × ( b + c ) = a × b + a × c Babylonians could use the formula � If they wanted to know, say, 47 p , they ( ) = + 2 − 2 − 2 ÷ ab [ a b a b ] 2 added 40 p and 7 p . � Sometimes the tables finished by giving � Or, an even better one is the square of the number p as well. � Since they had tablets containing ( ) ( ) 2 2 = + − − ÷ ab [ a b a b ] 4 squares, they could also find products another way. Lewinter & Widulski The Saga of Mathematics 19 Lewinter & Widulski The Saga of Mathematics 20 Babylonian Multiplication Babylonian Multiplication 10 1,40 19 6,1 � Using the table at � Multiplication can 10 ; 50 the right, find 11 × 12. also be done like it is 11 2,1 20 6,40 in our number � Following the × 30; 20 12 2,24 21 7,21 system. formula, we have 13 2,49 22 8,4 11 × 12 = � Remember that 3, 36, 40 14 3,16 23 8,49 (23 2 – 1 2 ) ÷ 4 = carrying centers + 5 , 25, 0 15 3,45 24 9,36 around 60 not 10. (8, 48) 60 ÷ 4 = 16 4,16 25 10,25 � For example, 5 , 28 ; 36 , 40 (2, 12) 60 . 17 4,49 26 11,16 5 1 10 × 30 18 5,24 27 12,9 6 3 21 22 Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski The Saga of Mathematics Babylonian Division Babylonian Division � 44 ÷ 12 = 44 × (1/12) = 44 × (0;5) 60 = � Correctly seen as multiplication by the reciprocal of the divisor. (3;40) 60 . � Note: 5 × 44 = 220 and 220 in base-60 is � For example, 3,40. 2 ÷ 3 = 2 × (1/3) = 2 × (0;20) 60 = (0;40) 60 � 12 ÷ 8 = 12 × (1/8) = 12 × (0;7,30) 60 = � For this purpose they kept a table of (1;30,0) 60 . reciprocals (see Table 1, page 29). � 25 ÷ 9 = 25 × (1/9) = 25 × (0;6,40) 60 = � Babylonians approximated reciprocals (2;46,40) 60 . which led to repeating sexagesimals. 23 24 Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski 4
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