Babylonian & Egyptian Mathematics

Transcription

1 Babylonian/Egyptian Mathematics from the Association of Teachers of Mathematics Page 1 Babylonian & Egyptian Mathematics The Babylonians lived in Mesopotamia, a fertile plain between the Tigris and Euphrates rivers. They developed an abstract form of writing based on cuneiform (i.e. wedge-shaped) symbols. Their symbols were written on wet clay tablets which were baked in the hot sun and many thousands of these tablets have survived to this day. It was the use of a stylus on a clay medium that led to the use of cuneiform symbols since curved lines could not be drawn. Perhaps the most amazing aspect of the Babylonian's calculating skills was their construction of tables to aid calculation. The Babylonians had an advanced number system, in some ways more advanced than our present system. It was a positional system with base 60 rather than the base 10 of our present system. Now 10 has only two proper divisors, 2 and 5. However 60 has 10 proper divisors so many more numbers have a finite form. The Babylonians divided the day into 24 hours, each hour into 60 minutes, each minute into 60 seconds. This form of counting has survived for 4000 years. To write 5h 25' 30", i.e. 5 hours, 25 minutes, 30 seconds is just to write the base 60 fraction, 5 25/60 30/3600 or as a base 10 fraction 5 4/10 2/100 5/1000 which we write as in decimal notation. Two tablets found at Senkerah on the Euphrates in 1854 date from 2000 BC. They give squares of the numbers up to 59 and cubes of the numbers up to 32. The table gives 82 = 1 4 which stands for 82 = 1 4 = = 64 and so on up to 592 = 58 1 (= = 3481). One major disadvantage of the Babylonian system however was their lack of a zero. This meant that numbers did not have a unique representation but required the context to make clear whether 1 meant 1, 61, 3601, etc. The Babylonians used the formula ab = ((a + b)2 - a2 - b2)/2 to make multiplication easier. Even better is the formula ATM, 7 Shaftesbury St, DERBY DE23 8YB

2 Babylonian/Egyptian Mathematics from the Association of Teachers of Mathematics Page 2 ab = (a + b)2/4 - (a - b)2/4 which shows that a table of squares is all that is necessary to multiply numbers, simply taking the difference of two numbers that were looked up in the table. Division is a harder process. The Babylonians did not have an algorithm for long division. Instead the based their method on the fact that a.b = a.(1/b) so what was necessary was a table of reciprocals. We still have their reciprocal tables going up to the reciprocals of numbers up to several billion. Of course the tables are in their number notation, but translating into our notation, but leaving the base as 60, the beginning of one of their tables would look like Now the table had gaps in it since 1/7, 1/11, 1/13, etc. do not have terminating base 60 fractions. This did not mean that the Babylonians could not compute 1/13, say. They would write 1/13 = 7/91 = 7.(1/91) =(approx) 7.(1/90) and these values were given in the tables. One of the Babylonian tablets (Plimpton 322) which is dated from between 1900 and 1600 BC contains answers to a problem containing Pythagorean triples, i.e. numbers a, b, c with a2 + b2 = c2. It is said to be the oldest number theory document in existence. A translation of another Babylonian tablet which is preserved in the British Museum goes as follows

3 Babylonian/Egyptian Mathematics from the Association of Teachers of Mathematics Page 3 4 is the length and 5 the diagonal. What is the breadth? Its size is not known. 4 times 4 is times 5 is 25. You take 16 from 25 and there remains 9. What times what shall I take in order to get 9? 3 times 3 is 9. 3 is the breadth. The Egyptians and the Romans had number systems which were not well suited for arithmetical calculations. Addition of Roman numerals is not too bad but multiplication is essentially impossible. The Egyptian system had similar drawbacks. The Egyptians were very practical in their approach to mathematics. This image is an example of Egyptian mathematics - it is the Rhind papyrus. The Rhind papyrus is named after the Scottish Egyptologist, Henry Rhind, who purchased it in Luxor in The papyrus, a scroll about 6.000m long and 0.333m wide, was written around 1650BC by the scribe Ahmes who is copying a document which is 200 years older. This makes the original papyrus and the Moscow papyrus both date from about 1850BC. Unlike the Greeks who thought abstractly about mathematical ideas, the Egyptians were only concerned with practical arithmetic. In fact the Egyptians probably did not think of numbers as abstract quantities but always thought of a specific collection of 8 objects when 8 was mentioned. To overcome the deficiencies of their system of numerals the Egyptians devised cunning ways round the fact that their numbers were unsuitable for multiplication as is shown in the Rhind papyrus which date from about 1700 BC. The Rhind papyrus recommends that multiplication be done in the following way. Assume that we want to multiply 41 by 59. Take 59 and add it to itself, then add the answer to itself and continue

4 Babylonian/Egyptian Mathematics from the Association of Teachers of Mathematics Page 4 Since 64 > 41, there is no need to go beyond the 32 entry. Now go through a number of subtractions = 9, 9-8 = 1, 1-1 = 0 to see that 41 = Next check the numbers in the right hand column corresponding to 32, 8, 1 and add them X X X 2419 Notice that the multiplication is achieved with only additions, notice also that this is a very early use of binary arithmetic. Reversing the factors we have X 2 82 X X X X 2419 References A Aaboe, Episodes from the Early History of Mathematics (1964). A E Berriman, The Babylonian quadratic equation, Math. Gaz. 40 (1956), J K Bidwell, A Babylonian geometrical algebra, College Math. J. 17 (1) (1986), E M Bruins, Egyptian arithmetic, Janus 68 (1-3) (1981),

5 Babylonian/Egyptian Mathematics from the Association of Teachers of Mathematics Page 5 A B Chace, L S Bull, H P Manning and R C Archibald, The Rhind Mathematical Papyrus (Oberlin, Ohio, ). J Friberg, Methods and traditions of Babylonian mathematics. Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations, Historia Mathematica 8 (1981), J Friberg, Methods and traditions of Babylonian mathematics. II : An old Babylonian catalogue text with equations for squares and circles, J. Cuneiform Stud. 33 (1) (1981), P Gerdes, Three alternate methods of obtaining the ancient Egyptian formula for the area of a circle, Historia Mathematica 12 (3) (1985), R J Gillings, Mathematics in the Time of the Pharaohs (Cambridge, MA., 1982). R J Gillings, The Recto of the Rhind Mathematical Papyrus and the Egyptian mathematical leather roll, Historia Mathematica 6 (4) (1979), R J Gillings, The Egyptian 2/3 table for fractions : The Rhind mathematical papyrus (B.M ), Austral. J. Sci. 22 (1959), R J Gillings and C L Hamblin, Babylonian sexagesimal reciprocal tables, Austral. J. Sci. 27 (1964), J Hoyrup, Babylonian mathematics, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), D E Knuth, Ancient Babylonian algorithms, Twenty-fifth anniversary of the Association for Computing Machinery. Comm. ACM 15 (7) (1972), M Linton, Babylonian triples, Bull. Inst. Math. Appl. 24 (3-4) (1988), K Muroi, The expressions of zero and of squaring in the Babylonian mathematical text VAT 7537, Historia Sci. (2) 1 (1) (1991), K Muroi, Extraction of cube roots in Babylonian mathematics, Centaurus 31 (3-4) (1988), O Neugebauer and A Sachs, Mathematical Cuneiform Texts (New Haven, CT., 1945). C S Rees, Egyptian fractions, Math. Chronicle 10 (1-2) (1981), C S Roero, Egyptian mathematics, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), A J Sachs, Babylonian mathematical texts. I : Reciprocals of regular sexagesimal numbers, J. Cuneiform Studies 1 (1947),

6 Babylonian/Egyptian Mathematics from the Association of Teachers of Mathematics Page 6 G Sarton, Remarks on the study of Babylonian mathematics, Isis 31 (1940), G J Toomer, Mathematics and Astronomy, in J R Harris (ed.), The Legacy of Egypt (Oxford, 1971), T Viola, On the list of Pythagorean triples ("Plimpton 322") and on a possible use of it in old Babylonian mathematics (Italian), Boll. Storia Sci. Mat. 1 (2) (1981), B L van der Waerden, Science Awakening (Groningen, 1954). B L van der Waerden, Geometry and Algebra in Ancient Civilizations (New York, 1983). B L van der Waerden, On pre-babylonian mathematics. I, Archive for History of Exact Sciences 23 (1) (1980/81), B L van der Waerden, On pre-babylonian mathematics. II, Archive for History of Exact Sciences 23 (1) (1980/81), I Yaglom, Number systems : Mayans, Romans, Babylonians - lend us your calculators, Quantum 5 (6) (1995),

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