CLAY TABLETS. or better that a b = (a+b)2 (a b) 2

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1 CLAY TABLETS CHIARA GUI Maye one of the main reason why Baylonians are now widely known is ecause of their clay talets. Sumerians and Baylonians were people who lived in Mesopotamia etween 3000 B.C and 1000 B.C. They used to write and record all computations on dump clay talets with a stick, they then let them dry in the hot sun or they acked them in oven. So till nowadays we can deal with a lot of talets, as we could look at an ancient lirary. They used the so called cuneiform writing (wedge-shaped writing) and a positional sexagesimal (ase 60) numer system. Many of the mathematical talets concern practical topics related to management of the reign or palaces. Some are related to commerce and other report geometrical prolems related to irrigation systems or architectural prolems; nally some are recorded y archeologists as simple writing exercises of scries. In this assay will e presented only few examples, enought to let e understand aylonian mathematical skills. First need to e specied that, dierently from the modern (greek) way of studying mathematics, racking one's rain over theorems and astract prolems, Baylonians used mathematics only to deal with concrete prolems. Their tools to nd answers to prolems, instead of theorems, were algorithms with the aid of tales which let them repeat quickly standard computations. As J J O'Connor and E F Roertson say Perhaps the most amazing aspect of the Baylonian's calculating skills was their construction of tales to aid calculation. [JOC-EFR, 000] The most easy example of that, is the use of tales to compute the product of two numers. They used the property that a = (a+) a or etter that a = (a+) (a ) 4, so instead of our classic column calculation, they only needed to e ale to comine results from tales of squares. The most complete talets with squares tale had een founded at Senkerah on the Euphratesare, they are from around 000 BC. The two talets contains the squares of all the numers from 1 to 59 and the cues from 1 to 3. Aout division we know that Baylonians even did not consider it as an operation, they just read a as the product of a and 1, so, all what they needed, was reciprocals tales which listed the inverse of integer numers 1. Reciprocal tales, even more than the square tales, were the most important to e own y a Baylonian scrie. It has een found reciprocal tales going up to the reciprocals of numers up to several illion. In Figure 0.1 there is an example of such tale. Very useful for computations were also the single multiplication tales, which list the multiples of a single numer. If we call that numer p, the tale gave all the multiples from 1p up to 0p, and then it went up in steps of 10, writing the 1 Numer as 1 7 or 1 are missing in this tale since, as Baylonian scries used to write 13 in computation, 7 does not divide. Instead of periodical or illimitate fraction they just used approximations of the required numer. 1

2 CLAY TABLETS (a) () Figure 0.1. Talet CBS 8536, in the museum of the university of Pennsylvania, from the ook [Cajori,198]

3 CLAY TABLETS 3 products 30p, 40p and 50p. So if one needed to know, for example, 38p, he added 30p with 8p. Sometimes single multiplication tales nished y giving also p.

4 CLAY TABLETS 4 Figure 0.. Talet MS 3048 on clay, Baylonia, 19th c. BC, 1 talet, 7,6x4,4x,3 cm [Moyer] Beside single computations, tales were used also to solve prolems and equations. For example it has e found talets giving a list of the sums n 3 + n. This list permited to solve several equations of the form ax 3 + x = c doing the following steps: multiply the equation y a and y 1 to get ( ax 3 )3 + ( ax ) = ca ; which, 3 calling y = ax, is an equation of the form y3 + y = ca 3 look up in the n 3 + n tale, to nd the value which satises n 3 + n = ca ; 3 if such numer is found, then the solution x could e easily computed multiplying that numer y y a. Cuic prolems were generally related to volumes. Another study of cuic prolems can e seen in the talet in Figure 0.. Its description reports: Every line of this talet says, "m has the root n". The numers n at right take the values from 1 to30. The numers m at left take the corresponding values n (n+1) (n+) = n 3 +3n +n. In the 6 th line, for instance, n = 6 and m = = 336 = Here the prolems would have een of the form "An excavated room. Its length equals its width plus 1 cuit. Its height equals its length. Its volume plus its ottom area is... (a given numer)."[moyer] For clarity in the following explanations we will use moden notation and terminology, althought formally is not correct. In fact, in Baylonian times, there was nothing similar to our modern notation, which uses algeraic symolic representation of questions. This means that Baylonians did not deal with equations. Nevertheless Bailonians used to study typical methods, useful to solve typical prolems, which is 'de facto' the aim of modern equations.

5 CLAY TABLETS 5 Surely Baylonians could also solve all linear equation as ax = looking at reciprocals and multiples tales. Aout quadratic equations we know that they considered separately two types of equation, namely x + x = c and x x = c where here, c have to e thought positive ut not necessarily integers. To solve ) + c ( them they asically used the standard formula for the solutions x = ( and x = ) + c+ respectively. Notice that they would take only the positives root from the two roots of quadratic equations, since it is the only which makes sense while tolking aout areas and lenght in uildings. Eectively quadratic prolems were usually related to computations with areas. On an old aylonian talet, we can, for example, read the attempt to solve the equation x + 7x = 1 using the formula x = ( ) + c. The aim of this talet was to explain how to nd the dimensions of a rectangle knowing that its length exceeds its readth y 7 and its area is 1, 0. The answer is written as follows: Compute half of 7, namely 3; 30, square it to get 1; 15. To this the scrie adds 1, 0 to get 1; 1, 15. Take its square root (from a tale of squares) to get 8; 30. From this sutract 3; 30 to give the answer 5 for the readth of the triangle.[joc-efr, 000] As nal example of the use of taes we report the prolem from the Tell Dhiayi talet and its method to nd a solution. This talet is signicative ecause is one of the examples that let us elieve that Baylonians eside squares and roots knew also the Pythagorean rule. The tale asks for the sides of a rectangle whose diagonal is 1, 15 and whose area is 0, 45. To make the explanation more understandale let us continue to use modern notation calling x and y the rectangle's dimensions. The scrie now reports the following steps for the computation: Compute the product xy = 1; 30. Sutract it fromx + y = 1; 33, 45 to getx + y xy = 0; 3, 45. Take the square root to otain x y = 0; 15. Divide y to get (x y) = 0; 7, 30. Divide x + y xy = 0; 3, 45 y 4 to get x 4 + y 4 xy Add xy = 0; 45 to get x 4 + y 4 + xy = 0; 45, 56, 15. Take the square root to otain (x+y) = 0; 5, 30. Add (x+y) = 0; 5, 30 to (x y) = 0; 7, 30 to get x = 1. Sutract (x y) = 0; 7, 30 from (x+y) = 0; 0, 56, 15. = 0; 5, 30 to get y = 0; 45. Hence the rectangle has sidesx = 1 and y = 0; 45.[JOC-EFR, 000] Let conclude this short excursus aout clay talets and its use quoting again J J O'Connor and E F Roertson Rememer that this is 3750 years old. We should e grateful to the Semitic scrie for recording this little masterpiece on talets of clay for us to appreciate today.[joc-efr, 000] References [JOC-EFR, 000] J J O'Connor and E F Roertson, 000, [JOC-EFR, 000] J J O'Connor and E F Roertson, 000, [Cajori,198] Florian Cajori, A history of mathematical notations,198, open court Pu. Co., Chicago [Moyer] Ernest Moyer,

6 CLAY TABLETS 6 [Melville,001] Duncan J. Melville, 001,