BABYLONIANS AND (QUADRATICS In this task, you investigate the cultural contribution of the Babylonians to the solving of quadratic equations. A short paper on some of their methods is available at https: //www.jstor.org/stable/27963851, but you can refer to other papers and/or books too. (1) Identify the period of time when the cultural contribution was made. Remember to reference the paper(s) you use. (2) The Bablyonian tablet YBC6967 describes the problem of finding the width and length of a rectangle with a given area. What type of practical activities in Ancient Babylon would have required this type of problem? (3) The paper describes how the Babylonians found solution to quadratics of the form z? + pz = , > 0. (a) State the formula the Babylonians used to solve quadratics of this form. (b) State the quadratic formula used now, that is, the one that was given in this unit. Express this formula in terms of the variables p and g so it can be easily applied to quadratics of the form z? + px = gq. (c) For each of the following quadratic equations explain why the equation is, or is not, in the form z? + pz =g, > 0? o 22442 =10 e z2+ 52 =-10 (d) Give your own example of a quadratic in the form 2+ pr=gq, q>0. e Identify what p and g are in your example. e Solve this example using the Babylonian formula in (3a). e Solve this example using the quadratic formula in (3b). (e) Compare the solutions found by the two methods. Explain any differences. (f) Given your answer for Question 2, can you suggest why the Babylonians developed their method compared to the current day methods? (4) The Babylonians also solved quadratics of the form az? + bz = . Their method involved a change of variable which gave them a quadratic of the form u? + bu = ac which could be solved using the Babylonian method for solving z* + 1 px = q. This is briefly described in the paper https : //www. jstor.org/stable/ 27963851 (see page 65). Demonstrate how this method can be applied to find a solution to the qua- dratic 3x2 + 6x = 9. (5) Identify the co-ordinates of the points where the line y = 15 intersects with the quadratic y = 3x2 + 6x + 6.Babylonian Quadratics By ROBERT D. McMILLAN, Oklahoma Christian College, Oklahoma City, OK 73111 Ithough most mathematics teachers be- lieve that teaching mathematical history is an important objective, they find it hard to incorporate into teaching al- gebra. One idea that seems to fit in the classroom and that students find interest- ing, however, is the method used by the an- cient Babylonians to solve quadratic equa- tions. They solved quadratic equations 3600 years agobefore the Greeks had a written language, when the Hittites were first in- troducing iron, and when the children of Israel were finding favor with the pharaohs of Egypt. Their problems and solutions appear on clay tablets that have withstood the years in amazingly good condition. It is interesting to see how their methods and solutions are similar to the approaches used today. The ideas presented here can be used after students become familiar with the quadratic formula or possibly as motivation for learning the quadratic formula. We shall limit our study to the front and reverse side of one tablet from the Babylo- nian collection, YBC 6967 at Yale Univer- sity (fig. 1). The middle photograph indi- cates the thickness of the tablet. The front and back include one quadratic equation and its solution comprising about sixteen lines with dimensions of approximately 4.5 em x 7 em x 1 em. This tablet was written sometime between 1900 B.c. and 1600 B.c. The problem is part of a series of tablets that assume the product of the length and width of a rectangle is 60, although the as- sumption is not stated in the beginning. Today it would be written as follows: The length of a rectangle exceeds the width by 7, and the area is 60. What is the length and width? The translation to English and base ten is given in figure 1. The Babylonian solution is completely without algebraic notation and serves as a sample problem to teach students how to solve similar problems. (The use of symbols in algebra as we know it did not come into general use until the middle 1600s.) The standard form of one type of Babylonian quadratic in our algebraic notation is 2 +pr=q,qg>0. Their one positive solution is given by P\\ p 1) x= (2) +g =75 The Babylonian solution using modern no- tation and the modern solution appear in figure 2. As a class project, one or two similar problems can be solved using the Babylo- nian method without algebraic notation, then formula (1), and finally the quadratic formula. The different solutions can then be compared, giving the advantages and disad- vantages of each. For example, the quadra- tic formula is more general. Then students can use the quadratic formula to verify that equation (1) is actually the positive solu- tion of x? 4+ px = 0. Some sample prob- lems are as follows: The length of a rectangle exceeds the width by 10. The area is 600. Find the length and width of the rectangle by Babylonian methods without using alge- braic notation. The length of a rectangle exceeds the width by 2/3. The area is 7/12. Find the length and width of the rectangle by Babylonian methods without using alge- braic notation. The Babylonians also solved quadra- tics of a more general nature, such as 2) ax* + bx =c. In our terminology the technique in- 63 January 1984 Line Front Side The length exceeded the width by 7. 2 What are the length and width? 3-5 As for youhalve 7, by which the length exceeded the width, and the result is 3.5. 6-7 Multiply together 3.5 with 3.5, and the result is 12.25. 8 To 12.25 which resulted for you, 9 add 60, the product, and the result is 72.25. 10 What is the square root of 72.257 8.5. 11 Lay down 8.5 and 8.5 its equal. Top Reverse Side 1-2 Subtract 3.5, the takiltum,* from the one; 3 additto the other. 4 Oneis 12, and the other is 5. 5 12is the length, 5 the width. *This term has several meanings. In this context it means \"a term that is to be added and subtracted from the square root.\" In other tablets it is a value to be subtracted only. Photos are courtesy of the Yale Babylonian Col- lection, Sterling Memorial Library, Yale University. Fig. 1 The Babylonian Solution The Modern Solution Using Modern Notation Let y be the length and x be the width. Let y be the length and x be the width. Then y = x +7, and xy = 60, or Then y = x +7, and xy = 60, x (x + 7) =60. Thus, x2 + 7x - 60 =0. x (x + 7) =60. Thus, x2 + 7x =60. From From the quadratic formula we find equation (1), 5 . -7+49 -4(1)(-60) X : + 60 - X = - 2 2 The length is then obtained by adding 7/2 to the square root rather than adding 7 to Thus, x = 5, and x = -12. The -12 is dis- the result 5. Thus, 12 is the length, and 5 regarded; so x = 5, and y = 5 + 7 = 12. is the width. Hence, 12 is the length, and 5 is the width. Fig. 2 volved a change of variable. First, multi- solving quadratics has been considered ply (2) by a to get important for many years and lends a2x2 + abx = ac. credit to the emphasis we place on solv- ing them today. Substitute u = ax to obtain BIBLIOGRAPHY u2 + bu = ac, Aaboe, Asger. Episodes from the Early History of Math- which is the standard Babylonian form. ematics. New York: Random House, 1964. Gandz, Solomon. Origin and Development of the Qua- After finding u, then divide by a to get x. dratic Equations in Babylonian, Greek, and Early The techniques presented here give Arabic Algebra. Osiris Series 3. Kingsway, London: students practice in solving quadratics as Saint Catherine Press, 1938. well as some of the history of their solu- Neugebauer, O., and A. Sachs. Mathematical Cunei- orm Texts, vol. 29. New Haven, Conn.: American tions. In addition, it teaches them that Oriental Society, 1945