Polynomials and Equations

CHAPTER THREE: NOTES ON THE STUDY OF EQUATIONSIN ANCIENT CIVILIZATIONS

CHAPTER THREE NOTES ON THE STUDY OF EQUATIONS IN ANCIENT CIVILIZATIONS Equations are among the topics of mathematics that have been studied extensively for thousands of years. As equations will be the main subject for the rest of the present course, we shall begin here with a brief description of a small selection of results obtained by mathematicians in the antiquity. 3.1 Ancient Egyptian and Babylonian algebra In the nineteenth century archaeologists found very old Egyptian manuscripts at burial sites in the Nile valley. These manuscripts were written in ink on a kind of paper made from the papyrus plants. Among these ancient manuscripts there were books on mathematics. Of these early books on mathematics the most famous is probably the Rhind papyrus now kept in the British Museum. The Rhind papyrus was written some time between 2000 B.C. and 1800 B.C. and contains numerous mathematical problems of the day; they are presented in the form of teacher's questions and pupil's answers. A very large part of this oldest surviving mathematics textbook of the world consists of practical problems of the daily life similar in mathematical content to the present-day primary school arithmetic. But there are also problems that could very well belong to secondary school algebra. These problems do not concern specific concrete objects such as bread and beer, nor are they exercises of operation on known numbers. They are actually problems on equations. The unknown (x in our notation) is usually called aha that means heap. Problem 24 of the Rhind papyrus is an example of the aha calculation. It asks the tJalue of heap if heap and a setJenth of heap is 19. Written in our notation, it is to solve for x in the equation 55 Polynomials and Equations 1 x+ 7x = 19. The Egyptian way of solving this linear equation by the method of false position proceeds as follows. If the value of heap is 7 then heap and a seventh heap is 8. Now 8 multiplied by 2 + i + 1 (this is the Egyptian way of writing the faction ~9) is 19. Therefore the correct value of heap is 7 multiplied by 2 + i +1which is 16 + ! + 1(= 1~3). The solution may look extremely cumbersome today, however if we were only allowed the use of fractions with numerator 1 we would not be able to do better. While there is no material support to think that the Egyptians knew much about algebra beyond linear equations, the ancient Babylonianswere accomplished algebraists. The Egyptian way of writing is very much like our own except that the ink and the paper were different from ours; the Babylonians 'wrote' differently. They 'wrote' on clay. Wedge-shaped marks were impressed with a stylus upon soft clay tablets which were then baked hard in an oven or by the heat of the sun. This type of writing is known as cuneiform because of the shape of individual impressions. Clay tablets survived much better than papyrus manuscripts and thousands were found by archaeologists in the last two hundred years, now preserved in museums . From this material historians are able to study the civilization of Mesopotamia between 1500 B.C. and 1000 B.C.. Many of these tablets were identified as mathematical tables and texts. Besides being able to solve linear equations, the Babylonians were also proficient in coping with quadratic equations and various systems of equations. For example, one of the tablets contains the following problem. To find the side of a square if the area less than the side is the given number 14 X 60 + 30 (this is the way in which numbers are written in the ancient hexagesimal numeral system in which the place values are powers of 60 instead of being powers of 10 as in our decimal system). In modern notation this is a quadratic equation of the form x2 - px = q with positive p and q . For this type of equation, the solution given in the tablet IS x v'(~)2 + q + ~ . 56 The Study of Equations in Ancient Oivilizations The ancient Babylonians also studied the general solution of quadratic equations of the form x2 + bx = q x 2 - q = px where p and q were positive numbers. Naturally the equation x2 + px + q = 0 was omitted because it may have no positive root for some positive p and q and negative numbers were not known...