Lecture 1. The Dawn of Mathematics

Lecture 1. The Dawn of Mathematics The Dawn of Mathematics In ancient times, primitive people settled down in one area by water, built homes, and relied upon agriculture and animal husbandry. At some point, they began to use simple concepts of numbers such as 1, 2, simple counting and simple measuring. As trading, distributing and measuring became more and more complicated, people began to record a variety of things by using script and writing on bones and stones, or using knotted cords. Figure 1.1 One of the earliest fossilized bones and knotted cords. One of the earliest dated occurrences of the representation of numbers is on a fossilized bone discovered at Ishango in Zaire, near the headwaters of the Nile River, and carbon-dated to sometime around 20,000 B.C.E. (see Figure 1.1, left). In the Inca Empire, which arose from the highlands of Peru sometime in early 13th century, knotted cords called quipu were developed as a means to keep records (see Figure 1.1, right). The Incas recorded population, production, or other specified meanings by different types of knots, positions of knots, and colors of cords. Knotted cords are also found in China, West Africa and Australia. Gradually, ancient mathematics began to grow. It is of interest that the word mathematics comes from the Greeks, which means learning, study, and science. 1

Most of these primitive civilizations did not get more advanced mathematics beyond distinguishing among one, two, and many. For example, certain Australian aboriginal tribes counted to two only, with any number larger than two called simply much or many. 1 However, a very few possessed advanced mathematical ideas. Let us discuss mathematics in the four major ancient civilizations as follows. 2 Mesopotamian (Babylonian) Mathematics Based on records, the oldest of the world s civilizations is that of Mesopotamia which emerged in the Tigris and Euphrates river valleys, now the southern part of modern Iraq, around 5000 B.C. This was a rich area where there were ocean coasts, rivers, mountains, plains, deserts, swamps and meadows. Many kingdoms arose in this area over the next 3000 years, including one based in the city of Babylon 3 around 1700 B.C. Hammurabi, the ruler of Babylon, one of the city-states, had expanded his rule to much of Mesopotamia. The term Babylonian covers a series of peoples, who concurrently or successively occupied this area. A professional army had come into existence, and a middle class of merchants and artisans had grown up between peasants and officials. Figure 1.2 Babylonia tablets. 1 cf., The History of Mathematics, David M. Burton, sixth edition, McGral-Hill, 2007, p.1. cf., Victor J. Katz, A History of Mathematics - an introduction, 2nd edition, Addison -Wesley, 1998, p.1- p.3. 3 The remains of Babylon can be found in present-day Al Hillah, Babil Province, Iraq, about 85 kilometers (55 mi) south of Baghdad. 2 2

Clay Tablets As early as 3000 B.C., writing began and was done by means of a stylus on clay tablets. These tablets were inscribed when the clay was still soft and then baked. Hence, the tablets became very hard and well preserved. Our main information about Babylonia, over the course of many years, comes from these clay tablets. They were dated from two periods: some from about 2000 B.C., and a large number from the period 600-300 B.C. The written tradition died out under Greek domination in 100 B.C. and remained totally lost until the 19th century. It was Henry Rawlinson (1810-1895) who was first able to translate some of them. Among all discovered Babylonian clay tablets, about three hundred are on mathematics. There are two kinds: one kind gave mathematical tables such as a table of multiplication, and tables of square roots, etc.; another kind is collections of mathematical problems. Counting The Babylonia number system is a mixture of base 10 and base 60. A number less than 60 was represented by 1 and 10 and the base 10 was used. For a number larger than 60, base 60 was used. [Example] 1. = 23. 2. = 2 60 3 + 23 60 2 + 22 60 + 31. Figure 1.3 Babylonia whole numbers. The Babylonians never had symbol to indicate the absence of a number at the righthand side (as in our 20, 30, 200,...). As a result, their numbers were ambiguous; one had 3

to figure it out by comparing with other content. For example, Another possibility: = 1 602 + 0 60 + 20 = 3620. = 1 60+20 = 80. The Babylonians also used fractions such as 1/2, 1/3 and 2/3. We do not know how the base 60 came to be used. It could have come from weight measures with values involving 1/2, 1/3 and 2/3. It could reflect their understanding on time and angles. Arithmetic To indicate addition, the Babylonians just joined numbers together. To indicate subtraction, they used the symbol. To indicate multiplication, they used a complicated symbol. To divide by an integer a is to multiply by the reciprocal 1/a. Square or cube root calculation relied on the tables. When the root was a whole number, it was given exactly; otherwise, it was given approximately. How did they make such a table? We do not know. From a problem about a rectangle with a diagonal, it seems that the Babylonians used the approximate formula: b a2 + b a + 2a when b a is very small. Figure 1.4 The Babylonians. Algebra The Babylonians had more advanced knowledge about algebra than the Egyptians. They used special terms and symbols to represent unknowns. For example, one problem from a tablet was as follows: 4 4 Morris Kline, Mathematical Thought from Ancient to Modern Times, volume 1, New York Oxford, Oxford University Press, 1972, p.9. 4

I have multiplied length and breadth and the area is 10. I have multiplied the length by itself and have obtained an area. The excess of length over breadth I have multiplied by itself and this result by 9. And this area is area obtained by multiplying the length by itself. What are the length and breadth? By using today s notation, the problem is { xy = 10, 9(x y) 2 = x 2. (x = the length, y = the breadth) The algebraic problems were solved by describing only the steps required to execute the solution, without reasoning. Geometry The Babylonians knew how to find areas for triangles and trapezoids, and volumes for cylinders. The area of a circle was obtained by using 3 for π. From one clay tablet, 15 sets of Pythagorean numbers (see Lecture 4) were discovered by the Babylonians. There was evidence that the Babylonians knew a relation between the side of a square and its diagonal, which is a special case of the Pythagorean Theorem (see Lecture 4) and was known at least 1000 years before Pythagoras. In 330 B.C., Alexander the Great conquered Mesopotamia. The period from 300 B.C. to the birth of Christ is called Seleucid 5. Most of what the Babylonians contributed to mathematics was in the Seleucid period. Egyptian Mathematics Agriculture emerged in the Nile valley in Egypt about 7000 years ago, and the first dynasty to rule Egypt dates from 3100 B.C. Egypt is a gift of the Nile. Once a year the river, drawing its water from the south, floods the territory all along its banks and leaves behind rich soil. Such periodical events forced Egyptians to study astronomy and calendar to determine the dates of flooding. Also it forced Egyptians to measure their land as soon as flooding retreated, which involved geometric and arithmetic knowledge. The pyramid is a symbol of Egyptian culture, and also is a symbol of their mathematical knowledge. Certainly it required high level measuring skills. The earliest known Egyptian 5 The name is after the Greek general who first took control of the region following the death of Alexander in 323 B.C. 5

pyramid is the Pyramid of Djoser which was built about 2500 B.C. Figure 1.5 Rhind papyrus and Moscow papyrus, about 1700 B.C. Papyri Writing began in Egypt at about the same time as in Mesopotamia. The scribes gradually developed the hieroglyphic writing which dots the tombs and temples. Much of our knowledge of the mathematics of ancient Egypt comes not from the hieroglyph but from two papyri 6. The first papyrus was purchased by A.H. Rhind in 1858; it was copied about 1650 B.C. and is approximately 18 feet long and 13 inches high. The second papyrus (i.e., the Moscow papyrus) was purchased by V.S. Golenishchev in 1893; it was dated roughly the same period and is over 15 feet long but 3 inches high. It is very fortunate that the dry Egyptian climate allowed these mathematical papyri to be preserved. These two papyri discussed many concrete mathematical problems with solutions involving multiplication, division, fractions, finding area, etc. The Rhind papyrus contained 85 mathematical problems, and the Moscow papyrus contained 25 mathematical problems. Counting There were two different number systems for the Egyptians: the hieroglyphic system and hieratic system. The hieroglyphic number system was the one written on temple walls or carved on columns. When the scribes wrote on papyri, the hieratic system was used. In the hieroglyphic system, 10 was represented by and 10,000 by. 6, 100 by, and 1000 by, Hieroglyphs was a formal writing system used by the ancient Egyptians. Papyrus is a thick paper-like material produced from the pith of the papyrus plant, a wetland sedge that was once abundant in the Nile Delta of Egypt. 6

Figure 1.6 Egyptians hieroglyphic and hieratic number systems. [Example] To represent 12,443, the Egyptians would write Arithmetic Multiplication and division were reduced to the additive process. For example, to find 12 12, what the Egyptians did was 1 12 2 24 4 48 8 96 so that 12 12 = 48 + 96 = 144. And, to find 19/8, what they did was so that 19/8 = 2 + 1/4 + 1/8. 1 8 2 16 1/2 4 1/4 2 1/8 1 Algebra Basically the Egyptians were able to solve linear equations with one unknown. One of the problems in papyri runs as follows 7 7 Morris Kline, Mathematical Thought from Ancient to Modern Times, volume 1, New York Oxford, Oxford University Press, 1972, p.19. 7

Directions for dividing 700 breads among four people, 2/3 for one, 1/2 for the second, 1/3 for the third, 1/4 for the fourth, i.e., 2 3 x + 1 2 x + 1 3 x + 1 4 x = 700. Geometry The Egyptians could calculate area and volume of many simple geometric objects. They even had a striking rule for the volume of a truncated pyramid of square base: V = h 3 (a2 + ab + b 2 ) where h is the height and a, b are sides of the top and the bottom. The Egyptian civilization went its own way until Alexander the Great conquered Egypt in 332 B.C. As a consequence, until about A.D. 600, its history and mathematics belonged to the Greek civilization. Chinese Mathematics 8 Chinese civilization dates back 5000 years or more. The Xia Dynasty of China was the first dynasty to be described in ancient historical records, from 2100 B.C. to 1600 B.C. It was near the Yellow River. In the Shang dynasty (1600 B.C.-1046 B.C.), it was very popular for people to use oracle bones, which were pieces of bone inscribed with ancient writing. In fact, Oracle bones in China were totally lost and were only discovered near at the end of the 19th century. There were about 150,000 pieces of such bones discovered, which contained about 45,000 distinct words. Up to now only 1/3 of these words are understood. About 57% of these bones were dated around 1212 B.C. The bones are the source of our knowledge of early Chinese number systems. Among these oracle bones, there were 13 words that represented 10, 100, 1000, 10000 so that they can combine to form any number in the range of 100000. The Chinese used the base 10. p.3. 8 cf., Victor J. Katz, A History of Mathematics - an introduction, 2nd edition, Addison -Wesley, 1998, 8

Figure 1.7 Number system on oracle bones, China. Around the beginning of the first millennium B.C., the Shang dynasty was replaced by the Zhou dynasty, which in turn dissolved into numerous warring feudal states. In the sixth century B.C., there occurred a great intellectual flowering, including the famous philosopher Confucius. The feudal period ended as the weaker states were gradually absorbed by the stronger ones, until ultimately China was unified under the Emperor Qin Shi Huang in 221 B.C. Under his leadership, China was transformed into a highly centralized bureaucratic state. He enforced a legal code, levied taxes evenly, and demanded the standardization of weights, measures, money and written script. One difference between the Near East and the Far East on preserving information is in climate. The dry climate and soil in Egypt and Babylonia is much more helpful in preserving materials than in China. The ancient China was a bamboo civilization. The bamboo was used to make books. The opened, dried, and scraped strips of bamboo were laid side by side, in vertical position, and they were joined by crosswise cords. Too often the joining cords rotted and broke so that the slips were lost and damaged. Another kind of material to make books is silk, which was also not easy to be preserved. The great majority of ancient books were lost over time. Indian mathematics 9 India s history began with the Indus Valley Civilization, which spread and flourished in the north western part of the Indian subcontinent from 3300 B.C. to 1300 B.C. Then a civilization called the Harappan arose in India on the banks of the Indus River between 1700-1300 B.C., but there is no direct evidence of its mathematical p. 4. 9 cf., Victor J. Katz, A History of Mathematics - an introduction, Second edition, Addison-Wesley, 1998, 9

production. The earliest Indian civilization for which there is such evidence was formed along the Ganges River by Aryan tribes migrating from the Asian steppes late in second millennium B.C. The literature of the Brahmins was oral for many generations, expressed in lengthy verse called Vedas. Although these verses probably achieved their current form by 600 B.C., there are no written records dating back beyond the current era. Figure 1.8 A Vedic manuscript in ancient Indian. Some of the material from the Vedic era describes the intricate sacrificial system of the priests. It is in these works, the Sulvasutras, that we find mathematical ideas. Sulvasutras is the source for our knowledge of ancient Indian mathematics. Remarks Most elementary mathematics made its first steps very slowly. It took a long time to make even small progress. Mathematics in the above civilizations all came from direct result of physical evidence, trial and error, and insight. On the whole, their arithmetic, algebra and geometry were very elementary, but they were enough to solve practical problems in daily life. The concept of proof, the notion of a logical structure, and the ability to formulate general questions from special ones were not found in the Babylonian, Egyptian, Chinese and Indian civilizations. Mathematics as an organized, independent, and reasoned discipline did not exist before the classical Greeks of the period from 600 to 300 B.C. 10