M100 M1 Early Number Systems and Symbols.pptx

Mathematics

1. Early Number Systems and Symbols History of Mathematics Miss Louella P. Mamuyac

2. C. J. K E Y S E R To think the thinkable—that is the mathematician’s aim

3. Primitive Counting The root of the term mathematics is in the Greek word mathemata , which was used quite generally in early writings to indicate any subject of instruction or study.

4. Primitive Counting The Pythagoreans are said to have used it to describe arithmetic and geometry; previously, each of these subjects had been called by its separate name, with no designation common to both.

5. Primitive Counting Three or four thousand years ago, in ancient Egypt and Babylonia, there already existed a significant body of knowledge that we should describe as mathematics.

6. Alexander Pope “This mighty maze is not without a plan.”

7. Primitive Counting Certain Australian aboriginal tribes, for instance, counted to two only, with any number larger than two called simply “much” or “many.” South American Indians along the tributaries of the Amazon were equally destitute of number words.

8. Primitive Counting Although they ventured further than the aborigines in being able to count to six, they had no independent number names for groups of three, four, five, or six. In their counting vocabulary, three was called “ two-one ,” four was “ two-two ,” and so on

9. Primitive Counting A similar system has been reported for the Bushmen of South Africa, who counted to ten (10=2+2+2+2+2) with just two words; beyond ten, the descriptive phrases became too long

10. Primitive Counting The earliest and most immediate technique for visibly expressing the idea of number is tallying.

11. Primitive Counting The idea in tallying is to match the collection to be counted with some easily employed set of objects—in the case of our early forebears, these were fingers, shells, or stones. Sheep, for instance, could be counted by driving them one by one through a narrow passage while dropping a pebble for each.

12. Primitive Counting The term tally comes from the French verb tailler , “to cut,” like the English word tailor ; the root is seen in the Latin taliare , meaning “to cut.” It is also interesting to note that the English word write can be traced to the Anglo-Saxon writan , “to scratch,” or “to notch.”

13. Primitive Counting Counts were maintained by making scratches on stones, by cutting notches in wooden sticks or pieces of bone, or by tying knots in strings of different colors or lengths.

14. Notches as Tally Marks Bone artifacts bearing incised markings seem to indicate that the people of the Old Stone Age had devised a system of tallying by groups as early as 30,000 B.C.

15. Notches as Tally Marks The most impressive example is a shinbone from a young wolf, found in Czechoslovakia in 1937; about 7 inches long, the bone is engraved with 55 deeply cut notches, more or less equal in length, arranged in groups of five.

16. Notches as Tally Marks Another arresting example of an incised bone was unearthed at Ishango along the shores of Lake Edward, one of the headwater sources of the Nile. The best archeological and geological evidence dates the site to 17,500 B.C., or some 12,000 years before the first settled agrarian communities appeared in the Nile valley.

17. Notches as Tally Marks This fossil fragment was probably the handle of a tool used for engraving, or tattooing, or even writing in some way.

18. Notches as Tally Marks The English language has taken note of the peculiar quality of the double tally stick. Formerly, if someone lent money to the Bank of England, the amount was cut on a tally stick, which was then split. The piece retained by the bank was known as the foil , whereas the other half, known as the stock , was given the lender as a receipt for the sum of money paid in.

19. Notches as Tally Marks Another arresting example of an incised bone was unearthed at Ishango along the shores of Lake Edward, one of the headwater sources of the Nile. The best archeological and geological evidence dates the site to 17,500 B.C., or some 12,000 years before the rst settled agrarian communities appeared in the Nile valley.

20. Notches as Tally Marks Another arresting example of an incised bone was unearthed at Ishango along the shores of Lake Edward, one of the headwater sources of the Nile. The best archeological and geological evidence dates the site to 17,500 B.C., or some 12,000 years before the rst settled agrarian communities appeared in the Nile valley.

21. Notches as Tally Marks Three views of a Paleolithic wolfbone used for tallying. (The Illustrated London News Picture Library.)

22. The Peruvian Quipus: Knots as Numbers In the New World, the number string is best illustrated by the knotted cords, called quipus , of the Incas of Peru.

23. The Peruvian Quipus: Knots as Numbers The Incas became renowned for their engineering skills, constructing stone temples and public buildings of a great size. A striking accomplishment was their creation of a vast network (as much as 14,000 miles) of roads and bridges linking the far- flung parts of the empire.

24. The Peruvian Quipus: Knots as Numbers When the Spanish conquerors arrived in the sixteenth century, they observed that each city in Peru had an “official of the knots,” who maintained complex accounts by means of knots and loops in strands of various colors.

25. The Peruvian Quipus: Knots as Numbers

26. The Peruvian Quipus: Knots as Numbers

27. The Peruvian Quipus: Knots as Numbers

28. The Peruvian Quipus: Knots as Numbers

29. The Peruvian Quipus: Knots as Numbers Recalling that ascending positions carry place value for successive powers of ten, let us suppose that a particular cord contains the following, in order: a long knot with four twists, two single knots, an empty space, seven clustered single knots, and one single knot. For the Inca, this array would represent the number

30. The Peruvian Quipus: Knots as Numbers The Mayan calendar year was composed of 365 days divided into 18 months of 20 days each, with a residual period of 5 days. This led to the adoption of a counting system based on 20 (a vigesimal system).

31. The Peruvian Quipus: Knots as Numbers

32. The Peruvian Quipus: Knots as Numbers Thirteenth-century British Exchequer tallies. (By courtesy of the Society of Antiquaries of London.)

33. The Peruvian Quipus: Knots as Numbers

34. Number Recording of the Egyptians and Greeks The History of Herodotus The writing of history, as we understand it, is a Greek invention; and foremost among the early Greek historians was Herodotus . Herodotus (circa 485–430 B.C.) was born at Halicarnassus, a largely Greek settlement on the southwest coast of Asia Minor.

35. Number Recording of the Egyptians and Greeks Herodotus became a citizen of Thurium in southern Italy, a new colony planted under Athenian auspices. In Thurium , he seems to have passed the last years of his life involved almost entirely in finishing the History of Herodotus, a book larger than any Greek prose work before it

36. Hieroglyphic Representation of Numbers As soon as the unification of Egypt under a single leader became an accomplished fact, a powerful and extensive administrative system began to evolve. The census had to be taken, taxes imposed, an army maintained, and so forth, all of which required reckoning with relatively large numbers.

37. Hieroglyphic Representation of Numbers

38. Hieroglyphic Representation of Numbers Special pictographs were used for each new power of 10 up to 10,000,000: 100 by a curved rope, 1000 by a lotus ower , 10,000 by an upright bent nger , 100,000 by a tadpole, 1,000,000 by a person holding up two hands as if in great astonishment, and 10,000,000 by a symbol sometimes conjectured to be a rising sun.

39. Egyptian Hieratic Numeration In this so-called “hieratic” (sacred) script, the symbols were written in a cursive, or free-running, hand so that at first sight their forms bore little resemblance to the old hieroglyphs.

40. The Greek Alphabetic Numeral System Around the fth century B.C., the Greeks of Ionia also developed a ciphered numeral system, but with a more extensive set of symbols to be memorized. They ciphered their numbers by means of the 24 letters of the ordinary Greek alphabet, augmented by three obsolete Phoenician letters

41. The Greek Alphabetic Numeral System

42. The Greek Alphabetic Numeral System Tens of thousands were indicated by using a new letter M, from the word myriad (meaning “ten thousand”). The letter M placed either next to or below the symbols for a number from 1 to 9999 caused the number to be multiplied by 10,000, as with

43. The Greek Alphabetic Numeral System Another number replacement that occurs in early theological writings concerns the word amen, which is _____in Greek. These letters have the numerical values totaling 99. Thus, in many old editions of the Bible, the number 99 appears at the end of a prayer as a substitute for amen.

44. Hieroglyphic Representation of Numbers Express each of the given numbers in Egyptian hieroglyphics. (a) 1492. (b) 1999. (c) 12,321. (d) 70,807. (e) 123,456.

45. Babylonian Cuneiform Script Shortly after 3000 B.C., the Babylonians developed a system of writing from “pictographs”—a kind of picture writing much like hieroglyphics. But the materials chosen for writing imposed special limitations of their own, which soon robbed the pictographs of any resemblance to the objects they stood for.

46. Babylonian Cuneiform Script Whereas the Egyptians used pen and ink to keep their records, the Babylonians used first a reed and later a stylus with a triangular end. With this they made impressions (rather than scratches) in moist clay.

47. Babylonian Cuneiform Script The sharp edge of a stylus made a vertical stroke (|) and the base made a more or less deep impression ( ), so that the combined effect was a head-and-tail figure resembling a wedge, or nail ( ). Because the Latin word for “wedge” is cuneus , the resulting style of writing has become known as “ cuneiform .”

48. Babylonian Cuneiform Script

49. Writing in Ancient China “In China,” wrote the Italian Jesuit Matteo Ricci (died 1610), “it is forbidden under pain of death to study mathematics, without the Emperor’s authorization.”

50. Writing in Ancient China Chinese bamboo or counting-rod numerals, which may go back to 1000 B.C., originated from bamboo sticks laid out on at boards. The system is essentially positional, based on a 10-scale, with blanks where we should put zeros. There are two sets of symbols for the digits 1, 2, 3,…,9, which are used in alternate positions. The first set is used for units, hundreds, ten thousands:

51. Writing in Ancient China 5*10,000 + 2*1000 + 100 + 7*10 + 4 =52174

52. Writing in Ancient China Translate each of these numerals from the Chinese system to our numerals.

53. Quiz Time! Access the quiz link posted in the google classroom.

54. Reference Burton, D.M.(2011). The History of Mathematics: An Introduction. N.Y.: McGraw-Hill.

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