Beginnings of Counting and Numbers

1. Beginnings of Counting and Numbers

2. Tallies and Tokens

3. Bone Tallies • The Lebombo Bone is a portion of a baboon fibula, discovered in the Border Cave in the Lebombo mountains of Swaziland. It dates to about 35,000 years ago, and has 29 distinct notches. It is assumed that it tallied the days of a lunar month. • Picture Link • The radius bone of a wolf, discovered in Moravia, Czechoslovakia in 1937, and dated to 30,000 years ago, has fifty-five deep notches carved into it. Twenty-five notches of similar length, arranged in-groups of five, followed by a single notch twice as long which appears to terminate the series. Then starting from the next notch, also twice as long, a new set of notches runs up to thirty. • Picture link

4. Ishango Bone • Ishango Bone, discovered in 1961 in central Africa. About 20,000 years old.

5. Ishango Bone Patterns • Prime numbers? • Doubling? • Multiplication? • Who knows? 11 13 17 19 11 21 19 9 3 6 4 8 10 5 5 7

6. Lartet Bone • Discovered in Dodogne, France. About 30,000 years old. It has various markings that are neither decorative nor random (different sets are made with different tools, techniques, and stroke directions). Some suggest that the marks are meant to record different observations of the moon.

7. Lartet Bone

8. Medieval Tally Sticks

9. “Split” Tally Stick

10. Split Tally Sticks from England • Tally Sticks were used until comparatively modern times. • Stopped use in 1724, but remained legally valid. • England abolished the use of tally sticks in 1826, and most were burned in 1834, setting Parliament (the Palace of Westminster) on fire. • Picture Link

11. Token Counting • Around 10 to 11 thousand years ago, the people of Mesopotamia used clay tokens to represent amounts of grain, oil, etc. for trade. These tokens were pressed into the surface of a clay “wallet” then sealed inside as a record of a successful trade contract. These impressions in clay eventually became stylized pictographs, and later, symbols representing numerosities.

12. Clay Tokens

13. Clay Wallet

14. Impressions in Clay

15. Pressing Tokens into Clay

16. Knot Systems

17. Knot Counting Among the Incas • Quipus – knotted strings using place value. • Three kinds of knots: • Figure 8 knots were units – ones. • Long slip knots represented 2 – 9 depending on number of loops • Single knots represented 10’s, 100’s, 1000’s. (Sometimes long slip knots were also used for 10’s and 100’s.)

18. Example of Quipu Counting 2,154 306 31 2,060

19. Quipus

20. Inca Quipu

21. Counting Boards and Abaci

22. Yupanas – Incan Counting Boards Still being figured out, but there are some hypotheses.

23. Yupana Example • Stone box with dividers. Lightly shaded areas are raised one level; darker shaded areas raised two levels.

24. Yupana Example • Counters (of different colors or types, maybe) were put in different locations, and their values were multiplied as follows:

25. Yupana Example • Another hypotheses is based on powers of 10 and Fibonnaci numbers. • Picture link

26. Roman Abacus

27. Chinese Suanpan

28. Japanese Soroban

29. Counting Boards – Basically Abaci MMDCCXXXVII + MMMDCCCLXXIIII= MMMMMMDCXI

30. Counting Systems: • Body Counting • One-two- … - many • Two-counting • More complicated counting systems • Five-, Five-ten, and Five-twenty counting

31. Body Counting • 1        little finger •                                2        ring finger •                                3        middle finger •                                4        fore finger •                                5        thumb •                                6        hollow between radius and wrist •                                7        forearm •                                8        inside of elbow joint •                                9        upper arm •                              10        point of shoulder •                              11        side of neck •                              12        ear •                              13        point on the head above the ear •                              14        muscle above the temple •                              15        crown of the head

32. Body Counting • Counting in Foe (http://www.youtube.com/watch?v=H13Se4nBPDA)

33. One-Two- … -Many • Some systems have only 1, 2, and “many.” • Will trade two sheep for a tin of tobacco twice, but not all at the same time. • Examples: • Pirahã, Brazil: hoi, hói, baágiso • Djauan, Australia: jirriyn, jatkorrng, gulpan, malnguyn

34. Grouping and Cycles • Counting systems can sometimes be best described in terms of the cycles (rather than the base) that they use. For example, the counting system might feature a 2-cycle (as with two-counting) with six objects being thought of as three groups of two. Many systems have a second cycle combining number words. The second cycles are commonly cycles of five so that, for example, the number 14 might be two fives and two twos. Other common cycles involve twenty and ten.

35. Two-counting • Two-counting: • Examples from Australia, South America, South Africa, and Papua New Guinea • Examples: • Imonda, PNG: mugasl, sabla, sablamugõ, sablasabla, sablasablamugõ. . . . • Western Arrernte, Australia: ŋinta, tařa, tařamiŋinta, tařamatařa. • One, two, two-one, two-two, two-two-one, two-two-two, and so on.

36. Other Simple Counting Systems • Aboriginal Australian (Gamilaraay): one (mal) two-two (bularr-bularr) two (bularr) two-three (bularr-guliba) three (guliba) three-three (guliba-guliba) • Toba tribe of Paraguay: one two-three two-fours-and-one two two-threes two-and-two-fours three one-(&)-two-threes four two-fours

37. More Complicated Counting Systems • Counting systems based on composite units/cycles of 5 and 20 are common. In Papua New Guinea, for example, the 800 different language groups have their own counting systems with a variety of basic number words. Commonly used number words are hand as 5, and person (10 fingers and 10 toes) as 20. A few groups have a hand as 4 (without the thumb) or as 6 (with the thumb as two knuckles).

38. Kâte Language from PNG Moc = one, jajahec = two, me-moc = one hand (five), ngic-moc = one man (twenty) So the name for 8 means literally “one hand and fingers two-and-one”

39. Roro Language from PNG

40. Other systems of counting in Oceana & Papua New Guinea • A few 3-, 4-, and 6- cycles with various other groupings (probably explained by how the thumb is treated). • 10-cycles, including some in which 7 is denoted by10-3, 8 by10-2, 9 by 10-1; in others, 6 is denoted by 2X3, 8 by 2X4, 7 by 2X3+1; • 5-cycles, typically using groups of 10, 20, and/or 100 as well

41. Five-counting • A Pure Example: Betoya, South America: 1. tey. (masc.; teo fem.)2. cayapa.3. toazumba.4. cajezea = 2 with plural termination (i.e, “twos”)5. teente = hand.6. teyentetey = hand + 1.7. teyentecayapa = hand + 2.8. teyentetoazumba = hand + 3.9. teyentecaesea = hand + 4.10. cayaente, or cayahuena = 2 hands.11. cayaente-tey = 2 hands + 1.15. toazumba-ente = 3 hands.16. toazumba-ente-tey = 3 hands + 1.20. caeseaente = 4 hands.

42. Five-Ten Counting • The Pure Structure: • Different number words up to five, then: • Five • Ten • Ten-and-five • Two-tens • Two-tens-and-five • Three-tens • Three-tens and five • Etc.

43. Five-ten Counting Example • Luo of Kenya:

44. (Five)-ten Counting Example • Secoya, Ecuador and Peru 1. tee, tei, teo (inanimate, masculine, feminine ) 2. kaja 3. toaso 4. kahese -e/i/o, ( inanimate, masculine, feminine ) 5. te-hɨtɨ ( lit ''a hand of X exists'' ) 6. ɨha-tupɨ (lit: ''thumb [from the other hand] (exists)'' ) 7. ɨha-tupɨseŋã-maka-jo (lit: ''after the thumb'' ) 8. hopoajo (lit: ''middle finger (exists)'' ) 9. hopoajokɨno-make-jo (lit: ''close to middle finger'' ) 10. sia-hɨ-ŋa (lit: ''all hands (exist'' ) 11. siahɨŋate- e/i/o 12. siahɨŋakaja 20. siahɨŋa siahɨŋa

45. Five-Twenty Counting • The Pure Structure: • Different counting words up to five, then: • Five • Two-fives • Three-fives • Twenty • Twenty-and-five • Twenty-and-two-fives • Twenty-and-three-fives • Two-twenties • Two-twenties-and-five • Etc.

46. Five-Twenty Counting Example: Aztecs

47. Five-Twenty Counting in Welsh

48. Five-Ten-Twenty Counting • Different Numbers words for 1-5, then: • Five • Ten • Ten-and-five • Twenty • Twenty-and-five • Twenty-and-ten • Twenty-and-ten-and-five • Two-twenties • Two-twenties-and-five • Two-twenties-and-ten • Etc.

49. Summary of Counting Systems

50. Counting Words • Often derived from body parts or other associations.