Beginnings of Counting and Numbers Tallies and Tokens Picture Link (http://www.flickr.com/photos/quadrofonic/834667550/) Bone Tallies • The Lebombo Bone is a portion of • The radius bone of a wolf, a baboon fibula, discovered in the discovered in Moravia, Border Cave in the Lebombo Czechoslovakia in 1937, and mountains of Swaziland. It dates dated to 30,000 years ago, has to about 35,000 years ago, and fifty‐five deep notches carved has 29 distinct notches. It is into it. Twenty‐five notches of assumed that it tallied the days of similar length, arranged in‐groups a lunar month. of five, followed by a single notch twice as long which appears to • terminate the series. Then Picture Link starting from the next notch, also (http://www.historyforkids.org/learn/africa/science/numbers.htm) twice as long, a new set of notches runs up to thirty. • Picture link (http://books.google.com/books?id=C0Wcb9c6c18C&pg=PA41&lpg=PA41 &dq=wolf+bone+moravia&source=bl&ots=1z5XhaJchP&sig=q_8WROQ1Gz l4‐6PYJ9uaaNHLhOM&hl=en&ei=J8D‐ TZSgIuPTiALxn4CEBQ&sa=X&oi=book_result&ct=result&resnum=4&ved=0 CCsQ6AEwAw) Ishango Bone • Ishango Bone, discovered in 1961 in central Africa. About 20,000 years old. Ishango Bone Patterns • Prime 11 13 17 19 numbers? • Doubling? 11 21 19 9 • Multiplication? • Who knows? 3 6 4 8 10 5 5 7 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. Lartet Bone Medieval Tally Sticks “Split” Tally Stick 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 (http://preoccupations.tumblr.com/post/5368524205/tally‐stick‐the‐antiquated‐debt‐counter‐ measures) Charles Dickens on Tally Sticks: • “… it took until 1826 to get these sticks abolished. In 1834 … there was a considerable accumulation of them. … [W]hat was to be done with such worn‐out worm‐eaten, rotten old bits of wood? The sticks were housed in Westminster, and it would naturally occur to any intelligent person that nothing could be easier than to allow them to be carried away for firewood by the miserable people who lived in that neighborhood. However [the sticks were no longer] useful and official routine required that they never should be,. Charles Dickens on Tally Sticks: • ... and so the order went out that they should be privately and confidentially burned. It came to pass that they were burned in a stove in the House of Lords. The stove, overgorged with these preposterous sticks, set fire to the paneling; the paneling set fire to the House of Commons; the two houses [of government] were reduced to ashes; architects were called in to build others; and we are now in the second million of the cost thereof.” • Dickens, Charles and Shepherd, Richard Herne; The Speeches of Charles Dickens. With an introduction by Bernard Darwin. Edited and prefaced by R. H. Shepherd.; Publisher: Michael Joseph, London 1937 Token Counting Clay Tokens • 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. Clay Wallet Impressions in Clay Pressing Tokens into Clay Knot Systems Knot Counting Among the Incas • Quipus – knotted strings using place value. Different colored strings represent different objects being counted – yellow for llamas, blue for sheep, etc. • 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.) Example of Quipu Counting 2,154 306 31 2,060 Quipus Inca Quipu Counting Boards and Abaci Yupanas – Incan Counting Boards Still being figured out, but there are some hypotheses. Yupana Example • Stone box with dividers. Lightly shaded areas are raised one level; darker shaded areas raised two levels. Yupana Example • Counters (of different colors or types, maybe) were put in different locations, and their values were multiplied as follows: x 12 x 1 x 1 x 1 x 1 x 6 x 1 x 1 x 2 x 3 x 2 x 1 x 1 x 6 x 1 x 1 x 1 x 1 x 12 Yupana Example • Another hypotheses is based on powers of 10 and Fibonnaci numbers. • In fact, we really don’t know how they were used, or even if (for sure) they were used in counting. • However, “Yupana” is Quechua for “counting tool.” Roman Abacus Chinese Suanpan Japanese Soroban Counting Boards – Basically Abaci • MMDCCXXXVII + MMMDCCCLXXIIII= MMMMMMDCXI Counting Systems: • Body Counting • One‐two‐ … ‐ many • Two‐counting • More complicated counting systems • Five‐, Five‐ten, and Five‐twenty counting Body Counting • 1 little finger • 2ring finger • 3 middle finger • 4fore finger • 5 thumb • 6 hollow between radius and wrist • 7forearm • 8inside 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 Body Counting • Counting in Foe (http://www.youtube.com/watch?v=H13Se4nBPDA) One‐Two‐ … ‐Many • Some systems have only 1, 2, and “many.” – This may be accompanied by a general difficulty with larger numbers. For examples, members of some tribes will trade two sheep for a tin of tobacco twice, but won’t trade four sheep for two tins. • Example from Australia: The Story of 1 (starting at 7:20) (http://video.google.com/videoplay?docid=‐1957179570191443503) Pirahã, Brazil: hoi, hói, baágiso. • In one task, the word hoi was used to describe one object, hoí to describe two or more objects, and baáɡiso to refer to quantities of three or more, consistent with meanings of “one”, “roughly two” and “many.” • In a second task, hói was used to refer to quantities as large as six, hoí was used for quantities between four and ten, and baáɡiso was used for quantities between seven and ten. None of the three words were used consistently to refer to any particular quantity across the two tasks. Thus they are much more likely to be relative or comparative terms like “few” than absolute terms like “one.” Djauan, Australia: jirriyn, jatkorrng, gulpan, malnguy • One‐ two‐ several‐ many‐ – one: root is ‐jirriyn (preceded by appropriate gender prefix masculine, feminine, neuter or zero) – two: jatkorrang – three or several: gulpam – many: malnguyn (or various other expressions meaning `big crowd', etc.) • In some situations, there were ways that the notion of 'hand' was used, for example: “I will stay away for ngan‐barrak‐jirriyn (one hand) of days.” 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 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. Two‐counting • Two‐counting: – Examples from Australia, South America, South Africa, and Papua New Guinea • Examples: • Imonda, PNG: mugasl, sabla, sabla mugõ, sabla sabla, sabla sabla mugõ. • Western Arrernte, Australia: ŋinta, tařa, tařamiŋinta, tařamatařa. • One, two, two‐one, two‐two, two‐two‐one, two‐two‐ two, and so on. 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 (Four‐cycle, with multiplicative idea: not four‐and‐four, but two‐fours) More Complicated Counting Systems • Counting systems based on composite units or 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). Kâte Language from PNG Moc = one, jajahec = two, me‐moc = one hand (five), ngic‐moc = one man (twenty). Thus, the name for 8 means literally “one hand and fingers two‐and‐one” English Equivalent Kâte number Kâte operative pattern for numeral in word each counting number figures words 1moc 1 2 jajahec 2 3Jahec‐a‐moc 3=2+1 4Jahec‐a‐jahec 4=2+2 5Me‐moc 5 6Me‐moc‐a‐moc 5+1 7Me‐moc‐a‐jajahec 5+2 8Me‐moc a jahec‐a‐moc 8=5+(2+1) 13 Me‐jajahec a jahec‐a‐moc 13=10+(2+1) or (5+5)+(2+1) 15 Me‐jajahec a kike‐moc 15=10+5 or 15=5+5+5 20 ngic‐moc 20 (or 20=4x5) 23 ngic‐moc a jahec‐a‐moc 23=20 +(2+1) 26 ngic‐moc‐a‐me‐moc‐a‐moc 20+5+1 Roro Language from PNG English numeral Equivalent Roro number word Roro operative pattern for each in figures counting number word 1hamomo 1 2rua 2 3aihau 3 4bani, 4 5ima 5 6 abaihau 2x3 7 abaihau hamomo 2x3+1 8 ababani 2x4 9 ababani hamomo 2x4+1 10 harau haea ten, one of 11 harauhaea hamomo 1 ten + 1 12 harauhaea rua 1 ten + 2 15 harauhaea ima 1 ten + 5 20 harau rua ten, two of 26 harau rua abaihau 2 tens + 6 30 harau aitau 3 tens 40 harau bani, 4 tens 100 sinabu, hinabu a new word for hundred 200 sinabu rua 2 hundreds Counting in Kuriti/Kurti • Video • (http://www.youtube.com/watch?v=p2Qr4fv2eDY&feature=related) 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).