Beginnings of Counting and Numbers Tallies and Tokens

Picture Link (http://www.flickr.com/photos/quadrofonic/834667550/) Bone Tallies

• The is a portion of • The radius bone of a wolf, a baboon fibula, discovered in the discovered in , Border 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, 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 , techniques, and stroke directions). Some suggest that the marks are meant to record different observations of the moon. Lartet Bone Medieval Tally Sticks “Split” 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 ) 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 .” 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 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). • 10‐cycles, including some in which 7 is denoted by10 ‐ 3, 8 by 10 ‐2, 9 by 10 ‐ 1; in others, 6 is denoted by 2 x 3, 8 by 2 x 4, 7 by 2 x 3 + 1; • 5‐cycles, typically using groups of 10, 20, and/or 100 as Five‐counting • A “pure” example: Betoya, South America (reported in 1892, Proceedings of the American Philosophical Society): 1. tey. (masc.; teo fem.) 2. cayapa. 3. toazumba. 4. cajezea = 2 with plural termination (i.e, “twos”) 5. teente = hand. 6. teyente tey = hand + 1. 7. teyente cayapa = hand + 2. 8. teyente toazumba = hand + 3. 9. teyente caesea = hand + 4. 10. caya ente, or caya huena = 2 hands. 11. caya ente‐tey = 2 hands + 1. 15. toazumba‐ente = 3 hands. 16. toazumba‐ente‐tey = 3 hands + 1. 20. caesea ente = 4 hands. 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. Five‐ten Counting Example

• Luo of Kenya:

1: achiel …. (5 + N pattern)

2: ariyo 10: apar

3: adek 11: apar‐achiel

4: angwen …. (10 + N pattern)

5: abich 20: piero‐ariyo

6: ab‐chiel …. (20 + Npattern)

7: ab‐ariyo 30: piero‐adek (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‐tupɨ seŋã‐maka‐jo (lit: ''after the thumb'' ) 8. hopoajo (lit: ''middle finger (exists)'' ) 9. hopoajo kɨno‐make‐jo (lit: ''close to middle finger'' ) 10. sia‐hɨ‐ŋa (lit: ''all hands (exist)'' ) 11. siahɨŋate‐ e/i/o 12. siahɨŋa kaja 20. siahɨŋa siahɨŋa 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. Five‐Twenty Counting

• Five‐Twenty counting systems sometimes also have a grouping at 10 superimposed on them. For example, they may have 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. Five‐(Ten)‐Twenty Counting Example: Aztecs 1: ce 9: chic‐naui 30: cem‐poualli‐om‐ matlacti 2: ome 10: matlacti …. 3: yey 11: matlacti‐on‐ce 40: ome‐poualli 4: naui …. …. 5: macuilli 15: caxtulli 50: ome‐poualli‐om matlacti 6: chica‐ce 16: caxtulli‐on‐ce 7: chica‐ome …. 8: chicu‐ey 20: cem‐poualli Five‐(Ten)‐Twenty Counting in Welsh

1 un 16 un ar bymtheg = 1 + 5 + 10. 2 dau 17 dau ar bymtheg = 2 + 5 + 10 3 tri 18 tri ar bymtheg = 3 + 5 + 10. (also sometimes deunaw = 2x9) 4 pedwar 19 pedwar ar bymtheg = 4 + 5 + 10. 5 pump 20 ugain. 6 chwech 30 deg ar hugain 7 saith 40 Deugain 8 wyth 50 Hanner cant 9 naw 60 Trigain (3x20) 10 deg 70 deg a thrigain 11 un ar ddeg = 1 + 10. 80 pedwar ugain 12 deuddeg = 2 + 10. 90 deg a pedwar ugain 13 tri ar ddeg = 3 + 10. 100 Cant 14 pedwar ar ddeg = 4 + 10 200 dau cant 15 pymtheg = 5 + 10 1000 Mil Geography of Counting Systems Counting Words

• Often derived from body parts or other associations, or from acts of counting on body parts. Example: Pumé, Venezuela

• The number four literally means “has a partner.” • The number five means “one‐side hand only.‘’ • The number six means “one‐side hand only, one.” • The number ten literally means “all hands.” • The number sixteen means “all hands, from one‐side foot, one.” The number twenty literally means “all feet.” • The number forty literally means “all feet of two people.” Example: Greenlandic Inuktitut

• Greenlandic Inuktitut has a traditional counting system based on the hands and feet. • 'Six' means something like 'crossing over to the edge of the other hand', then 'seven' is '6‐1', eight '6‐2', etc. • 11 means roughly 'moving down there (to the feet)' • 16 means roughly 'going across to the other edge again' • 20 is 'man finished' Ainu Counting Words

Number Meaning of Ainu word Number Meaning of Ainu word 1 Beginning-to-be 40 2 X 20 4 Much 60 3 X 20 5 Hand 80 4 X 20 6 4 from 10 30 10 from 2 X 20 7 3 from 10 50 10 from 3 X 20 8 Two steps down 70 10 from 4 X 20 9 One step down 90 10 from 5 X 20 10 Two sided (i.e. both hands) 100 5 X 20 20 Whole (man) 110 10 from 6 X 20 Counting Words Derived from Body Parts:

The word for the number... is derived from a phrase meaning...

15 Three fists 10 Two hands 20 Man complete 100 Five men finished 9 Hand and hand less one 2 Raise a separate finger 6 To cross over 6 Take the thumb 9 One in the belly 40 A mattress Inca Counting Words

• Separate words occur for the idea of : – ... the two together that make a pair ... – ... the one together with its mate ... – ... two ‐ in reference to one thing that is divided into two parts ... – ... a pair of two separate things bound intimately together, such as two bulls yoked together for plowing ... Written Numeration Systems Sumerian Cuneiform Value Counters Written Symbols 3500 BC 3200 BC 2650 BC 1

10

60

600

3600

36000 Babylonian Cuneiform Mayan Number System

• Base 20 Place‐value system with a zero!! • Written vertically Mayan Number System

The “Date” on the left is

8.5.16.9.7 Egyptian Number System

Based on powers of 10, but not positional.

• Link

(http://www.clas.ufl.edu/users/ufhatch/HIS‐SCI‐STUDY‐GUIDE/0010_egyptainNumeralsIntValues.html)

• The Story of 1 (at about 12:45 – 14:45)

(http://video.google.com/videoplay?docid=‐1957179570191443503) Egyptian Number System Roman Number System

Symbol Value I1 V5 X10 L50 C 100 D 500 M 1000

A bar can be placed over a symbol to indicate multiplication by 1000: Greek Number System

• Early Attic System

ΙΠΔ Η Χ Μ

1 5 10 50 100 500 1000 5000 10000 (5x10) (5x100)

• Powers of 10, and their halves • 2012 = XXΔΙΙ Greek Number System

• Each unit (1, 2, …, 9) was assigned a separate letter, each tens (10, 20, …, 90) a separate letter, and each hundreds (100, 200, …, 900) a separate letter. This requires 27 letters, so 3 obsolete characters were added. • A ‘ was used after a letter to indicate a numeral, and a , was used before a letter to multiply its value by 1000. Greek Number System

• For even greater • Υνγ’ = 453 numbers, the “myriad” • ,δωοβ = 4,872 symbol M from Attic numeration was used; its value was 10,000 • Mωμθ =10,849 and the number of 10,000’s was put above • , = the M 71,755,875 • Based on powers of 10 • Not Positional Hebrew Number System

• As in Greek, every letter in the alphabet is used to form numbers. • Larger hundreds written as sums of 100 – 400. • Larger numbers written by repetition using larger powers of 10. • Not positional Gematria

• Note that every word in both Hebrew and Greek can be thought of as a number, because letters were used to designate numerals. • This gives rise to translating certain words into numbers. For example, the word amen in Greek is αμην, which interpreted as a number is 1 + 40 + 8 + 50 = 99. In some early Christian manuscripts, the number 99 appears at the end of prayers. Gematria

• Another favorite game in the middle ages: Take an innocent name like St. Claire –or in Greek, στ κλαιρε –and find that it equates to: 200 + 300 + 20 + 30 + 1 + 10 + 100 + 5 = 666

• An interesting hobby, I suppose. • BTW Williams is off‐limits because there’s no equivalent for “w” in Greek so Ha‐Ha! Chinese Number System

• Four basic systems evolved, based on powers of 10. • Not positional. Chinese Stick Numerals Summing Up

• Various written systems were developed, some more advanced than others. • We’ll talk more about the now‐dominant Hindu‐Arabic numeration system later. • We’ll play around with some arithmetic in a few of these systems soon.