Beginnings of Counting and Numbers

Tallies and Tokens

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

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

Ishango Bone Patterns Prime numbers? Doubling? Multiplication? Who knows?

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

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.

Clay Tokens

Clay Wallet

Impressions in Clay

Pressing Tokens into Clay

Knot Systems

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.)

Example of Quipu Counting 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:

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

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 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

Body Counting Counting in Foe (

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

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.

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-threetwo-fours-and-one two two-threestwo-and-two-fours three one-(&)-two-threes four two-fours

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).

Kâte Language from PNG EnglishEquivalent Kâte numberKâte operative pattern for numeral inwordeach counting number figures words 1moc1 2jajahec2 3Jahec-a-moc3=2+1 4Jahec-a-jahec4=2+2 5Me-moc5 6Me-moc-a-moc5+1 7Me-moc-a-jajahec5+2 8Me-moc a jahec-a-moc8=5+(2+1) 13Me-jajahec a jahec-a-moc13=10+(2+1) or (5+5)+(2+1) 15Me-jajahec a kike-moc15=10+5 or 15= ngic-moc20 (or 20=4x5) 23ngic-moc a jahec-a-moc23=20 +(2+1) 26ngic-moc-a-me-moc-a-moc 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”

Roro Language from PNG English numeral in figures Equivalent Roro number wordRoro operative pattern for each counting number word 1hamomo1 2rua2 3aihau3 4bani,4 5ima5 6abaihau2x3 7abaihau hamomo2x3+1 8ababani2x4 9ababani hamomo2x4+1 10harau haeaten, one of 11harauhaea hamomo1 ten harauhaea rua1 ten harauhaea ima1 ten harau ruaten, two of 26harau rua abaihau2 tens harau aitau3 tens 40harau bani,4 tens 100sinabu, hinabua new word for hundred 200sinabu rua2 hundreds

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

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. teyente tey = hand teyente cayapa = hand teyente toazumba = hand teyente caesea = hand caya ente, or caya huena = 2 hands. 11. caya ente-tey = 2 hands toazumba-ente = 3 hands. 16. toazumba-ente-tey = 3 hands 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: ariyo10: apar 3: adek11: apar-achiel 4: angwen…. (10 + N pattern) 5: abich20: piero-ariyo 6: ab-chiel…. (20 + N pattern) 7: ab-ariyo30: 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. siahɨŋa te- 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 Example: Aztecs 1: ce9: chic-naui30: cem-poualli-om-matlacti 2: ome10: matlacti…. 3: yey11: matlacti-on-ce40: ome-poualli 4: naui…. 5: macuilli15: caxtulli50: ome-poualli-om matlacti 6: chica-ce16: caxtulli-on-ce 7: chica-ome…. 8: chicu-ey20: cem-poualli

Five-Twenty Counting in Welsh 1 un16 un ar bymtheg = dau17 dau ar bymtheg = tri18 tri ar bymtheg = (also sometimes deunaw = 2x9) 4 pedwar19 pedwar ar bymtheg = pump20 ugain. 6 chwech30 deg ar hugain 7 saith40 Deugain 8 wyth50 Hanner cant 9 naw60 Trigain (3x20) 10 deg70 deg a thrigain 11 un ar ddeg = pedwar ugain 12 deuddeg = deg a pedwar ugain 13 tri ar ddeg = Cant 14 pedwar ar ddeg = dau cant 15 pymtheg = Mil

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.

Summary of Counting Systems

Counting Words Often derived from body parts or other associations.

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 NumberMeaning of Ainu wordNumberMeaning of Ainu word 1Beginning-to-be402 X 20 4Much603 X 20 5Hand804 X from from 2 X from from 3 X 20 8Two steps down7010 from 4 X 20 9One step down9010 from 5 X 20 10Two sided (i.e. both hands)1005 X 20 20Whole (man)11010 from 6 X 20

Counting Words Derived from Body Parts: The word for the number...is derived from a phrase meaning... 15Three fists 10Two hands 20Man complete 100Five men finished 9Hand and hand less one 2Raise a separate finger 6To cross over 6Take the thumb 9One in the belly 40A mattress

Inca Counting Words For example 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 ValueCountersWritten Symbols 3500 BC3200 BC2650 BC

Babylonian Cuneiform

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

Mayan Number System The “Date” on the left is

Egyptian Number System Based on powers of 10, but not positional. Link

Egyptian Number System

Roman Number System SymbolValue I1 V5 X10 L50 C100 D500 M1000 A bar can be placed over a symbol to indicate multiplication by 1000:

Greek Number System Early Attic System 2011 = XXΔΙ ΙΠΔΗΧΜ (5x10) (5x100)

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 numbers, the “myriad” symbol M from Attic numeration was used; its value was 10,000 and the number of 10,000’s was put above the M

Hebrew Number System Like 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 So: Every word in both Hebrew and Greek can be thought of as a number. Which explains, to some extent, the fascination with numerology. Just sayin’.

Chinese Number System Four basic systems evolved, based on powers of 10. Not positional.

Chinese Stick Numerals

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.