Beginnings of Counting and Numbers

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?

11 13 17 19

11 21 19 9

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

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

2,154 306 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.

• 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(http://www.youtube.com/watch?v=H13Se4nBPDA)

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-three two-fours-and-onetwo two-threes two-and-two-foursthree one-(&)-two-threesfour 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

English Equivalent Kâte number Kâte operative pattern for numeral in word each counting number figures words

1 moc 12 jajahec 23 Jahec-a-moc 3=2+1 4 Jahec-a-jahec 4=2+2 5 Me-moc 56 Me-moc-a-moc 5+17 Me-moc-a-jajahec 5+28 Me-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

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 PNGEnglish numeral in figures

Equivalent Roro number word Roro operative pattern for each counting number word

1 hamomo 12 rua 23 aihau 34 bani, 45 ima 56 abaihau 2x37 abaihau hamomo 2x3+18 ababani 2x49 ababani hamomo 2x4+1

10 harau haea ten, one of11 harauhaea hamomo 1 ten + 112 harauhaea rua 1 ten + 215 harauhaea ima 1 ten + 520 harau rua ten, two of26 harau rua abaihau 2 tens + 630 harau aitau 3 tens40 harau bani, 4 tens

100 sinabu, hinabu a new word for hundred200 sinabu rua 2 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 + 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 + N pattern)

7: ab-ariyo 30: piero-adek

(Five)-ten Counting Example

• Secoya, Ecuador and Peru

1. tee, tei, teo (inanimate, masculine, feminine )2. kaja3. toaso4. 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/o12. siahɨŋa kaja20. 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: Aztecs1: 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-Twenty Counting in Welsh1 un 16 un ar bymtheg = 1 + 5 + 10.2 dau 17 dau ar bymtheg = 2 + 5 + 103 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 hugain7 saith 40 Deugain8 wyth 50 Hanner cant9 naw 60 Trigain (3x20)10 deg 70 deg a thrigain11 un ar ddeg = 1 + 10. 80 pedwar ugain12 deuddeg = 2 + 10. 90 deg a pedwar ugain13 tri ar ddeg = 3 + 10. 100 Cant14 pedwar ar ddeg = 4 + 10 200 dau cant15 pymtheg = 5 + 10 1000 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 WordsNumber 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 fists10 Two hands20 Man complete

100 Five men finished9 Hand and hand less one2 Raise a separate finger6 To cross over6 Take the thumb9 One in the belly

40 A 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 CuneiformValue 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

Egyptian Number System

Roman Number SystemSymbol Value

I 1

V 5

X 10

L 50

C 100

D 500

M 1000

A bar can be placed over a symbol to indicate multiplication by 1000: 𝑉

Greek Number System

• Early Attic System

• 2011 = XXΔΙ

Ι Π Δ Η Χ Μ1 5 10 50

(5x10)100 500

(5x100)1000 5000 10000

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

• Υνγ’ = 453• ,δωοβ = 4,872

• Mωμθ =10,849

• =71,755,875

• Based on powers of 10• Not Positional

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.