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 ingroups 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
11
13
11
3 6
17
21
4
8
19
19
10
5
9
5
7
• 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
2,154 306
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
x1
x1
x1
x6
x1
x1
x2
x3
x1
x1
x1
x2
x1
x6
x1
x1
x1
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
2
3
4
5
6
7
8
9
10
11
12
13
14
15
little finger
ring finger
middle finger
fore finger
thumb
hollow between radius and wrist
forearm
inside of elbow joint
upper arm
point of shoulder
side of neck
ear
point on the head above the ear
muscle above the temple
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-twotwo, 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
four
two-three
two-fours-and-one
two-threes
two-and-two-fours
one-(&)-two-threes
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
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”
English
numeral in
figures
1
2
3
4
5
6
7
8
13
15
20
23
26
Equivalent Kâte number
word
moc
jajahec
Jahec-a-moc
Jahec-a-jahec
Me-moc
Me-moc-a-moc
Me-moc-a-jajahec
Me-moc a jahec-a-moc
Me-jajahec a jahec-a-moc
Me-jajahec a kike-moc
ngic-moc
ngic-moc a jahec-a-moc
ngic-moc-a-me-moc-a-moc
Kâte operative pattern for
each counting number
words
1
2
3=2+1
4=2+2
5
5+1
5+2
8=5+(2+1)
13=10+(2+1) or (5+5)+(2+1)
15=10+5 or 15=5+5+5
20 (or 20=4x5)
23=20 +(2+1)
20+5+1
Roro Language from PNG
English numeral
in figures
1
2
3
4
5
6
7
8
9
10
11
12
15
20
26
30
40
100
200
Equivalent Roro number word
hamomo
rua
aihau
bani,
ima
abaihau
abaihau hamomo
ababani
ababani hamomo
harau haea
harauhaea hamomo
harauhaea rua
harauhaea ima
harau rua
harau rua abaihau
harau aitau
harau bani,
sinabu, hinabu
sinabu rua
Roro operative pattern for each
counting number word
1
2
3
4
5
2x3
2x3+1
2x4
2x4+1
ten, one of
1 ten + 1
1 ten + 2
1 ten + 5
ten, two of
2 tens + 6
3 tens
4 tens
a new word for hundred
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
2. kaja
3. toaso
4. kahese -e/i/o,
5. te-hɨtɨ
6. ɨha-tupɨ
7. ɨha-tupɨ seŋã-maka-jo
8. hopoajo
9. hopoajo kɨno-make-jo
10. sia-hɨ-ŋa
11. siahɨŋa te- e/i/o
12. siahɨŋa kaja
20. siahɨŋa siahɨŋa
(inanimate, masculine, feminine )
( inanimate, masculine, feminine )
( lit ''a hand of X exists'' )
(lit: ''thumb [from the other hand] (exists)'' )
(lit: ''after the thumb'' )
(lit: ''middle finger (exists)'' )
(lit: ''close to middle finger'' )
(lit: ''all hands (exist'' )
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: 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 Welsh
1 un
2 dau
3 tri
16 un ar bymtheg = 1 + 5 + 10.
17 dau ar bymtheg = 2 + 5 + 10
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
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
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...
15
10
20
100
9
2
6
6
9
40
is derived from a phrase meaning...
Three fists
Two hands
Man complete
Five men finished
Hand and hand less one
Raise a separate finger
To cross over
Take the thumb
One in the belly
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 Cuneiform
Value
Counters
3500 BC
1
10
60
600
3600
36000
Written Symbols
3200 BC
2650 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
8.5.16.9.7
Egyptian Number System
Based on powers of 10, but not positional.
• Link
Egyptian Number System
Roman Number System
Symbol
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
Ι
Π
Δ
1
5
10
• 2011 = XXΔΙ
Η
50
(5x10)
100
Χ
500
(5x100)
1000
Μ
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
• 𝑀,ζροε , εωοε =
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.