CHAPTER-III
A COMPARATIVE STUDY OF SOME ANCIANT
NUMBER SYSTEMS
[A part of this work is published in Int. Jour. of Math. Arc.-2(9), Sept
- 2011, page: 1550-1561. ISSN 2229-5046.]
.
40
Chapter-III
3.1 Introduction
Ancient Egypt was an civilization of eastern North Africa, concentrated along the
lower reaches of the Nile River.
Some of the achievements of the ancient Egyptians include the quarrying, surveying
and construction techniques that facilitate the building of monumental pyramids, temples
and obelisks, a system of mathematics, a practical and effective system of medicine,
irrigation systems and agricultural production techniques, new forms of literature etc.
The Babylonians (2300 BC to 1600 BC) lived in Mesopotamia about 5,000 years
ago, Babylonians began a numbering system. It is one of the oldest numbering system.
The Mayans(Indians may see this civilization as related our ancient scriptures like
Ramayana where we find relative of great King Ravana, in Particular Maya-Ravana, in
PATAAL (down the Earth) meaning America (3113BC to 900 AD) lived in Central
America. The Mayans were highly skilled mathematicians, astronomers, artists and
architects. The Maya civilization collapsed mysteriously around 900 AD.
The Mayans had several calendars. There was 360 days civil year, a 260 days
religious year and the complicated Long Count Calendar which measured time from the
start of Maya civilization (August 12, 3113 BC) and completes a full cycle on December
21, 2012.
Babylonia was situated on the western Asia and in the area known as Mesopotamia.
Mesopotamia was in the same geographical position as modern Iraq. Two great rivers
flowed through this land- the Tigris and the Euphrates. The Babylonians (2300 BC to 300
BC) lived in Mesopotamia about 5,000 years ago. Babylonians began a numbering system.
It is one of the oldest numbering system.
Greece is a country in South Eastern Europe. Greece is a peninsula, surrounded on
three sides by water. The country Greece is also covered with mountains. Geeks were very
advanced in mathematics.
41
Chapter-III
China, a vast country is also located on the Asia continent.. In the west of China
there are the Himalayas, the highest mountains in the world. China’s lowest point is in the
Turfan Depression, at 154 meters below sea level. Chinese (14th-11th centuries B.C )
invented some number system- traditional and rod numeral system.
Arabia is located on the traditional continent of Asia, in the extreme south west
portion known as the Middle East. It is surrounded by the Persian gulf on the north east, the
strait of Hormuz and the gulf of Oman on the east, the Arabian sea on the south east and
south, the gulf of Aden on the south, the Bab-el-Mandeb strait on the south-west and the
Red sea on the south-west and west. The northern portion of the peninsula merges with the
Syrian Desert with no clear boarder line.
India is also situated at Asia continent. In the northern part of India stand the
Himalayan Mountains. The southern region of India is surrounded by three bodies of water.
They are the Arabian sea to the south west, the Indian ocean on the southern side and to the
south east lay the Bay of Bengal. In ancient times India was much more extended to the
North West and west, consisting of modern Pakistan and Afghanistan.
It is really interesting that modern English numerals, that most of the part of the
world used today, invented in India. Through Arabs the modified form of these numerals
spread throughout the world.
The early Egyptian, Babylonian, Mayan, Greek, Chinese and Indian were very rich
in mathematics. They developed their own number system. But there were some differences
between the number system of Egyptian, Babylonian and Mayan.
3.2. ANCIENT EGYPT, BABYLON AND MAYAN NUMBER SYSTEM.
3.2.1. In ancient Egypt (2300 BC to 1600 BC) the numerical notation was very
simple. They used some symbols to represent the numbers called hieroglyphics. Some
Egyptian numbers were as tabulated below,
English
1
2
3
4
5
6
7
8
9
10
Egyptian
|
||
|||
||||
|||||
||||||
|||||||
||||||||
||||||||
I
42
Chapter-III
English
11
15
20
22
Egyptian
I |
I || || |
II
I I ||
English
1000
10,000
Egyptian
♣
or
60
90
II I
II II
II I
II II I
1,00,000
100
Or
1,000,000
or
Thus we observe that in ancient Egyptian number system there was a special sign
for every power of ten.
1=
10 = I Heel bone or string.
100 =
Or
coil of rope.
1000 = ♣ flower.
10,000 =
or
Pointing finger.
or
100,000 =
1,000,000 =
surprised man.
Tad pole.
43
Chapter-III
Generally, the Egyptian numbers were written from left to right as the modern form
of express a number. Such as
II I
= 360
II I
I I I I ||| = 2343
♣♣
♣♣♣♣ I I ||| = 34023
The Egyptian numbers were also written from right to left i.e. the largest decimal order
would be written first. In this case the symbols themselves were reversed. Below some of
the Egyptian numbers written from right to left and their modern English form
II
= 350
II I
||| I I I I
||| I I
♣♣♣♣
♣♣ = 2343
= 34023
It is to be noted that in Egyptian number system if we write a number from left to
right (i.e. the largest decimal order would be written last) or we write the same number from
right to left (i.e. the largest decimal order would be written first) the value of the number
remain same.
It is possible because in ancient Egyptian number system there was a special sign for
every power of ten.
3.2.2. The Babylonians lived in Mesopotamia about 5,000 years ago. Babylonians
began a number system. It is one of the oldest numbering system. The Babylonians
developed a form of writing based on cuneiform in between 2300 BC to 1600 BC .In Latin
the meaning of cuneiform is “wedge shape”. They used only two symbols to represent their
44
Chapter-III
all numbers.. These two symbols are ∇ =1, < =10 . Some of the Babylonian numbers were as
tabulated below,
English
1
2
5
Babylonian
∇
∇∇
9
∇∇
10
∇ ∇∇ ∇
∇ ∇∇
<
∇ ∇∇ ∇∇
14
15
∇∇
<
<
∇∇
∇∇
∇ ∇∇
Englis
20
24
30
36
40
42
59
<< <<
<< << ∇∇
h
< << ∇ ∇∇
Babylo
<< ∇ ∇
<<
∇∇
-nian
∇∇∇
<< <
<< << <
∇∇∇∇∇
∇∇∇∇
3.2.3. The Maya number system used a combination of two symbols. A dot (.) was
used to represent the units and a dash (─) was used to represent five.
Some of Mayans numbers were
English
1
2
3
4
Mayan
•
••
•••
••••
English
11
12
13
14
Mayan
•
•••
••••
••
5
15
6
7
8
9
•
••
•••
••••
10
16
17
18
19
•
••
•••
••••
45
Chapter-III
3.3. THE DECIMAL, SEXAGESIMAL AND VIGESIMAL SYSTEM
The Egyptian number system was decimal based, Babylonian number system was
sexagesimal and Mayan number system was vigesimal based. Before going to discuss about
decimal, sexagesimal and vigesimal system we must know about place value notation. Place
value notation is the use of numerals in different position to represent numbers.
3.3.1 A number represents a quantity. Number is an abstract idea of collection of
things but numerals are man-made symbols that represent the numbers. Numbers (Quantity)
are always of same value, no matter what symbol or word is used to represent them. As for
example, 1, I, i all three numerals represent the same number we know as “one”.
Our system uses place value notation, for example, 43 means “four tens and three
ones.”
Tens
Ones
4
3
In place value notation, the places are the exponents of the base. In our decimal
(base 10) system we have places for 1, 10, 10 squared (100), 10 cubed (1000) and so on. In
a sexagesimal (base 60) system, the places are 1, 60, 60 squared (3600) etc. In vigesimal
(base 20) system, 1, 20, 20 squared (400), 20 cubed (800) and so on.
Value “Two hundred and twelve” in Base 10 & Base 60
DECIMAL
Value = 212
SEXAGESIMAL
Value=212
10 squared place
(100)
2
60 Squared place
(602 =3600)
Tens place (10)
Ones place (1)
1
2
Sixties place (60)
Ones place (1)
3
32
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Chapter-III
3.3.2 The Egyptian number system was decimal base. Some of the Egyptian
numbers with their place values and decimal values are given below.
Place Value
10,00000
100,000
10000
Decimal Value
1000
100
♣
10
1
I
|
460
||||
||||||
IIIIII
3234
|||
||
|||
||||
I I I ||||
♣♣♣
22022
||
||
||
||
♣ ♣ I I ||
1,23,564
|
||
|||
|||||
||||||
||||
♣♣♣
I I I I I I ||||
2, 11, 2522
||
|
|
||
|||||
||
||
♣♣
I I ||
3.3.3 In between 2300 BC and 1600 BC the Babylonians had a very advanced
number system. It was the base 60 system or sexagesimal system rather than the base ten
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Chapter-III
system or decimal system. In this system numbers above the units (from 1 to 59) are
arranged according to power of 60. At that time 60 itself was called “sussu” or “soss” and
602 was called “sar”. The Babylonians divided the day into twenty four hours, each hours
into sixty minutes and each minute to sixty second. They used only two symbols ∇ and <
to represent all the numbers from 1 to 59. By grouping these two symbols together, they
created symbols for all 59 numbers.
Some of the Babylonian numbers in sexagesimal system are given below.
Place value
603
602
<<
<
Sexagesimal Value
60
1
∇
∇
∇ ∇ = 61
<< < ∇ ∇ ∇ ∇ ∇
∇∇
∇∇∇
= 72, 37, 723
∇∇
<
∇
<<<
∇∇< ∇
<<<
<<< ∇ ∇
<<< ∇ ∇
= 4, 37, 792
∇∇
< ∇∇
∇∇ < ∇ ∇ ∇ ∇ ∇
∇∇∇
= 7, 923
∇
∇∇
<<
∇∇∇
<<
= 3, 722
3.3.4 The Mayans used a vigesimal system, which had a base 20. This system is
believed to have been used because 20 was the total number of fingers and toes. The
Mayans wrote their numbers vertically with the lowest denomination on the bottom. Their
system was set up so that the first five place values were based on the multiples of 20. The
Mayan numbers are read from bottom to top. The Mayans departed from a pure base 20
48
Chapter-III
system by letting the symbols in the third position (from the bottom) represent the number
of 360=18×20 instead of 400=202. According to some scholars the reason may be that their
year consisted of 360 days. In the table below are represented some Mayan numbers.
Place 18×202 = 7200
•••
value
18×20 = 360
20
•
•
••
•
•
Unit
•
••
•
••
•••
•••
••••
••••
•••
•
•
••
••
•
•••
••
••••
•
= 468
•••
•
Vigesimal
Value
= 25
= 44
= 873
= 396
•••
•
••••
= 27,374
•
= 66
••
= 405
= 40
•••
= 1962
••
••
•
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Chapter-III
3.4 Position of zero
In Egyptian number system there was no symbol to represent the number zero. In
Egyptian number system the symbol for zero was not necessary to write a number. In
modern number system we cannot write 306 or 2010 without the zero.
But the Egyptian wrote
306 =
| | |
| | |
2010 = ♣ ♣ I
Babylonians also did not have a symbol for zero. But they used the idea of zero.
When it was necessary to express zero, they just left a blank space in the number they were
writing or used a wedge mark. When they wrote “60”, they would put a single wedge mark
in the second place of the numeral.
∇ / = 60 ,
∇ ∇ / =120 ,
< ∇ / = 660 ,
The Mayans symbolized the concept of ’nothing’or zero. The most common symbol
for zero was
→ a head.
•••
= 60
3.5 A comparative study of sexagesimal, decimal and vigesimal system
3.5.1 Place Value
Numerals 1 – 2 – 3 – 2 in different place value notation.
50
Chapter-III
Place Value
603
602
60
1
1
2
3
2
103
102
10
1
1
2
3
2
203
202
20
1
1
2
3
2
Sexagesimal
1× 603 + 2 × 602 + 3 × 60 + 2 ×1
= 2, 23, 382
Decimal Value
1×103 + 2 ×10 2 + 3 ×10 + 2 ×1
= 1232
Vigesimal Value
1× 203 + 2 × 202 + 3 × 20 + 2 ×1
= 8862
Value “Three thousand seven hundred twenty one” in Base 60, Base 10 and Base 20.
Place Value
603
602
60
1
Sexagesimal
1
2
1
Value = 3721
103
102
10
1
Decimal Value
3
7
2
1
Value = 3721
203
202
20
1
Vigesimal
9
6
1
Value = 3721
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Chapter-III
Numerals 2 – 1 – 6 – 3 in different place value notation.
Place Value
603
602
60
1
2
1
6
3
103
102
10
1
Sexagesimal
2 × 603 + 1× 602 + 6 × 60 + 3 ×1
= 4,35,963
Decimal
Value
2
1
6
3
(200 + 100 + 60 + 3)
= 2163
203
202
20
1
Vigesimal
Value
2
1
6
3
2 × 203 + 1× 202 + 6 × 20 + 3 ×1
= 16, 523
Value “Four thousand three hundred twenty one” in Base 60, Base 10 and Base 20.
Place Value
603
602
60
1
Sexagesimal
1
12
1
Value = 4321
103
102
10
1
Decimal
4
3
2
1
Value = 4321
203
202
20
1
Vigesimal
10
16
1
Value = 4321
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Chapter-III
Observation:
From above example, we observe that the sexagesimal and vigesimal number
system is more time consuming than decimal system. In decimal system it is easy to find
out the place value of any digit in a number but in sexagesimal and vigesimal system it is
quite difficult. Due to such type of drawbacks of sexagesimal and vigesimal system modern
world give more preference to decimal system then sexagesimal and vigesimal system.
3.5.2 ADDITION IN DECIMAL, SEXAGESIMAL AND VIGESIMAL SYSTEM
Let us take two numbers 7866 and 3727
In our decimal system if we add these two numbers, we will get
7866
+ 3727
11593
If we add these two numbers in Babylonian number system then we will get
+
602
60
∇∇
<∇
∇
∇∇
∇∇∇
∇∇∇
< ∇ ∇∇
= 3 × 60 2 + 13 × 60 + 13 ×1
= 10800 + 780 + 13
= 11593
1
∇∇∇
∇∇∇
∇∇∇
∇∇∇∇
< ∇ ∇∇
Number in English form
3600 × 2 + 60 × 11 + 1× 6 = 7866
60 2 ×1 + 60 × 2 + 1× 7 = 3727
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Chapter-III
Now we will add these two numbers in Mayan number system,
360
•
•
=31×360=1160
+
+
20
•
Equal to
•
=21×20=420
+
1
Number in
English form
•
••
•••
360 × 21 + 20 ×15
360 × 10 + 20 × 6
+ 1× 6 = 7866
+ 1× 7 = 3727
=13×1=13
Total 11593
The result (11593) is same as result of decimal system and sexagesimal system.
Now let us consider three numbers
8930, 887 and 532.
In decimal system, if we add them, we will get
8930
887
+
532
10,349
Now their place value notation in Babylonian number system is
54
Chapter-III
602
60
∇∇
<<
<<
∇∇∇∇
< <<
∇∇
< < <<
∇∇
∇∇∇∇
60 2 × 2 + 60 × 28 + 1× 50 = 8930
∇∇∇
60 ×14 + 1× 4 7 = 887
∇∇∇∇
60 × 8 + 1× 52 = 532
< < <<< ∇∇
∇∇∇∇
< < ∇ ∇ ∇∇
∇∇∇∇∇
< <<<<∇ ∇
∇∇
Number in English form
∇∇∇∇
<
+
1
= 2 × 60 2 + 52 × 60 + 29 ×1
= 7200 + 3120 + 29
= 10349
Now we will add these three numbers in Mayan number system,
7200
•
360
••••
20
••••
••
+
1
Number in
English form
8930
•
+
•
= 7200
•••
= 2880
Equal to
•••
•••
•
= 260
••
••
••••
=
887
532
9
Total = 10349
[In vigesimal system we cannot write the numbers greater than 19 in unit places so,
in case of 29 we have to distribute 29 as 9 in unit place and 20 in 20th place. Since the
system is vigesimal therefore 20 becomes 1 in 20th place]
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Chapter-III
Observation;
The above examples make it clear that in case of addition the results are same in
decimal, sexagesimal and vigesimal system.
3.5.3 Subtraction in Decimal, Sexagesimal and Vigesimal system
Let us take two numbers 7866 and 3727. In our decimal system if we subtract 3727
from 7866, we will get
3866
- 3727
4139
If we subtract 3727 from 7866, in Babylonian number system, then we will get
602
60
1
number in English
< <<<<<
∇∇
< ∇
∇
∇∇
∇
∇∇∇∇
∇∇∇∇
∇∇∇
∇∇∇
∇∇∇
∇∇∇∇
< <<<<
∇∇∇∇
∇∇∇∇ ∇
7866
3727
= 60 2 ×1 + 60 × 8 + 1 × 59
= 3600 + 480 + 59
= 4139 , same as the result of
decimal system.
[We cannot subtract 7 from 6, therefore we carry over 1 from 60th place. Since the
system is sexagesimal, therefore in unit place 1 becomes 60]
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Chapter-III
Now we will subtract 3727 from 7866 in Mayan number system
360
•
•
20
1
Number in
English form
= 360 × 11 = 3960
•
•
••
7866
3727
=
•••
= 20 × 8 = 160
••••
= 1×19 = 19
Total = 4139
So in Mayan number system when we subtract 3727 from 7866, the result is 4139.
The result is same as decimal system and sexagesimal system. (We cannot subtract 7from 6,
therefore we carry over 1 from 20th place. Since the system is vigesimal, therefore in unit
place 1 becomes 20). Now, let us take another example
Suppose we need to subtract 887 from 8930. In decimal system
8930
- 887
8043
Now we will subtract 887 from 8930 in Babylonian number system or sexagesimal
system
602
∇∇
60
<<
<<<
∇∇∇∇
<<
∇∇∇∇
<
∇∇
1
∇∇
∇∇
< ∇∇∇∇
< <<<
∇∇∇
∇∇∇∇
∇∇∇
number in English form
8930
887
= 60 2 × 2 + 60 ×14 + 1 × 3
= 7200 + 840 + 3 = 8043
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Chapter-III
The result is same as decimal system. Now we will subtract 887 from 8930 in
Mayan number system or vigesimal system.
7200
•
360
••••
20
••••
••
-
•
=7200×1=7200
••
= 360×2 =720
•
= 20×6 =120
•••
= 1×3 =3
=
•••
••
1
Total = 8043
Observation;
Thus from the above examples we have observed that in case of subtraction the
results are same in decimal, sexagesimal and vigesimal system.
The Multiplication and division in sexagesimal and vigesimal system are left for the
readers.
3.6 GREEK NUMERATION SYSTEMS
(600 BC to 500 AD)
3.6.1 In ancient time the Greeks used two systems of numeration. In earlier times
they developed the ‘Herodianic’ system. The system is called ‘Herodianic’ because it is
described in a fragment attributed to Herodian, a grammarian of the latter half of the second
century AD. The writer says that he has seen the signs used in various ancient inscriptions,
decrees and laws. At a later time, the Greeks used the Ionic system to write their numerals.
In Herodianic system the Greeks used the following symbols to represent different
numbers.
English
1
2
3
4
5
6
7
8
9
10
Greek
I
II
III
IIII
Γ
ΓI
ΓII
ΓIII
ΓIIII
∆
58
Chapter-III
English
11
12
13
14
15
16
17
18
19
20
Greek
∆I
∆I I
∆I I I
∆I I II
∆Γ
∆Γ I
∆Γ I I
∆Γ I I I
∆Γ I I II
∆∆
English
30
40
50
Greek
∆ ∆∆
∆∆∆∆
60
∆∆
English
200
300
Greek
HH
HHH
English
3000
Greek
XXX
70
80
90
100
∆ ∆∆
∆ ∆∆∆
∆ ∆∆∆∆
H
500
600
5000
1000
2000
X
XX
10,000
50,000
M
Like the ancient Chinese number system ancient Greek number system (Herodianic)
was also both additive and multiplicative in nature. Additive in nature because some
numbers were constructed from the symbols I, Γ,∆, H, X, M, additively by repeating them,
such as
∆∆∆
= 30
HHHH
= 400
M M M M X X X H H ∆ ∆ I = 432 21
HHH∆∆∆∆III
= 5343
Multiplicative in nature mean that 50 is represented by the symbol of 5 and symbol of 10.
= 5. 10
= 50
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Chapter-III
5000 is represented by the symbol of 5 and symbol of 1000
= 5. 1000 = 5000
Therefore we have observed that in ancient Greece there were six simple
symbols which represents 1, 5, 10, 100, 1000, 10000 and four compound symbols which
represents 50, 500, 5000, 50000. They used these ten symbols to construct any other
numbers. For example –
55=
Γ
79=
∆∆Γ I I I I
102= HH
134= H∆∆∆I I I I
229= HH∆∆Γ I
567=
746=
823=
HH∆∆∆∆ΓI
∆Γ I I
HHH ∆ ∆ I I I
1298= X HH ∆ I I I I
2345= XXHHH I I I I Γ
1764= X
2471= XXHHHH ∆∆ I
HH
∆IIII
13481=MX X X H H H H ∆∆∆ I
55555=
Γ
The five symbols Γ, ∆, H, X and M are the initial letters of Greek number words.
The symbol Γ stood for Greek word “Pente” (five). Therefore “ Pentagon”, “Pentameter”
and “ Pentathlon” derived from Greek word “Pente”. The symbol ∆ represents “deka”
(10) and leads to “decagon” and “decameter” (10 meters). Simibrly H stood for “hekaton”
(100) , which gives us “hectare”. The symbol X represents “Kilioi” (1000), which leads
“Kilometer” (1000 meter) and the symbol M, stood for “myrioi” which leads “myriad”.
In Herodianic member system we observed that Greeks used a special symbol to
represent 1 (1), 10 (∆), 100 (H), 1000 (X) and 10,000 (M). Therefore, from my point of
view they followed decimal number system. Some of the Greek numbers (Herodianic ) with
their place values and decimal values are given below.
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Chapter-III
Place value
Decimal Value
10,000
1000
100
10
II
I
I
III
1
MM X
H ∆∆∆Γ
= 21635
II
Γ
ΓII
ΓIII
Γ
XX
HH
ΓIII
= 52758
I
IIII
Γ
II
M
I
HHHH∆∆I
= 15421
II
ΓI
ΓIIII
X HH
ΓII
∆∆∆∆Γ I I
= 6297
IIII
IIII
II
III
ΓIII
IIII
II
HHHH ∆∆ III= 423
XXX X
HHH∆
∆∆∆∆ II
= 4842
3.6.2 The Ionic System
In the Ionic number system the Greeks used the alphabet as digits. Therefore how
they distinguished numerals from words? They solved this problem by drawing lines over
numerals or by adding an accent at the end. Some of the Greek numerals in the Ionic
number system were.
English
1
2
3
4
5
6
7
8
9
61
Chapter-III
Greek
α
β
γ
δ
ε
ς
ζ
η
θ
English
10
20
30
40
50
60
70
80
90
ι
κ
λ
µ
ν
ξ
ο
π
100
200
300
400
500
600
700
800
900
ρ
σ
τ
υ
φ
χ
ψ
ω
λ
Greek
English
Greek
ϕ
English
12
25
37
43
56
Greek
ιβ
κε
λζ
µγ
νς
English
67
74
81
99
102
Greek
εζ
οδ
πα
ςθ
ρβ
English
150
237
596
670
999
Greek
ρυ
σλζ
φςζ
χο
λςθ
A problems arose in this system; how to write symbols for numbers larger than 999.
This problem was solved by drawing a stroke before a numeral, multiplied the number
represented by the numeral by 1000. Some numbers larger than 999 are
English
Greek
English
Greek
1000
2000
3000
4000
5000
6000
7000
α
β
γ
δ
ε
ς
ζ
8000
9000
3473
6781
9899
η
θ
γυογ
ςψπα
θπςθ
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Chapter-III
Observation:
Thus we observed that in Ionic number system Greeks used symbols to represent many
more number. It is really difficult to remember so many symbols.
In ancient Greek number system (both Herodianic and Ionic) there was no symbol
for Zero. In Greek number system the symbol of Zero was not necessary to write a number.
Without symbol of Zero they can expressed every members such as
6000= ς or
707=
X
ψζ or
9009 = θθ or
HH Γ II
XXXX Γ IIII
500= φ or
3.7 THE NUMBER SYSTEMS OF ANCIENT CHINA (14th – 11th Centuries BC)
In ancient China in between 14th -11th centuries BC the numerical notation was very
simple. Chinese used some symbols to represent each number. The ancient Chinese number
system was both additive and multiplicative in nature. Here additive nature of the system
was that symbols were juxtaposed to indicate addition. So, they used the symbol of 1 twice
to represent 2, similarly to represent 4 they repeated symbol of 1 up to four times, 3496 was
represented by the symbol for 3000 followed by the symbol of 400 followed by the symbol
of 90 followed by the symbol of 6. Where Multiplicative in nature means that 200 is
represented by the symbol for 2 and the symbol for 100, 300 is represented by the symbol
for 3 and the symbol for 100 and so on. Similarly 2000 is represented by the symbol for 2
and the symbol for 1000, 4000 is represented by the symbol for 4 and the symbol for 1000
etc. The largest number in between 14th to 11th century BC has been found 30000.
There are two theories about the number system of ancient China. According to first
theory the symbols are phonetic. By this we mean that since the number nine looks like a
fish hook, therefore in ancient China, perhaps the word for nine was close to the sound of
the word for ‘fish hook’. Similarly the symbol for 1000 is a ‘man’, therefore perhaps the
word for “thousand” in ancient Chaina was close to the sound of the word for ‘man’.
63
Chapter-III
According to the second theory, the number symbols are of religious significance.
This theory explained that symbol of scorpion used to represent 10,000 since swarms of
scorpions meant “a large member” to people at that time. Perhaps the symbol for 100
represents a toe and one might explain this if people at the time counted up to ten on their
fingers then 100 for each toe, and then 1000 for the ‘man’ having counted ‘all’ parts of the
body.
Some ancient Chinese numerals were
English
1
2
3
4
5
6
7
11
12
13
14
15
16
8
9
10
Chinese
English
17
18
19
20
Chinese
English
30
40
50
200
300
400
60
70
80
90
100
Chinese
English
500
600
700
800
900
Chinese
English 1000
Chinese
2000
3000
4000
5000
6000
7000
8000
9000
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Chapter-III
English
10,000
20,000
30,000
Chinese
1.
2.
= 140
3.
= 258
= 351
4.
= 3565
Sometimes the Chinese used Ψ for ‘and’ between tens and units or hundreds and tens.
For example.
659:
Ψ
Ψ
Six hundred and five ten and nine.
924:
Ψ
Ψ
Nine hundred and twenty and four.
Observation:
So, like ancient Egyptian number system, in ancient Chinese number system also,
there was a special sign for every power of ten.
1-
10-
65
Chapter-III
100100010,000-
3.7.1. Base of the numbers
The ancient Chinese number system was decimal base. Below some of the
Chinese numbers with their place values and decimal values and modern English form.
Place Value
10,000
1000
100
Chinese Form
10
English Form
1
21457
6531
895
5613
475
3.7.2. Position of Zero
In ancient Chinese number system there was no symbol for zero. To express any
Chinese number it had no need for a Zero.
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Chapter-III
For example.
8060
represented by
8800 represented by
3.7.3. Chinese Rod Numerals (400 BC)
In 4th century BC the Chinese numbers were represented by little rods made from
bamboo or ivory. The Chinese used counting board. A counting board consisted of a
checker board with rows and columns. The numerals made from rods were called rod
numerals.
The nine signs of the rod numerals corresponding to the first nine numbers are of
two types, such as,
1
2
3
4
5
6
7
8
A
B
The biggest problem with Chinese Rod numerals was that it could lead to some
possible confusion. For example
=3
=4
= 21
= 22
= 12
=1111
9
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Chapter-III
= 111
= 13
= 31
= 211
= 121
= 112
Therefore what was the value of
value of
? It could be 3, 21,12 or 111. Also what was the
? It could be 4, 22, 13, 31, 211, 121, 112 or 1 1 1 1. The Chinese adopted a
clever way to avoid this problem. The ‘A’ numerals are put in the places for units,
hundreds, ten thousands etc. and ‘B’ numerals are in the places for tens, thousands, hundred
thousand etc.
For example-
3467=
or
46283=
or
62=
or
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Chapter-III
21=
or
111=
or
121=
or
112=
1111=
or
or
In rod numeral notation, there was no specific symbol for zero. Zero was
represented by a blank space. For example,
50290=
706=
60=
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Chapter-III
3.8.The Hindu Arabic Numerals
The early numerals of India were of various types. The Brahmi numerals (300 BC to
400 AD) have been found in inscriptions in coves and on coins in region near Poona,
Bombay and Uttar Pradesh. Since in Brahmi numerals there was not a place value system so
there were symbols for four to nine, for ten and multiples of ten up to ninety, and for
hundred and thousand. Two and three were represented by repetitions of the horizontal
stroke for one. There were distinct symbols for many more numbers. Some of Brahmi
numerals are.
1
2
3
4
5
6
7
8
9
10
20
30
40
50
60
70
80
90
100
200
400
700
1000
4000
6000
10000
20000
Brahmi Numerals
( 300 BC to 400 AD )
After Brahmi numerals the next numerals found in India were the Gupta numerals.
The Gupta period is that during which the Gupta dynasty ruled. It was from the early 4th
century AD to the late 6th century AD. There were many similarities between Brahmi
numerals and Gupta numerals. Therefore it may be said that Gupta numerals developed
from Brahmi numerals.
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Chapter-III
1
2
3
4
5
6
7
8
9
10
20
40
70
100
200
500
1000
2000
3000
4000
8000
70,000
GUPTA NUMERALS (400 ad TO 600 AD )
In Gupta numerals also there was not a place value system so there were symbols for
many more numbers. Since each symbol represent a particular number therefore in Brahmi
and Gupta number system there was not necessesity of symbol for zero.
Nagari numerals or Devanagiri numerals evolved from the gupta numerals
beginning around the 7th century AD and continued to developed from 11th century
onwards. In Nagari numerals the separate strokes in the numerals for two and meaning of
the word “Deva” is “God”. Therefore the word “Devanagiri” means “writing of the god”.
1
2
3
4
5
6
7
8
9
0
Nagari or Devanagiri numerals
(700 AD to 1100 AD)
In Nagari or Devangiri numerals we have observed the symbol for zero. So symbol
of Zero was invented in India in between 700 AD to 1100 AD. After invention of the
symbol for zero the Hindu number system because purely decimal base.
Some of the Hindu numbers ( Nagari numerals ) with their place value and decimal
values are
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Chapter-III
Place Value
104
103
Decimal Value
102
10
1
=790
=1469
=25908
=85371
=60
=908
One of the famous Indian astronomical text book the Siddhanta by Brahmagupta
where he used decimal place value system was brought to Baghdad and translated into
Arabic towards the end of the eight century. At that time Arabs used the Greek numeral
system. Therefore Indian or Hindu numeral system was known in the Arab world as early as
the middle of the seventh century. The great Arab mathematician al-Khawarizmi had
become familiar with the Hindu system through study of the Siddhanta. At that time, their
notation had the following form:
9
1
2
3
4
5
6
Hindu or Indian numerals
7
8
9
0
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Chapter-III
These numerals are called Indian or Hindu numerals. We observed that there are
some similarities between Indian numerals with Brahmi numerals and Indian numerals with
Nagari numerals. Therefore it can be assumed that the Indian numerals are proceeded from
the Brahmi and Nagari numerals.
There were two types of numerals evolved from the Indian numerals namely,
1
(i)
The West Arabic or Gobar numerals.
(ii)
The East Arabic numerals.
2
3
4
5
6
7
8
9
0
8
9
0
West Arabic or Gobar numerals
1
2
3
4
5
6
7
East Arabic numerals
Observation:
Thus the west Arabic or Gobar numerals reveal our modern numerals.
The Arabs introduced the West Arabic numerals or Gobar numerals into Spain.
From Spain these numerals gradually became known in Western Europe. Gerbert (later
Pope Sylvester II) was the first person who attempted to spread the use of the Gobar
numerals in Western Europe in the tenth century. During one of his journey in Spain he
acquainted with the Arabic numerals. He wrote a booklet in which he described about the
use of the new symbols and their advantages. In this way the west Arabic numeral system
occupies a prime place in the modern numeral system. Since West Arabic numerals evolved
from Hindu numerals it is also called as Hindu-Arabic numerals. Today in most part of the
modern world the system of numeration used is the Hindu-Arabic numeration system.
The journey of numerals has taken us from 2300 BC to present day. In the world
different races used different numeral system and we still use some of the old numeral
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Chapter-III
systems and symbols. Our system of numeric’s is ever changing and who knows what it
will look like in 2500 A.D.?
3.9 A Comparative study of the numeral system of ancient Egypt, Babylon, Mayan,
China, Greek and Hindu Arabic
The following observations are being made out during my studies.
(i) In Egyptian number system a number is expressed without using any symbol
representing zero. Similarly in China, Greek and Hindu (Brahmi, Gupta) system also a
number can be expressed without symbol of zero. The reason is that in these number
systems there were separate symbols to indicate many more numbers.
(ii) In Egyptian number system the same number was written from right to left as well as
from left to right which is not normally seen in other member systems.
(iii) The Egyptian, Chinese and Greek ( Herodianic ) number system had a bases 10 system
for numerals. Therefore they have separate symbols for one unit, one ten, one hundred,
one thousand, one ten thousand etc. With the help of these symbols they expressed any
number. In Indian (Brahmi, Gupta) number system and Greek (Ionic) number system
also there were separate symbols to indicate the powers of ten. But in these two
systems there were separate symbols to indicate the other numbers also. Therefore in
both system there was not a place value system.
(iv) The symbols used in Egyptian, Greek, Indian and Chinese number system to represent
numbers are very laborious to write and also time consuming. For example in Egyptian
number system to frame the number 4086, eighteen symbols were required: four
hundred symbols, eight “ten” symbols and six “unit” symbols.
In Egyptian number system
486 =
IIIIIIII
(Total number of symbols= 18)
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Chapter-III
In Greek ( Herodianic ) number system.
486 = HHHH
∆∆∆ΓI
(Total number of symbols = 9)
In Greek (Ionic) number system
486 = υ π ς
In Indian (Brahmi) number system
486 =
(Separate symbol for 400, 80 and 6 )
(Separate symbol for 400,80 and 6 )
In Chinese number system
486 =
(Separate symbol for 400, 80 and 6 )
Since in Greek (Ionic), Indian (Brahmi, Gupta) and Chinese number system there
were symbols for many more numbers, therefore it is really difficult to remember so many
symbols.
(v)
The Hindu and Chinese system of writing numerals in fundamentally different from
that of the ancient Egyptians and Greeks numeral system. The Hindus and Chinese
have special symbols for the individual numbers from one to nine. The Hindu
(Nagari) system is a pure place value system. Only a pure place value system needs
a symbol for the zero.
(vi)
In Babylonian and Mayan number system, any number is expressed by using only
two symbols (as in case of binary system). One may take the opportunity for
comparison of this system with that of well-known binary system.
(vii)
The process of expressing any number in Babylonian and Mayan is time consuming.
(viii) In decimal system we can easily find out the place value of any digit from a number,
but in sexagesimal and vigesimal system it is quite difficult. This is the reason that
the modern number system follows the decimal system.
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Chapter-III
3.10 Contribution of zero to the number system
The symbols of the numbers like one, two, three etc. developed a long time before
the symbol of zero. The idea of “nothing” came into the mind of ancient people but there
was not a specific symbol to indicate zero.
The Babylonians used Sexagesimal number system and they were the first culture to
invent the place value system. We count sixty minutes in an hour in Sexagesimal system.
Already we have discussed that in Babylonian number system there was a gap between the
numbers of unit place and 60 place, 60 place and 602 place and so on. Therefore how they
indicate zero or “nothing”. If the blank places indicate zero then there was confusion like.
∇ ∇∇∇ = 63 or 3604
To avoid this confusion they invented (3000 BC) the symbol to represent zero.
When it was necessary to express zero they used a wedge mark ( or two wedge marks ). For
example
∇ ∇∇∇ = 63
∇∇ ∇∇∇ = 3603
Therefore credit goes to Babylonian who for the first time invented the symbol of
zero and more complex notion of the abstract ideas of “nothingness.”
Mayan invented the symbol of zero in 350 AD. They used the symbol “
”to
indicate zero. In Egyptian, Chinese, Greek and Hindu (Brahmi, Gupta) Number system
there were symbols to represent many more numbers; therefore symbol for zero was not
necessary in these number system. But Babylonian and Mayan used only two symbols to
express any number. Therefore a symbol for zero was necessary for them.
The final independent invention of the zero was in India. Using a vocabulary of
symbolic words to note zero is known from the text entitled the “Lokavibhaga” (458 AD).
Aryabhata (499 AD) used the word “Kha” to represent zero. But in India the symbol of zero
(0) was invented between 700 AD to 1100 AD. Since, in Hindu numeral systems there were
different symbols to indicate one to nine and also a symbol for zero therefore most historian
agree that the development of decimal place value system of numeration originated from
Indian subcontinent.
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Chapter-III
The Hindu introduced the zero to the Arabs. Arabs developed the modern day
numerals and passed them, along with zero, to the Europeans in the 12th century. Fibonacci
was one of the main mathematicians who accepted the concepts of zero in Europe. In his
treatise “Liber Abaci” (a tract about the abacus) published in 1202, he described the nine
Indian symbols together with the symbol 0 for zero.
Also, the Hindu introduced the symbol “O” for zero to China also. Two Chinese
scholars Chin Chin Shao and Chu Shih-Chieh used the symbol 0 for a places- based system
in the 12th and 13th centuries respectively.
Since in ancient Egypt, Greek Hindu (Brahmi, Gupta) and Chinese number systems
there were symbols for many more numbers, therefore without help of the symbol zero they
expressed any number easily. But it was really difficult to remember so many symbols.
Also, the symbols to represent numbers were very laborious to write. But invention of the
symbol for zero makes it easy. Now in the decimal number system we can be expressed any
number with the help of only ten symbols (digits). The path that leads to the discovery of
“O” lies only in the most advanced type of number system. This number system is called
“positional” because the value of a character depends on its position. Our modern way of
counting is positional (decimal system). For example the figure “6” has a different value in
614 and 263 determined by its position.