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Marie Trudelle
Professor Rosentrater
Fundamentals of Math I
14 November 2013
The Formation of Numeral Systems
Early primal civilization developed a means to keep track of days, food supply,
village members, deaths, etc. These basic principles evolved into impressive numeration
system with various rules and standards. Throughout this paper, the Chinese and Greek
numeration system will be analyzed in depth.
During an extensive archeological dig in Xiao dun, China, during the year 1899,
remnants of the ancient Chinese numeration system were revealed.1 Markings of different
numeral characters were inscribed into various mediums such as turtle shells, bones, and
sticks.2 The data recorded, containing particulars about calendar dates, quantity of food
stored, or amount of individuals killed in war proved to be valuable for the ancient
society. Two structures of the Chinese numeration system were developed—the written
character arrangement and the rod numeral method. The written character system came
first and was followed by the rod approach in the 6th century B.C.3 Both procedures are
founded in the base ten decimal place-notion system.4
Chinese numerals are characters that express numbers in print. Comparable to
how numbers are sometimes written out in English, such as one hundred, the Chinese
J.J O’Connor and E.F. Robertson. Chinese Numerals, (January 2004).
J.J O’Connor and E.F. Robertson. Chinese Numerals, (January 2004).
3 Leo Lusk. History of Numbers. (2003).
4 Leo Lusk. History of Numbers. (2003).
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2
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characters ‘spell out’ the number.5 The first ten numbers would be pronounced, “yī èr sān
sì wǔ liù qī bā jiǔ shí.”6 Chinese symbols are printed vertically with the highest place
value at the top.7 Below is a Chinese numeral chart expressing Chinese characters and
numbers respectively.
J.J. O’Connor and E.F. Robertson. Chinese
Numerals. (2004)
The Chinese system is a slightly altered multiplicative procedure. Characters are
to be read in pairs—one represents a number 1-9 while the lower number is always a
power of ten.8 For example, 500 is illustrated by the character five and one hundred.
While this technique continued into the thousands (8,000 was eight and one thousand),
there are no signs of numbers greater than 30,000.9 Perhaps archeologists have been
unable to locate evidence of large numbers or perhaps the Chinese never had a need for
such numbers.
While this system was mostly based off the multiplicative theory, there are two
sectors that reveal refinement. The first alteration includes the one unit—it uses a single
5J.J
O’Connor and E.F. Robertson. Chinese Numerals, (January 2004).
Standard Pronunciation. Chinese Phrase- 1 to 10.
7 Leo Lusk. History of Numbers. (2003).
8 Leo Lusk. History of Numbers. (2003).
9 J.J O’Connor and E.F. Robertson. Chinese Numerals, (January 2004).
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sign as opposed to a set of two.10 Therefore, the multiplier, a number one through nine, is
recorded; however, the power of the base ten isn’t included.11 The second difference
states, “…in the pair indicating 10s, if the multiplier is 1, then that multiplier is omitted.
Just the symbol for 10 is written.”12 The written system also lacked the notion of zero.13
The Chinese numeral system was clear and used by members of society everyday. Hence,
‘basic’ calculators, called counting rods, were invented in the 6th century BC.14
The Chinese used bamboo rods and sticks to constitute numbers 1-9.15 The idea
of rods was simplified into small counting boards, which were used like calculators. The
boards were not only manipulated by the Chinese but also utilized by foreigners passing
through the Silk Road.16 The counting boards revealed a number that was, “…formed in a
row with the units placed in the right most column, the tens in the next column to the left,
the hundreds in the next column to the left etc.”17 The rod system was simple and didn’t
need to have a zero notion because the column would be left empty.18 Below is an image
of a Chinese rod board and its natural place values.
10Addison-Wesley.
Historical Numeration Systems. (2004)
Addison-Wesley. Historical Numeration Systems (2004)
12 Addison-Wesley. Historical Numeration Systems. (2004)
13 http://www.pantaneto.co.uk/issue5/arsham.htm
14 Leo Lusk. History of Numbers. (2003).
15 Leo Lusk. History of Numbers. (2003).
16 Leo Lusk. History of Numbers. (2003).
17 J.J O’Connor and E.F. Robertson. Chinese Numerals. (January 2004).
18J.J O’Connor and E.F. Robertson. Chinese Numerals. (January 2004).
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Peggy Cheung. Development of Mathematics in Ancient China.
The only confusion with the counting boards was misreading numbers because
sometimes they were horizontal while other times vertical. The Chinese solved this issue
by numbers in the units column were represented vertically and the numbers in the tens
column were expressed horizontally.19 They proceeded to alternate between horizontal
and vertical lines in higher number powers.20 In the image below the lower row would
depict the one’s column and the top row reveals the tens column.
J.J. O’Connor and E.F. Robertson. Chinese Numerals. (2004)
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20
J.J O’Connor and E.F. Robertson. Chinese Numerals. (January 2004).
J.J O’Connor and E.F. Robertson. Chinese Numerals. (January 2004).
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The number 45698, for example, would be represented as:
In conclusion, the Chinese and Hindu-Arabic numeral system are incredibly
similar. Both adopted a base ten decimal system as well as a place-notion system. The
Chinese similarly developed nine symbols to represent 1-9 and then use a combination of
those symbols to represent the remaining numbers.21Another similarity is the use of an
empty set for zero. The Hindu Arabic system invented a symbol called the sunya, which
ultimately conveys an empty set.22 The Hindu Arabic system was invented around 800
BC while the Chinese was formulated in 480 BC.23
Next, the Greek numeration system will be analyzed and highlight progress as
well as areas of confusion. The Phoenicians invented the base 10 system in the year 900
BC.24 It first began with the Attic System, which was comparable to a tally system. This
system became troublesome because the numbers were lengthy and individuals were
forced to carve numbers possessing more than thirteen characters into stone.25 The
Phoenicians then adopted a twenty-four letter alphabetic to represent various numerals.26
Unfortunately; however, in order for the Greek number system to function properly,
twenty-seven letters were necessary. Therefore, three ancient symbols that were used in
the old alphabet were reintroduced and represented the numbers six, ninety, and nine
Lam Lay-Yong. Linkages: Exploring the Similarities Between the Chinese Rod
Numeral System and Our Numeral System. (1987).
22 Dr. Marcel Finan. The Hindu Arabic System. (2012)
23 Lam Lay-Yong. Linkages: Exploring the Similarities Between the Chinese Rod
Numeral System and Our Numeral System. (1987).
24 Carey Eskridge Lybarder, Number Systems.
25M.Lahanas, R. Andews, and K. Hysick. The Greek Number System.
26 Carey Eskridge Lybarder, Number Systems.
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hundred.27 These old letter’s are revealed below. Digamma, xi, and sampi; however, are
no longer used in the Greek numeration system.28 Therefore, commas are now placed
before the letters when symbolizing numbers from 1,000 to 999, 999.29
(Represents 6 and its Greek name is digamma).
(Represents 60 and its Greek name is xi)
(Represents 900 and its Greek name is sampi)
Similar to the Chinese, the Greeks gave each number, one through nine, a
different symbol. They also gave higher numbers (10, 20, 30, 40… 100, 200, 300…) new
symbols.30 While this prevented people from mixing up numbers, the society only
invented a finite amount of numbers. Below is a chart of various Greek letters and their
value.
Carey Eskridge Lybarder, Number Systems.
R. Andews, and K. Hysick. The Greek Number System.
29M.Lahanas, R. Andews, and K. Hysick. The Greek Number System.
30 Carey Eskridge Lybarder, Number Systems.
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28M.Lahanas,
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J.J. O’Connor and E.F. Robertson. Chinese Numerals. (2004)
Although, the Greeks simplified their system from the original Attic method, it is
incredibly challenging to perform arithmetic. One must need to comprehend a large sum
of Greek symbols in order to perform basic addition such as iota + iota = kappa.31
Multiplication was even more complex and individuals were forced to use the aid of an
abacus.32
While the Babylonian system was established before the Greek, they opted to not
use a positional number system.33 Their numeration system was based on the additive
principle, so the number 241 would be composed by 200 + 40 + 1 or
Jo Edkins. Greek Numbers. (2006)
Jo Edkins. Greek Numbers. (2006)
33 Carey Eskridge Lybarder, Number Systems.
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. Hence, the Greeks lacked a zero notion.34It is noted that, “Greek
mathematicians did not need to name their numbers since they worked with numbers as
lengths of lines.”35
While the Chinese and Greek systems are different they possess a handful of
similar qualities. Both numeration systems are decimal base ten structures. They contain
distinct symbols for numbers 1-9; however, the Greek system also invented new symbols
to exhibit the tens, hundreds, etc. value. The Chinese system is remarkably easy to use
and was implemented in the everyday life of merchants, scribes, and army officials.
While it is mostly multiplicative is does possess a few additive properties. On the other
hand, the Greek system is entirely additive but lacks a place value system. Both the
Chinese and Greek have omitted the notion of zero, allowing other civilizations to ponder
the abstract idea.
Works Cited
34Carey
35
Eskridge Lybarder, Number Systems,
J.J O’Connor and E.F. Robertson. A History of Zero. (November 2000).
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Addison-Wesley. Historical Numeration Systems. http://wwwrohan.sdsu.edu/~ituba/math303s08/mathideas/mmi10_04_01.pdf
Carey Eskridge Lybarder, Number Systems,
http://www.math.wichita.edu/history/topics/num-sys.html#greek
Dr. Marcel Finan. The Hindu Arabic System.
http://faculty.atu.edu/mfinan/2033/section6.pdf (2012)
J.J O’Connor and E.F. Robertson. Chinese Numerals, http://www-history.mcs.st
and.ac.uk/HistTopics/Chinese_numerals.html (January 2004).
Jo Edkins. Greek Numbers. http://gwydir.demon.co.uk/jo/numbers/greek/ (2006)
Lam Lay-Yong. Linkages: Exploring the Similarities Between the Chinese Rod Numeral
System and Our Numeral System.
http://link.springer.com/article/10.1007/BF00417009#page-1 (1987).
Leo Lusk. History of Numbers.
http://mathematics.gulfcoast.edu/mgf1107ll/Chap1Sec7.htm (2003).
M.Lahanas, R. Andews, and K. Hysick. The Greek Number System.
Rwww.merkinms.org/.../18%20Greek%20Number%20System%20PP%20...
Peggy Cheung. Development of Mathematics in Ancient China.
http://www.sfcc.edu.hk/academic_subjects/mathematics/web/1999_2000_projects
/Peggy%20Cheung/Mathematica/MATHChina.htm (Image).
Standard Pronunciation. Chinese Phrase- 1 to 10.
http://www.standardmandarin.com/chinese-phrase/1-to-10