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Counting rods
(算筹)
were used in traditional Chinese mathematics calculations
from around 500 B. C. to 1500 A.D. According to the Book
of Music and the Calendar in the History of Han
Dynasty (《隋书·律历志》)
,
a counting rod was a round bamboo stick 1 fen (about
2.5 mm) in diameter and 6 cun (about 15cm) in length.
There were two ways to place the rods representing 1, 2, 3,
4, 5, 6, 7, 8, 9, namely, vertically and horizontally:
1
2 3
4 5
6 7
8 9
1
2 3
4 5
6 7
8
9
When the rods were used to
represent integers with more than two figures, these two
ways alternate between the two neighboring positions. The
Master Sun’s Mathematical Manual
(《孙子算经》)
in the 4th century
says:
“In making calculations we must
first know the positions. The units are vertical and the
tens horizontal, the hundreds stand while the thousands lie
down; thousands and tens therefore look the same, as also
the ten thousands and the hundreds….”
Thus, for
example, the number 8426 appeared as  . Before the 8th
century, the place where zero is required was always left
vacant. For example, the number 3054 appeared as  and 2005  , as you see in
the logo. In ancient China, the place numeration system was
used as early as the turning of the Spring and Autumn
(
春秋,
770-476 B.C. ).
In some works
on music in the Song
(
宋,
960-1279 )
and Yuan (
元,
1271-1368 )
Dynasties, zero was denoted by  . The circular
symbol for zero, namely  , first appeared
in the Daming Calendar (
大明历 ) in the Jin
Dynasty
(
金,
1115-1234 ) and it was
commonly used in the mathematical works in the Song and Yuan
Dynasties. For example, the number 105800000 appeared as  in Qin
Jiushao’s ( 秦九韶 )
Mathematical
Treatises in Nine Sections
(数书九章,
1247).
With this symbol, the number 2005 can be written as  .
In ancient
China, as old as the place numeration system is the magic
square of the 3rd order, the Luo Shu
Diagram
(
洛书 )
or the Nine Halls (
九宫图
),
which is the earliest one in the world. A magic square
consists of
n2
consecutive integers from 1 upwards arranged in the form of
a square, so that the sum of the numbers in every row,
column, and diagonal is equal to the 1/2n
(n2+1).
Mentions of
the River Diagram
(
河图 ) and Luo Shu
Diagram (
洛书 )
are made in many ancient Chinese books, including Book of
Changes (
易经 ),
Conversations and Discourses of Confucius (
论语),
Mo Tzu ( 墨子 ),
Historical Classic (
尚书 ),
A History of Former Han Dynasty ( 汉书
),
etc., all of which say that a dragon-horse came out of the
Yellow River (
黄河 )
with a diagram on its back and a turtle emerged from the
River Luo (
洛河 )
with some writings on its back, without any indication of
what the diagram and the wrings are.
Luo
Shu Diagram (Nine Halls)
The Nine
Halls was well known as early as the Former Han Dynasty.
The Record of Rites compiled by Dai De
(
大戴礼记 ) speaks of the
nine halls of the Ming Tang (
明堂 ),
where ancient emperors carried out the sacrificial rites, in
the order “two, nine, four; seven, five, three; six, one,
eight”. In his commentary of Hsü Yue’s (
徐岳 )
Memoire on some Traditions of Mathematical Art (
数术记遗 )
, Zhen Luan (
甄鸾
)
explains that “two and four are the shoulders, six and eight
the feet, there is three on the left and seven on the right;
it wears nine on its head and is shod with one, while five
is in the middle.” This plainly describes the arrangement of
the Nine Halls.
In the Song
Dynasty, the Luo Shu diagram was identified with the
Nine Halls by such scholar as Cai Yuanding
(
蔡元定 ),
whose proposal was approved by Zhu Xi (
朱熹
).
From then on, the form of the Luo Shu diagram was
established.
The
simplicity, harmony and symmetry incorporated in the Luo Shu
Diagram reflects the fascination of traditional Chinese
culture, and has long been the aims pursued by ancient
Chinese mathematicians.
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