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Chinese Mathematics History and MathematiciansLiu Hui and Zu ChongZhi
IDEA Publics Schools, Oct. 23, 2011
Chinese Mathematics History
This is a very brief review of ancient Chinese History in terms of the various dynasties and where
possible some brief descriptive phrases. Chinese History is a fascinating and diverse set of histories, and
worthy of an extended study. But for purposes of this article, it is enormously condensed into a listing of
dynasties.
1. XIA DYNASTY 夏朝 (Sage Emperors Yao and Shun, Yu the Great; 2205 - 2197 BCE)
2. SHANG DYNASTY 商朝-- Northern China (1700 - 1000 BCE)
Aristocratic, brilliant, luxurious, and savage (human/animal sacrifices). “Emerged with
relative suddenness”. Apparently stable for 700 years. Mature and technically perfect bronze
culture (from the near east?). Arrived following the fall of the Sumerian Civilization, circa 2000
BCE.
3. ZHOU DYNASTY 周朝-- Northern China (1000 /772/ 481 BCE: Western/Eastern Chou) King
Wu downed Di-xin, first 200 years the golden Age (King Wen=culture);
Mandate of Heaven, controlled press, obedience to Gods (see 600 B.C.E.)
4. SPRINGS AND AUTUMNS period 春秋時代-- Southern China (772 - 481 BCE)
Eight periods, Barbaric, life and limb cheap, philosophy versus war, i.e. Lao-Tzu (604-531
BCE), Tao de Ching (道德經), Confucius (580?-479 BCE)
5.WARRING STATES period 戰國時代(481 - 221 BCE)
Fragmentation of north into Yan, Zhao, and Wei (480-453-340); rise of Qin and Chu (340280), Qi neutral until overrun by Qin (284-221)
6. QIN DYNASTY 秦朝 (700 - 481 - 221 - 206 BCE)
“Relying on the magical powers of Heaven and Earth”, cyclical rise to power, book burning
(213), intellectuals persecuted (211), 1st emperor dies (210).
7. FORMER HAN DYNASTY 前漢 (202 BCE - 220 CE)
Tibetan Kingdom formed (150 BCE) (expansionist until 850 CE). Korea annexed 擴張版圖
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(108), conquers central Asia (101), Wang Mang (9-26 AD), invades Mongolia (91), reconquers
central Asia (c. 140), Tibetans become threatening (141), Eunuchs hold real power (175), partition
of empire into three parts
8. THREE KINGDOMS (220 - 280 CE) -- Kingdoms of Wei, Wu and Shu-Han WESTERN JIN
(280 - 316 CE)
Reunification of China until Barbarian invasion forces retreat to Yangtze.
9. NORTHERN AND SOUTHERN DYNASTIES 魏晉南北朝 (316 - 590 CE)
Southern Dynasty: Liu-Song (420-478), Qi (479-501), Liang (502-556) NORTHERN
CHOU (557 - 589 CE)
Founded (557), north China unified (577), brought into Sui Dynasty 隋朝(589) SUI
DYNASTY (581 - 618 CE)
Generalized uprisings bring Tang to power
10. TANG DYNASTY 唐朝 (600 - 902 CE) 盛唐
Central Asia, Korea and Manchuria come under Chinese control by 690, stretching as far as
modern Iran. Arab victory in 751. Tibetan Kingdom attacks on Gansu corridor from 763 (after
breaking alliance with Chinese) until decisive victory in 791, Tibet’s military expansion of Tibet
until 850
11. FIVE DYNASTIES/TEN KINGDOMS 五胡十國 (902 - 960) Founded by commanders of the
military regions
12. NORTHERN SONG DYNASTY 北宋 (960 - 1126 CE)
Struggle for influence between conservatives and reformers (1067-1100), The Chin
(Jurcheds) take over north China and Yellow River
13. SOUTHERN SONG DYNASTY 南宋 (1135 - 1270 CE)
After retreating to Yangtze, form capital at Hangchow, sign peace treaty with Chin (1142),
invade Chin after great victory (1162) (Chin attacking north)
14. JIN (RUZHEN女真人) 金 (1126 - 1225 CE) YUAN 元朝 (MONGOL蒙古人) DYNASTY
(1225 - 1368 CE)
Seize Chin capital (1225), allies with Song to destroy Chin Empire (1234), and takes over the
whole of China, the last Sung Emperor suicide (1279)
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14. MING DYNASTY 明朝 (1368 - 1644 CE) Construction of Great Wall 修築長城 (1470-80)
15. QING 清朝 (MANCHU滿洲人) DYNASTY (1644 - 1911 CE)
Occupation of Tibet (1731)
16. REPUBLIC OF CHINA (1912 - 1945 CE)
Kuomintang (1926), Civil War (1927-35), WWII (1935-1945), Taiwan (1949 to Present),
17. PEOPLE’S REPUBLIC OF CHINA (1949 CE - Present) [1]
Chinese Mathematicians
Even as mathematical developments in the ancient Greek world were beginning to falter during
the final centuries BC, the burgeoning trade empire of China was leading Chinese mathematics to evergreater achievements.
The simple but efficient of ancient Chinese numbering system, which developed way back to at
least the 2nd millennium BC, used small bamboo rods arranged to represent the numbers 1 to 9, which
were then places in columns representing units, tens, hundreds, thousands, etc. It was therefore a decimal
place value system, very similar to the one we use today - indeed it was the first number system, adopted
by the Chinese over a thousand years before it was adopted in the West - and it made even quite complex
calculations very quick and easy.
Written numbers, however, employed the slightly less efficient system of using a different
symbol for tens, hundreds, thousands, etc. This was largely because there was no concept or symbol of
zero, and it had the effect of limiting the usefulness of the written number in Chinese.
Ancient Chinese number system [2]
The use of the abacus is often thought of as a Chinese idea, although some type of abacus was in
use in Mesopotamia, Egypt and Greece, probably much earlier than in China (the first Chinese abacus, or
“suanpan 算盤”, we know of dates to about the 2nd Century BC).
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Lo Shu magic square, with its traditional graphical representation [2]
There was a pervasive fascination with numbers and mathematical patterns in ancient China, and
different numbers were believed to have cosmic significance. In particular, magic squares - squares of
numbers where each row, column and diagonal added up to the same total - were regarded as having great
spiritual and religious significance
The Lo Shu （六書）Square, an order three square where each row, column and diagonal adds
up to 15, is perhaps the earliest of these, dating back to around 650 BC (the legend of Emperor Yu’s
discovery of the square on the back of a turtle is set as taking place in about 2800 BC). But soon, bigger
magic squares were being constructed, with even greater magical and mathematical powers, culminating
in the elaborate magic squares, circles and triangles of Yang Hui （楊輝）in the 13th Century (Yang Hui
also produced a triangular representation of binomial coefficients identical to the later Pascal’s Triangle,
and was perhaps the first to use decimal fractions in the modern form).
Early Chinese method of solving equations [2]
But the main thrust of Chinese mathematics developed in response to the empire’s growing need
for mathematically competent administrators. A textbook called “Jiuzhang Suanshu” or “Nine Chapters
on the Mathematical Art” (written over a period of time from about 200 BC onwards, probably by a
variety of authors) became an important tool in the education of such a civil service, covering hundreds of
problems in practical areas such as trade, taxation, engineering and the payment of wages.
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It was particularly important as a guide to how to solve equations - the deduction of an unknown
number from other known information - using a sophisticated matrix-based method that did not appear in
the West until Carl Friedrich Gauss re-discovered it at the beginning of the 19th Century (and which is
now known as Gaussian elimination).
Among the greatest mathematicians of ancient China was Liu Hui （劉徽）, who produced a
detailed commentary on the “Nine Chapters 九章算術” in 263 AD, was one of the first mathematicians
known to leave roots unevaluated, giving more exact results instead of approximations. By an
approximation using a regular polygon with 192 sides, he also formulated an algorithm that calculated the
value of π as 3.14159 (correct to five decimal places), as well as developing a very early forms of both
integral and differential calculus.
The Chinese Remainder Theorem [2]
The Chinese went on to solve far more complex equations using far larger numbers than those
outlined in the “Nine Chapters”, though. They also started to pursue more abstract mathematical problems
(although usually couched in rather artificial practical terms), including what has become known as the
Chinese Remainder Theorem. This uses the remainders after dividing an unknown number by a
succession of smaller numbers, such as 3, 5 and 7, in order to calculate the smallest value of the unknown
number. A technique for solving such problems, initially posed by Sun Tzu in the 3rd Century AD and
considered one of the jewels of mathematics, was being used to measure planetary movements by Chinese
astronomers in the 6th Century AD, and even today it has practical uses, such as in Internet cryptography.
By the 13th Century, the Golden Age of Chinese mathematics, there were over 30 prestigious
mathematics schools scattered across China. Perhaps the most brilliant Chinese mathematician of this
time was Qin Jiushao （秦九韶）, a rather violent and corrupt imperial administrator and warrior, who
explored solutions to quadratic and even cubic equations using a method of repeated approximations very
similar to that later devised in the West by Sir Isaac Newton in the 17th Century. Qin even extended his
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technique to solve (albeit approximately) equations involving numbers up to the power of ten,
extraordinarily complex mathematics for its time. [2]
Liu Hui - Mathematician of Cao Wei 曹魏,
This is a Chinese name; the family name is Liu.
Liu Hui (3rd century) was a mathematician of the state of Cao Wei during the Three Kingdoms
period of Chinese history. In 263, he edited and published a book with solutions to mathematical
problems presented in the famous Chinese book of mathematic known as The Nine Chapters on the
Mathematical Art (九章算術).
He was a descendant of Marquis of Zixiang of Han dynasty, corresponding to now Zixiang
township of Shandong province. He completed his commentary to the Nine Chapters in the year 263. He
probably visited Luoyang, and measured the sun's shadow.
Contents of the Nine Chapters are as follows:
1. 方田 Fangtian - Rectangular fields. Areas of fields of various shapes; manipulation of vulgar
|ˈvəlgər| fractions.
2. 粟米 Sumi - Millet and rice. Exchange of commodities at different rates; pricing.
3. 衰分 Cuifen - Proportional distribution. Distribution of commodities and money at proportional
rates.
4. 少廣 Shaoguang - The lesser breadth. Division by mixed numbers; extraction of square and cube
roots; dimensions, area and volume of circle and sphere.
5. 商功 Shanggong - Consultations on works. Volumes of solids of various shapes.
6. 均輸 Junshu - Equitable taxation. More advanced problems on proportion.
7. 盈不足 Yingbuzu - Excess and deficit. Linear problems solved using the principle known later in
the West as the rule of false position.
8. 方程 Fangcheng - The rectangular array. Systems of linear equations, solved by a principle
similar to Gaussian elimination.
9. 勾股 Gougu - Base and altitude. Problems involving the principle known in the West as the
Pythagorean theorem.
Mathematical work
Along with Zu Chongzhi, Liu Hui was known as one of the greatest mathematicians of ancient
China. Liu Hui expressed all of his mathematical results in the form of decimal fractions (using
metrological units), yet the later Yang Hui (c. 1238-1298 AD) expressed his mathematical results in full
decimal expressions.
Liu provided commentary on a mathematical proof identical to the Pythagorean theorem of the
Greek Pythagoras (c. 580 BC-500 BC). Liu Hui called the figure of the drawn diagram for the theorem
the "diagram giving the relations between the hypotenuse and the sum and difference of the other two
sides whereby one can find the unknown from the known". In terms of plane areas and solid figures, Liu
Hui was one of the greatest contributors to 'empirical' solid geometry. For example, he figured out that a
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wedge with rectangular base and both sides sloping could be broken down into a pyramid and a
tetrahedral wedge. He also figured out that a wedge with trapezoid base and both sides sloping could be
made to give two tetrahedral wedges separated by a pyramid.
In his commentaries on the “JiuZhang SuanShu”, he presented (among others things):
An algorithm for calculation of pi (π) in the comments to chapter 1. He calculated pi to
3.141024 < π < 3.142074 with a 192 (= 25 × 6) sided polygon.
Archimedes used circumscribed 96 –sided polygon to obtain inequality, and then used an inscribed
96-gon to obtain inequality. Liu Hiu used only one inscribed 96-gon to obtain his π inequality, and his
results were a bit more accurate than Archimedes'. But he commented that 3.142074 was too large, and
picked the first three digits of π = 3.141024 ~3.14 and put it in fraction form. He later invented a quick
method and obtained π = 3.1416, which he doubled checked with 3072-gon (= 25 × 6), he was quite happy
about this result.
Nine Chapters had used the value 3 for π, but Zhang Heng 張衡(78-139 AD) had previously
estimated pi to the square root of 10.
Gaussian elimination.
Cavalier’s principle to find the volume of a cylinder, although this work was only finished by Zu
Gengzhi. The commentaries often include explanations why some methods work and why others do not.
Although his commentary was a great contribution, some answers had slight errors, which was later
corrected by the Tang mathematician and Taoist believer Li Chunfeng. Liu Hui also presented, in a
separate appendix of 263 AD called “Haidao suanjing” or “The Sea Island Mathematical Manual”,
several problems related to surveying. This book contained many practical problems of geometry,
including the measurement of the heights of Chinese pagoda towers. This smaller work outlined
instructions on how to measure distances and heights with "tall surveyor's poles and horizontal bars fixed
at right angles to them". With this, the following cases are considered in his work:
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The measurement of the height of an island opposed to its sea level and viewed from the sea
The height of a tree on a hill
The size of a city wall viewed at a long distance
The depth of a ravine (using hence-forward cross-bars)
The height of a tower on a plain seen from a hill
The breadth of a river-mouth seen from a distance on land
The depth of a transparent pool
The width of a river as seen from a hill
The size of a city seen from a mountain,
Liu Hui's information about surveying was known to his contemporaries as well. The cartographer
and state minister Pei Xiu (224–271) outlined the advancements of cartography, surveying, and
mathematics up until his time. This included the first use of a rectangular grid and graduated scale for
accurate measurement of distances on representative terrain maps.
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Liu Hui provided commentary on the Nine Chapter's problems involving building canal and river
dykes, giving results for total amount of materials used, the amount of labor needed, the amount of time
needed for construction, etc.
Although translated into English long beforehand, Liu's work was translated into French by Guo
Shuchun, a professor from the Chinese Academy of Sciences, who began in 1985 and took twenty years
to complete his translation. [4]
Brilliant achievements of ancient Chinese mathematics, was the first to create a decimal system
of the country, but also through the yin and yang, gossip first proposed the concept of the binary system
of the country, which are two processes affecting the history and modern life's great creation. Western
Han Dynasty or the early Western Han Dynasty by Zhao Shuang Note "Zhou Bi Suan Jing" arguments
about the Pythagorean Theorem of Pythagoras and Greece (Pythagoras) argument contemporaries, but
also each other without contact, invented their own . Three Kingdoms, the great mathematician Liu Hui's
"Nine Chapters on Arithmetic Note" seeking to create a polyhedral volume of critical theory - Liu Hui
principle, and limit infinitesimal division thinking that the area of circle formula, and seek to create a
scientific approximation of pi program to calculate the polygon's area is 192, find the approximate value
of pi is 3.14, with scores expressed as 157/50 and 3927/1250.
劉徽“割圓術”示意圖 [4]
Liu Hui and other mathematicians in creative work, to explore, based on another great
mathematician Zu (429-500) came into being. Zu, a scholar born in Southern family, childhood studious,
creative, diligent practice, 25-year-old into the Waring School in the province of academic research, and
later a senior official in the hands of any of the public and the government to join the army, to be
sufficient time for scientific research in astronomy, calendar, mathematics and mechanical engineering,
etc. have made significant achievements. Zu ancestral Zu's's mathematical achievements and his son are
concentrated in their math book, "augmented surgery," in this book, is listed as "considered by the 10
book" among the students in the Tang Dynasty and North Korea, Japanese students arithmetic textbooks ,
but unfortunately has been lost. Zu now known mathematical achievements are other writings left
incomplete records, mainly in pi, the ball from the cubic volume and open with three aspects. [4]
Zu ChongZhi and π
Calculating the value of pi is not only a very important in mathematics, but also very difficult to
study. Many ancient Chinese mathematicians are committed to the calculation of pi, and the 5th century
AD, Zu's achievements can be said that a leap calculated pi. Zu is a great ancient Chinese mathematician
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and astronomer. Zu was born in 429 AD in JienKang (now Nanjing), his family history is a study of the
astronomical calendar, and he was in contact with mathematical and astronomical knowledge, AD 464, at
the Zu 35 years old, he began to calculate the pi.
In ancient China, people from the practice recognized circumference of a circle is "a circular path
rather than on Wednesday," that is, the circumference of a circle is more than three times the diameter,
but how much more divergent views. In Zu, the Chinese mathematician Liu Hui made a scientific method
of calculating pi - "cut round operation," with inscribed regular polygon to approximate the circumference
of the perimeter, using this method, Liu Hui calculated pi to 4 digits after the decimal point. Zu on the
basis of previous method, after assiduously, repeated calculation, the calculated pi to 7 digits after the
decimal point (ie, between 3.1415926 and 3.1415927), and the scores obtained in the form of an
approximation of pi. How Zu exactly what methods that he used to reach this result, it’s undiscovered. If
we imagine him to use Liu Hui's "cut round operation" means to seek, then we would calculate the
polygon inscribed 16000, which requires much time and effort of great work!
Zu calculated pi with 7 digits of decimals. Some foreign mathematicians were also to obtain the
same result, but it was a thousand years later. To memorize Zu's outstanding contribution to mathematical
historians some foreign mathematicians suggested that pi π is called the "ancestral rate." In addition to
calculating pi achievements, Zu together with his son, with a clever solution to the sphere volume
calculations. In the West known as the "Cavalieri" principle by the Italian mathematician Cavalieri, but it
was a thousand years after the Zu was discovered. To memorize Zu and his son’s's significant
contribution to this principle, the principle of mathematics, also known as the "ancestral Zu's principles."
In fact, Chinese mathematical achievement is above the world before 14 centuries, China was the world's
most developed countries of mathematics. Such as the Pythagorean theorem in geometry, in the early
Chinese mathematical treatise 'Zhou Bi Suan Jing' (about the 2nd century BC, into a book) that is
discussed; a book in the 1st century BC Another important mathematical treatise. "Nine Chapters on
arithmetic, "the history of mathematics in the world, first to propose the concept of negative numbers and
add and subtract positive and negative rules; the Chinese already have the solutions for 10th power
equations in the 13th century, and European mathematicians was only found the solution of cubic
equations until the 16th century. [5]
Reference
 Chen, Stephen. "Changing Faces: Unveiling a Masterpiece of Ancient Logical Thinking." South
China Morning Post, Sunday, January 28, 2007. Guo, Shuchun, "Liu Hui"
(http://203.72.198.245/web/Content.asp?ID=43261&Query=1) . Encyclopedia of China
(Mathematics Edition), 1st Ed.
 Hsu, Mei-ling. "The Qin Maps: A Clue to Later Chinese Cartographic Development," Imago Mundi
(Volume 45, 1993): 90-100. Needham, Joseph & C. Cullen (Eds.) (1959). Science and
Civilization in China: Volume III, section 19. Cambridge University Press. ISBN 0-521-05801-5.
 Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the
Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd. Needham, Joseph (1986).
Science and Civilization in China: Volume 4, Physics and Physical Technology, Part 3, Civil
Engineering and Nautics. Taipei: Caves Books Ltd.
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Ho Peng Yoke: Liu Hui, Dictionary of Scientific Biography Yoshio Mikami: Development of
Mathematics in China and Japan. Crossley, J.M et al., The Logic of Liu Hui and Euclid,
Philosophy and History of Science, vol 3, No 1, 1994 this bo chen
http://www.halexandria.org/dward208.htm [1]
http://www.storyofmathematics.com/chinese.html [2]
http://en.wikipedia.org/wiki/Liu_Hui [3]
http://big5.showchina.org:81/gate/big5/www.showchina.org/rwzgxl/ [4]
http://big5.cri.cn/gate/big5/gb.cri.cn/chinaabc/chapter17 [5]
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