Why Question Report:

1
Why Question Report (41)What about Chinese
Mathematicians?Revised to What is the history
of Early Chinese Mathematics

• Mark Hovis
• History of Math
• July 16, 2008

2
Findings

• Chinese Mathematics developed in isolation from
  the rest of the world
• due to culture, geography, and politics.
• Much of early Chinese mathematics was produced
  because of the need to make calculations for
  constructing the calendar and predicting
  positions of the heavenly bodies. The Chinese
  word 'chouren' refers to both mathematicians
  and astronomers showing the close link between
  the two areas.
• When the country was conquered by foreign
  invaders, they were assimilated into the Chinese
  culture rather than changing the culture to
  their own. As a consequence, there was a
  continuous cultural development in China from
  around 1000 BC.
• There are periods of rapid advance, periods when
  a certain level was maintained, and periods of
  decline. This was due to the constant fighting
  and revision of the policies by succeeding
  Emperors .

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• Earliest artifacts were unearthed in 1984 Suan
  shu shu (A Book on Arithmetic) dating from
  around 180 BCE. It is a book written on bamboo
  strips and was found near Jiangling in Hubei
  province.
• before 100 BCE - tortoise shells.
• Chinese number system used number boards with
  rods (vertical horizontal)
• The system had place values and zero. (Some
  researchers believe that it even allowed for
  decimal places.)
• 100 BC to 100 AD - The oldest complete surviving
  text is the Zhoubi suanjing (Zhou Shadow Gauge
  Manual) It is an astronomy text, showing how to
  measure the positions of the heavenly bodies
  using shadow gauges which are also called
  gnomons.
• It contains important sections on mathematics.
  It contains a statement of the Gougu rule (the
  Chinese version of the Pythagorean theorem) and
  applies it to surveying, astronomy, and other
  topics. It is widely accepted that the work also
  contains a proof of Pythagorean theorem

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• 160 to about 227 -The most famous Chinese
  mathematics book of all time is the Jiuzhang
  Suanshu or, as it is more commonly called, the
  Nine Chapters of Mathematical Art. This important
  work came to dominate mathematical development
  and style for 1500 years. Many later developments
  came through commentaries on this text.
• Approx. 263 Liu Hui wrote his commentary on the
  Jiuzhang Suanshu or the Nine Chapters of
  Mathematical Art. Gave a more mathematical
  approach than earlier Chinese texts, providing
  principles on which his calculations are based.
  He found approximations to using regular polygons
  with 3 2n sides inscribed in a circle. His best
  approximation of was 3.14159 (pi), which he
  achieved from a regular polygon of 3072 sides. It
  is clear that he understood iterative processes
  and the notion of a limit. Gougu rule to
  calculate heights of objects and distances to
  objects which cannot be measured directly

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• 622 - The beginnings of Chinese algebra is seen
  in the work of Wang Xiaotong.
• He wrote the Jigu Suanjing (Continuation of
  Ancient Mathematics), a text with only 20
  problems which later became one of the Ten
  Classics. He solved cubic equations by
  extending an algorithm for finding cube roots.
  His work is seen as a first step towards the
  "tian yuan" or "coefficient array method".
• 6th thru 9th century - mathematics was taught as
  part of the course for the civil service
  examinations. Li Chungfeng (602 - 670) was
  appointed as the editor-in-chief for a collection
  of mathematical treatises to be used for such a
  course. The collection is now called the Ten
  Classics, a name given to them in 1084.
• 10th thru 12th century - very Little change
  stagnant period
• 1050 Jia Xian developed the first understanding
  of Pascal's triangle. Jia Xian is aware of the
  expansion of (a b)n and gives a table of the
  resulting binomial coefficients in the form of
  Pascal's triangle. Jia Xian appears to have
  calculated the binomial coefficients up to n 6
  and gave an accompanying table similar to
  Pascal's triangle which records the coefficients
  up to the row 1 6 15 20 15 6 1

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• 13th century Golden Age
• 1247 Qin Jiushao wrote his famous mathematical
  treatise Shushu Jiuzhang (Mathematical Treatise
  in Nine Sections). He was the first of the great
  thirteenth century Chinese mathematicians. This
  was a period of major progress during which
  mathematics reached new heights. The treatise
  contains remarkable work on the Chinese remainder
  theorem, gives an equation whose coefficients
  are variables and, among other results, Herons
  formula for the area of a triangle. Equations up
  to degree ten are solved using the
  Ruffini-Horner method.
• 1248 - Li Zhi (also called Li Yeh) was the next
  of the great thirteenth century Chinese
  mathematicians. His most famous work is the Ce
  yuan hai jing (Sea mirror of circle
  measurements). It contains the "tian yuan" or
  "coefficient array method" or "method of the
  celestial unknown" which was a method to work
  with polynomial equations. He also wrote Yi gu
  yan duan (New steps in computation) in 1259
  which is a more elementary work containing
  geometric problems solved by algebra.
• 1261 - Yang Hui (about 1238 - about 1298). He
  wrote the Xiangjie jiuzhang suanfa (Detailed
  analysis of the mathematical rules in theNine
  Chapters and their reclassifications), and his
  other works were collected into the Yang Hui
  suanfa (Yang Hui s methods of computation) which
  appeared in 1275. He described multiplication,
  division, root-extraction, quadratic and
  simultaneous equations, series, computations of
  areas of a rectangle, a trapezium, a circle, and
  other figures. He also gave a wonderful account
  of magic squares and magic circles.

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• 1280 Guo Shoujing produced the Shou shi li
  (Works and Days Calendar), worked on spherical
  trigonometry, and solved equations using the
  Ruffinni-Horner numerical method. He also
  developed a cubic interpolation formula
  tabulating differences of the accumulated
  difference as in Newtons forward difference
  interpolation method.
• 1299 Zhu Shijie (about 1260 - about 1320) who
  wrote the Suanxue qimeng (Introduction to
  mathematical studies) published in 1299, and the
  Siyuan yujian (True reflections of the four
  unknowns) published in 1303. He used an
  extension of the "coefficient array method" or
  "method of the celestial unknown" to handle
  polynomials with up to four unknowns. He also
  gave many results on sums of series. This
  represents a high point in ancient Chinese
  mathematics.
• 14th thru 17th century - Little change
  stagnant period

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• Other Questions
• Was the development of Chinese mathematics
  influenced by other
• mathematicians in other parts of the world and
  if so, when did this occur and how much
  influence was there?
• Connections
• Development of Astronomical investigations,
  Solving problems of the calendar, trade, land
  measurement, architecture, government records
  and taxes
• Why I picked this question
• I wanted to know why Chinese mathematics is
  regularly ignored.
• China is known as a cradle of civilization,
  with many highly
• developed areas of technology, philosophy, etc.
• (The Abacus has been around a long time.)
• Why dont we hear more about early Chinese
  mathematics?

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CHRONOLGY OF CHINESEDYNASTIES

• 400 BCE 1911 AD

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(No Transcript)
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Chinese Numbering System

• Chinese Numerals
• Now the numbers from 1 to 9 had to be formed from
  the rods and a fairly natural way was found.
• Here are two possible representations

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The biggest problem with this notation was that
it could lead to possible confusion. What was
? It could be 3, or 21, or 12, or even 111. Rods
moving slightly along the row, or not being
placed centrally in the squares, would lead to
the incorrect number being represented.

• The Chinese adopted a clever way to avoid this
  problem. They used both forms of the numbers
  given in the above illustration. In the units
  column they used the form in the lower row, while
  in the tens column they used the form in the
  upper row, continuing alternately. For example
  1234 is represented on the counting board by
•   and 45698 by  

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ZeroThere was still no need for a zero on the
counting board for a square was simply left
blank. The alternating forms of the numbers again
helped to show that there was indeed a space.For
example 60390 would be represented as 
14

• Ancient arithmetic texts described how to perform
  arithmetic operations on the counting board. For
  example Sun Zi, in the first chapter of the Sunzi
  suanjing (Sun Zi's Mathematical Manual), gives
  instructions on using counting rods to multiply,
  divide, and compute square roots.

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Nine Chapters on the Mathematical Art

• The Jiuzhang suanshu or Nine Chapters on the
  Mathematical Art is a practical handbook of
  mathematics consisting of 246 problems intended
  to provide methods to be used to solve everyday
  problems of engineering, surveying, trade, and
  taxation. It has played a fundamental role in the
  development of mathematics in China, not
  dissimilar to the role of Euclid's Elements in
  the mathematics which developed from the
  foundations set up by the ancient Greeks. There
  is one major difference which we must examine
  right at the start of this article and this is
  the concept of proof.
• It is well known what that Euclid, for example,
  gives rigorous proofs of his results. Failure to
  see similar rigorous proofs in Chinese works such
  as the Nine Chapters on the Mathematical Art led
  to historians believing that the Chinese gave
  formulas without justification. This however is
  simply an example of historians well versed in
  mathematics which is essentially derived from
  Greek mathematics, thinking that Chinese
  mathematics was inferior since it was different.
  Recent work has begun to correct this false
  impression and understand that there are
  different understandings of "proof". For example
  in 8 Chemla shows that Chinese mathematicians
  certainly understood how to give convincing
  arguments that their methodology for solving
  particular problems was correct.

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• Chapter 1 Land Surveying.
• Chapter 2 Millet and Rice.
• Chapter 3 Distribution by Proportion
• Chapter 4 Short Width.
• Chapter 5 Civil Engineering.
• Chapter 6 Fair Distribution of Goods.
• Chapter 7 Excess and Deficit.
• Chapter 8 Calculation by Square Tables.
• Chapter 9 Right angled triangles.

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• Let us leave the problem to whoever can tell the
  truth.

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The Ten Mathematical Classics

• The Sui dynasty was short lived, lasting from 581
  to 618, but it was important in unifying a
  country which had been divided for over 300
  years. Education became important and mathematics
  was taught at the Imperial Academy. The T'ang
  dynasty, which followed the Sui dynasty,
  continued the educational development which had
  already begun and formalised the teaching of
  mathematics.
• Li Chunfeng together with Liang Shu, an expert in
  mathematics from the ministry of education, and
  Wang Zhenru, a teacher from the national
  university and others were ordered by imperial
  decree to annotate the ten mathematical texts
  such as the Wucao suanjing or the Sunzi suanjing.
  Once their task was completed the Emperor Kao-tsu
  ordered that these books be used at the National
  University.
• Although called The Ten Mathematical Classics by
  later writers, there were more than ten books in
  the collection assembled by Li Chunfeng.

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The Ten Mathematical Classics

• Zhoubi suanjing (Zhou Shadow Gauge Manual)
• Jiuzhang suanshu (Nine Chapters on the
  Mathematical Art)
• Haidao suanjing (Sea Island Mathematical Manual)
• Sunzi suanjing (Sun Zi's Mathematical Manual)
• Wucao suanjing (Mathematical Manual of the Five
  Administrative Departments)
• Xiahou Yang suanjing (Xiahou Yang's Mathematical
  Manual)
• Zhang Qiujian suanjing (Zhang Qiujian's
  Mathematical Manual)
• Wujing suanshu (Arithmetic methods in the Five
  Classics)
• Jigu suanjing (Continuation of Ancient
  Mathematics)
• Shushu jiyi (Notes on Traditions of Arithmetic
  Methods)
• Zhui shu (Method of Interpolation)
• Sandeng shu (Art of the Three Degrees Notation
  of Large Numbers)

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• The way that mathematics was taught at the
  Imperial Academy was as follows. Thirty students
  were recruited from the lower ranks of society
  and divided into two classes each of 15 students.
  These two classes followed a different syllabus,
  with one class studying more basic practical
  mathematics while the other was the advanced
  class studying techniques. Teaching was done by
  doctors of mathematics and their assistants. The
  students spent seven years studying mathematics
  from The Ten Mathematical Classics and then took
  the civil service examinations. Examinations were
  held once a year and, as one would expect, they
  were different for the two classes. Questions
  taken from the texts had to be solved, and oral
  examinations were held for the advanced class in
  which the students had to complete sentences
  taken at random from these Ten Mathematical
  Classics. To pass the examinations a score of 6
  out of ten had to be achieved.

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Sources

• Katz, V. 2007
• http//www-history.mcs.st-andrews.ac.uk/HistTopics
  /Chinese_overview.html
• http//www-history.mcs.st-andrews.ac.uk/HistTopics
  /Chinese_numerals.html
• http//www-history.mcs.st-andrews.ac.uk/HistTopics
  /Ten_classics.html
• http//www-history.mcs.st-andrews.ac.uk/HistTopics
  /Nine_chapters.html