Characteristics of Chinese mathematics

1
Characteristics of Chinese mathematics

• Chinese mathematics is characterized by a
  practical tradition. Many scholars held that
  practical appliance prevented Chinese mathematics
  from developing into modern science like Greece
  mathematics that is characterized by a
  theoretical tradition. From the historical
  perspective, Chinese mathematics served the needs
  of the society that was geographically isolated
  from the outer world. The Chinese needed
  controlling the flood prone Yangtze and Yellow
  Rivers. Mathematics helped solve the problem of a
  safe environment in a water-dependent society.

2

• Particularly important was mathematical astronomy
  which attracted attention from rulers who had the
  royal observatory and employed mathematicians,
  astronomers, and astrologers. Mathematicians were
  responsible for establishing the algorithms of
  the calendar-making systems. So, mathematics
  served the needs of mathematical astronomy.
  Calendar-makers were required a high degree of
  precision in prediction. They worked hard at
  improving numerical method, which was the
  principal method of Chinese calendar-making
  systems. It was valued for high accuracy in
  prediction and computation.
• Some scholars think that Chinese mathematicians
  discovered the concept of zero, while others
  express the opinion that they borrowed it from
  the Hindus at the meeting place of the Hindu and
  Chinese cultures in south-east Asia. The Chinese
  symbol for zero developed from the circle to
  denote the empty space in a number. Although it
  is generally accepted that zero was first used by
  the Hindu, the Chinese had ling ( nothing)
  long before the Hindus hat their sunya

3
A brief outline of the history of Chinese
mathematics

• Numerical notation, arithmetical computations,
  counting rods
• Traditional decimal notation -- one symbol for
  each of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 100, 1000,
  and 10000. Ex. 2034 would be written with symbols
  for 2, 1000, 3, 10, 4, meaning 2 times 1000, plus
  3 times 10, plus 4.
• Calculations performed using small bamboo
  counting rods. The positions of the rods gave a
  decimal place-value system, also written for
  long-term records. 0 digit was a space. Arranged
  left to right like Arabic numerals. Back to 400
  B.C.E. or earlier.

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• Addition the counting rods for the two numbers
  placed down, one number above the other. The
  digits added (merged) left to right with carries
  where needed. Subtraction similar.
• Multiplication multiplication table to 9 times 9
  memorized. Long multiplication similar to ours
  with advantages due to physical rods. Long
  division analogous to current algorithms, but
  closer to "galley method."

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• Chinese Numerals
• In 1899 a major discovery was made at the
  archaeological site at the village of Xiao dun in
  the An-yang district of Henan province. Thousands
  of bones and tortoise shells were discovered
  there which had been inscribed with ancient
  Chinese characters. The site had been the capital
  of the kings of the Late Shang dynasty (this Late
  Shang is also called the Yin) from the 14th
  century BC. The last twelve of the Shang kings
  ruled here until about 1045 BC and the bones and
  tortoise shells discovered there had been used as
  part of religious ceremonies. Questions were
  inscribed on one side of a tortoise shell, the
  other side of the shell was then subjected to the
  heat of a fire, and the cracks which appeared
  were interpreted as the answers to the questions
  coming from ancient ancestors.

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• The importance of these finds, as far as
  learning about the ancient Chinese number system,
  was that many of the inscriptions contained
  numerical information about men lost in battle,
  prisoners taken in battle, the number of
  sacrifices made, the number of animals killed on
  hunts, the number of days or months, etc. The
  number system which was used to express this
  numerical information was based on the decimal
  system and was both additive and multiplicative
  in nature.

7
Zhoubi suanjing

• Zhoubi Suanjing was essentially an astronomy
  text, thought to have been compiled between 100
  BC and 100 AD, containing some important
  mathematical sections. The book was listed as the
  first and one of the most important of all the
  texts included in the Ten Mathematical Classics.
  The text measures the positions of the heavenly
  bodies using shadow gauges which are also called
  gnomons.

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• How a gnomon might be used is described in a
  conversation in the text
• Duke of Zhu How great is the art of numbers?
  Tell me something about the application of the
  gnomon.
• Shang Gao Level up one leg of the gnomon and use
  the other leg as a plumb line. When the gnomon is
  turned up, it can measure height when it is
  turned over, it can measure depth and when it
  lies horizontally it can measure distance.
  Revolve the gnomon about its vertex and it can
  draw a circle combine two gnomons and they form
  a square.

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• Zhoubi Suanjing contains calculations of the
  movement of the sun through the year as well as
  observations of the moon and stars, particularly
  the pole star.
• Perhaps the most important mathematics which is
  included in the Zhoubi Suanjing is related to the
  Gougu rule, which is the Chinese version of the
  Pythagoras Theorem.
• The big square has area (ab)2 a2 2ab b2.
  
• The four "corner" triangles each have area ab/2
• giving a total area of 2ab for the four
  added
• together. Hence the inside square (whose
• vertices are on the outside square) has area
  
• (a2 2ab b2) - 2ab a2 b2.
• Its side therefore has length ( a2 b2).
  Therefore the hypotenuse of the right angled
  triangle with sides of length a and b has length
  ( a2 b2).

10
Jiuzhang SuanshuThe Nine Chapters on the
Mathematical Art

• This book is the most influential of all Chinese
  mathematical works in the history of Chinese
  mathematics. It is the longest surviving and one
  of the most important in the ten ancient Chinese
  mathematical books. The book was co-compiled by
  several people and finished in the early

Eastern Han Dynasty (about 1st century),
indicating the formation of ancient Chinese
mathematical system. It became the criterion of
mathematical learning and research for
mathematicians of later generations ever since
then.
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• Afterwards, the Jiuzhang Suanshu have been
  annotated by many mathematicians, the most famous
  ones including Liu Hui (in 263AD) and Li Chunfeng
  (in 656AD). The edition published by the Northern
  Song government in 1084 was the earliest
  mathematical book in the world. The book was
  introduced to Korea and Japan during the Sui and
  Tang dynasties (581-907). Now, it has been
  translated into several languages, including
  Japanese, Russian, German, English and French,
  and become the basis for modern mathematics.
• The book is broken up into nine chapters
  containing 246 questions with their solutions and
  procedures. Each chapter deals with specific
  field of questions. Here is a short description
  of each chapter

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• Chapter 1, Field measurement(Fang tian)
  systematic discussion of algorithms using
  counting rods for common fractions for GCD, LCM
  areas of plane figures, square, rectangle,
  triangle, trapezoid, circle, circle segment,
  sphere segment, annulus. Rules are given for the
  addition, subtraction, multiplication and
  division of fractions, as well as for their
  reduction. Also, rules are given for the segment
  of a circle as
• A 1/2 (c s) s
• , where A is the area, c the chord s the
  sagitta of the segment.
• The same expression is found in the works of the
  Indian mathematician Mahavira about 850 AD.

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• Chapter 2, Cereals(Sumi) deals with
  percentages and proportions. It reflects the
  management and production of various types of
  grains in Han China.
• Chapter 3, Distribution by proportion(Cui fen)
  discusses partnership problems, problems in
  taxation of goods of different qualities, and
  arithmetical and geometrical progressions solved
  by proportion.
• Chapter 4, What width?(Shao guang) finds the
  length of a side when given th area or volume.
  Describes usual algorithms for square and cube
  roots.
• Chapter 5, Construction consultations(Shang
  gong) concerns with calculation for
  constructions of solid figures such as cube,
  rectangular parallelepiped, prism frustums,
  pyramid, triangular pyramid, tetrahedron,
  cylinder, cone, prism, pyramid, cone, frustum of
  a cone, cylinder, wedge, tetrahedron, and some
  others. It gives problems concerning the volumes
  of city-walls, dykes, canals, etc.

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• Chapter 6, Fair taxes(Jun shu) discusses the
  problems in connection with the time required for
  people to carry their grain contributions from
  their native towns to the capital. There are also
  problems of ratios in connection with the
  allocation of tax burdens according to
  population.
• Chapter 7, Excess and deficiency(Ying bu zu)
  uses of method of false position and double false
  position to solve difficult problems.
• Chapter 8, Rectangular arrays(Fang cheng)
  gives elimination algorithm for solving systems
  of three or more simultaneous linear equations.
  Introduces concept of positive and negative
  numbers (red reds for positive numbers, black for
  negative numbers). Rules for addition and
  subtraction of signed numbers.
• Chapter 9, Right triangles(Gou gu)
  applications of Pythagorean theorem and similar
  triangles, solves quadratic equations with
  modification of square root algorithm, only
  equations of the form x2 a x b, with a and b
  positive.

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The book's major achievements  1. Devising a
systematic treatment of arithmetic operations
with fractions, 1,400 years earlier than the
Europeans. 2. Dealing with various types of
problems on proportions, 1,400 years earlier than
the Europeans. 3. Devising methods for extracting
square root and cubic root, which is quite
similar to today's method, several hundred years
earlier than the Western mathematicians. 4.
Developing solutions for a system of linear
equations, about 1,600 years earlier than the
Western mathematicians.
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• 5. Introducing the concepts of positive and
  negative numbers, more than 600 years earlier
  than the West.
• 6. Developing a general solution formula for the
  Pythagorean problems (problems of Gou gu), 300
  years earlier than the West.
• 7. Putting forward theories of calculating areas
  and volumes of different shapes and figures.

17
The Abacus

• In about the fourteenth century AD the abacus
  came into use in China. Certainly this, like the
  counting board, seems to have been a Chinese
  invention. In many ways it was similar to the
  counting board, except instead of using rods to
  represent numbers, they were represented by beads
  sliding on a wire. Arithmetical rules for the
  abacus were analogous to those of the counting
  board (even square roots and cube roots of
  numbers could be calculated) but it appears that
  the abacus was used almost exclusively by
  merchants who only used the operations of
  addition and subtraction.

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• Here is an illustration of an abacus showing the
  number 46802.

For numbers up to 4 slide the required number of
beads in the lower part up to the middle bar.
For example on the right most wire two is
represented. For five or above, slide one bead
above the middle bar down (representing 5), and
1, 2, 3 or 4 beads up to the middle bar for the
numbers 6, 7, 8, or 9 respectively. For example
on the wire three from the right hand side the
number 8 is represented (5 for the bead above,
three beads below).
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• Sun Zi (c. 250? C.E.) Wrote his mathematical
  manual. Includes "Chinese remainder problem or
  problem of the Master Sun find n so that upon
  division by 3 you get a remainder of 2, upon
  division by 5 you get a remainder of 3, and upon
  division by 7 you get a remainder of 2. His
  solution Take 140, 63, 30, add to get 233,
  subtract 210 to get 23.
• Liu Hui (c. 263 C.E.)
• Commentary on the Jiuzhang Suanshu Approximates
  pi by approximating circles polygons, doubling
  the number of sides to get better approximations.
  From 96 and 192 sided polygons, he approximates
  pi as 3.141014 and suggested 3.14 as a practical
  approximation. States principle of exhaustion
  for circles Suggests Calvalieri's principle to
  find accurate volume of cylinder

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• Haidao suanjing (Sea Island Mathematical Manual).
  Originally appendix to commentary on Chapter 9 of
  the Jiuzhang Suanshu. Includes nine surveying
  problems involving indirect observations.
• Zhang Qiujian (c. 450?) Wrote his mathematical
  manual. Includes formula for summing an
  arithmetic sequence. Also an undetermined system
  of two linear equations in three unknowns, the
  "hundred fowls problem"
• Zu Chongzhi (429-500) Astronomer, mathematician,
  engineer.
• Collected together earlier astronomical writings.
  Made own astronomical observations. Recommended
  new calendar.
• Determined pi to 7 digits 3.1415926. Recommended
  use 355/113 for close approx. and 22/7 for rough
  approx.
• With father carried out Liu Hui's suggestion for
  volume of sphere to get accurate formula for
  volume of a sphere.

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• Liu Zhuo (544-610) Astronomer Introduced
  quadratic interpolation (second order difference
  method).
• Wang Xiaotong (fl. 625) Mathematician and
  astronomer. Wrote Xugu suanjing (Continuation of
  Ancient Mathematics) of 22 problems. Solved cubic
  equations by generalization of algorithm for cube
  root.
• Translations of Indian mathematical works. By
  600 C.E., 3 works, since lost. Levensita, Indian
  astronomer working at State Observatory,
  translated two more texts, one of which described
  angle measurement (360 degrees) and a table of
  sines for angles from 0 to 90 degrees in 24 steps
  (3 3/4 degree) increments.
• Hindu decimal numerals also introduced, but not
  adopted.

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• Yi Xing (683-727) tangent table.
• Jia Xian (c. 1050) Written work lost.
  Streamlined extraction of square and cube roots,
  extended method to higher-degree roots using
  binomial coefficients.
• Qin Jiushao (c. 1202 - c. 1261) Shiushu jiuzhang
  (Mathemtaical Treatise in Nine Sections), 81
  problems of applied math similar to the Nine
  Chapters. Solution of some higher-degree (up to
  10th) equations. Systematic treatment of
  indeterminate simultaneous linear congruences
  (Chinese remainder theorem). Euclidean algorithm
  for GCD.
• Li Chih (a.k.a. Li Yeh) (1192-1279) Ceyuan
  haijing (Sea Mirror of Circle Measurements), 12
  chapters, 170 problems on right triangles and
  circles inscribed within or circumscribed about
  them. Yigu yanduan (New Steps in Computation),
  geometric problems solved by algebra.

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• Yang Hui (fl. c. 1261-1275) Wrote sevral books.
  Explains Jiu Xian's methods for solving
  higher-degree root extractions. Magic squares of
  order up through 10.
• Guo Shoujing (1231-1316) Shou shi li (Works and
  Days Calendar). Higher-order differences (i.e.,
  higher-order interpolation).
• Zhu Shijie (fl. 1280-1303) Suan xue qi meng
  (Introduction to Mathematical Studies), and
  Siyuan yujian (Precious Mirror of the Four
  Elements). Solves some higher degree polynomial
  equations in several unknowns. Sums some finite
  series including (1) the sum of n2 and (2) the
  sum of n(n1)(n2)/6. Discusses binomial
  coefficients. Uses zero digit.