Question From the Nine Chapters on the Mathematical Art

A folio of The Ix Chapters on the Mathematical Art

The Nine Chapters on the Mathematical Art (simplified Chinese: 九章算术; traditional Chinese: 九章算術; pinyin: Jiǔzhāng Suànshù ; Wade–Giles: chiuⁱⁱⁱ chang^one suan⁴ shu^one ) is a Chinese mathematics book, equanimous by several generations of scholars from the 10th–2d century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest surviving mathematical texts from Cathay, the first being Suan shu shu (202 BCE – 186 BCE) and Zhoubi Suanjing (compiled throughout the Han until the belatedly 2nd century CE). It lays out an approach to mathematics that centres on finding the most general methods of solving problems, which may exist contrasted with the approach common to ancient Greek mathematicians, who tended to deduce propositions from an initial set of axioms.

Entries in the book usually accept the form of a statement of a problem, followed by the statement of the solution and an explanation of the procedure that led to the solution. These were commented on past Liu Hui in the 3rd century.

History [edit]

Original volume [edit]

The full title of The Nine Capacity on the Mathematical Art appears on two statuary standard measures which are dated to 179 CE, but in that location is speculation that the same book existed beforehand nether unlike titles.^[1]

Most scholars believe that Chinese mathematics and the mathematics of the ancient Mediterranean world had adult more or less independently up to the time when the Ix Chapters reached its final form. The method of chapter seven was not institute in Europe until the 13th century, and the method of chapter 8 uses Gaussian elimination earlier Carl Friedrich Gauss (1777–1855).^[2] There is also the mathematical proof given in the treatise for the Pythagorean theorem.^[iii] The influence of The Nine Capacity profoundly assisted the development of ancient mathematics in the regions of Korea and Nihon. Its influence on mathematical idea in Red china persisted until the Qing Dynasty era.

Liu Hui wrote a very detailed commentary on this book in 263. He analyses the procedures of the Nine Chapters stride by pace, in a manner which is clearly designed to give the reader conviction that they are reliable, although he is not concerned to provide formal proofs in the Euclidean fashion. Liu's commentary is of great mathematical interest in its own right. Liu credits the earlier mathematicians Zhang Cang (fl. 165 BCE - d. 142 BCE) and Geng Shouchang (fl. 75 BCE-49 BCE) (see armillary sphere) with the initial arrangement and commentary on the book, yet Han Dynasty records exercise not indicate the names of any authors of commentary, as they are not mentioned until the 3rd century.^[4]

The 9 Capacity is an anonymous work, and its origins are non articulate. Until recent years, there was no substantial evidence of related mathematical writing that might accept preceded it, with the exception of mathematical piece of work by those such every bit Jing Fang (78–37 BCE), Liu Xin (d. 23), and Zhang Heng (78–139) and the geometry clauses of the Mozi of the 4th century BCE. This is no longer the case. The Suàn shù shū (算數書) or writings on reckoning is an ancient Chinese text on mathematics approximately seven thousand characters in length, written on 190 bamboo strips. Information technology was discovered together with other writings in 1983 when archaeologists opened a tomb in Hubei province. It is among the corpus of texts known equally the Zhangjiashan Han bamboo texts. From documentary evidence this tomb is known to have been closed in 186 BCE, early in the Western Han dynasty. While its relationship to the Nine Chapters is still under discussion by scholars, some of its contents are clearly paralleled in that location. The text of the Suàn shù shū is nevertheless much less systematic than the Ix Chapters; and appears to consist of a number of more or less contained short sections of text drawn from a number of sources. The Zhoubi Suanjing, a mathematics and astronomy text, was as well compiled during the Han, and was even mentioned as a school of mathematics in and around 180 CE by Cai Yong.

Western translations [edit]

The championship of the book has been translated in a wide variety of ways.

In 1852, Alexander Wylie referred to it as Arithmetical Rules of the Nine Sections.

With but a slight variation, the Japanese historian of mathematics Yoshio Mikami shortened the championship to Arithmetic in 9 Sections. ^[5]

David Eugene Smith, in his History of Mathematics (Smith 1923), followed the convention used by Yoshio Mikami.

Several years later, George Sarton took note of the book, only only with limited attention and only mentioning the usage of red and blackness rods for positive and negative numbers.

In 1959, Joseph Needham and Wang Ling (historian) translated Jiu Zhang Suan shu as The Nine Chapters on the Mathematical Fine art for the outset time.

Later on in 1994, Lam Lay Yong used this title in her overview of the volume, as did other mathematicians including John North. Crossley and Anthony Westward.-C Lun in their translation of Li Yan and Du Shiran's Chinese Mathematics: A Concise History (Li and Du 1987).^[5]

Afterward, the name The Nine Capacity on the Mathematical Art stuck and became the standard English title for the book.

Table of contents [edit]

Contents of The Ix Chapters are as follows:

1. 方田 Fangtian - Bounding fields. Areas of fields of various shapes, such as rectangles, triangles, trapezoids, and circles; manipulation of vulgar fractions. Liu Hui's commentary includes a method for calculation of π and the approximate value of 3.14159.^[half-dozen]
2. 粟米 Sumi - Millet and rice. Exchange of commodities at different rates; unit pricing; the Rule of Three for solving proportions, using fractions.
3. 衰分 Cuifen - Proportional distribution. Distribution of commodities and money at proportional rates; deriving arithmetics and geometric sums.
4. 少廣 Shaoguang - Reducing dimensions. Finding the diameter or side of a shape given its volume or area. Sectionalization by mixed numbers; extraction of square and cube roots; diameter of sphere, perimeter and diameter of circle.
5. 商功 Shanggong - Figuring for construction. Volumes of solids of diverse shapes.
6. 均輸 Junshu - Equitable taxation. More than advanced discussion problems on proportion, involving piece of work, distances, and rates.
7. 盈不足 Yingbuzu - Excess and arrears. Linear problems (in two unknowns) solved using the principle known later in the Westward as the rule of false position.
8. 方程 Fangcheng - The two-sided reference (i.e. Equations). Bug of agricultural yields and the sale of animals that atomic number 82 to systems of linear equations, solved by a principle indistinguishable from the modern form of Gaussian emptying.^[vii]
9. 勾股 Gougu - Base and altitude. Problems involving the principle known in the West every bit the Pythagorean theorem.

Major contributions [edit]

Real number system [edit]

The Nine Chapters on the Mathematical Art does non hash out natural numbers, that is, positive integers and their operations, only they are widely used and written on the ground of natural numbers. Although it is not a volume on fractions, the meaning, nature, and iv operations of fractions are fully discussed. For example: combined division (addition), subtraction (subtraction), multiplication (multiplication), warp division (division), sectionalisation (comparing size), reduction (simplified fraction), and bisector (boilerplate).^[8]

The concept of negative numbers also appears in "Nine Capacity of Arithmetic". In order to cooperate with the algorithm of equations, the rules of addition and subtraction of positive and negative numbers are given. The subtraction is "split up by the same name, benefit by different names. The improver is "divide past different names, benefit from each other by the same name. Amongst them, "division" is subtraction, "benefit" is improver, and "no entry" means that in that location is no counter-party, just multiplication and division are non recorded.^[8]

The Nine Capacity on the Mathematical Art gives a sure discussion on natural numbers, fractions, positive and negative numbers, and some special irrationality. It basically has the prototype of real number system.

Gou Gu (Pythagorean) Theorem [edit]

The geometric figures included in The 9 Chapters on the Mathematical Art are more often than not directly and circular figures because of its focus on the applications onto the agricultural fields. In addition, due to the needs of civil architecture, The Nine Capacity on the Mathematical Art likewise discusses volumetric algorithms of linear and circular 3 dimensional solids. The arrangement of these volumetric algorithms ranges from simple to complex, forming a unique mathematical organisation.^[8]

Regarding the direct awarding of the Gou Gu Theorem, which is precisely the Chinese version of the Pythagorean Theorem, the volume divides it into four main categories: Gou Gu mutual seeking, Gou Gu integer, Gou Gu dual chapters, Gou Gu similar.

Gou Gu mutual seeking discusses the algorithm of finding the length of a side of the correct triangle while knowing the other two. Gou Gu integer is precisely the finding of some significant integer Pythagorean numbers, including famously the triple 3,four,five. Gou Gu dual chapters discusses algorithms for calculating the areas of the inscribed rectangles and other polygons in the circumvolve, which also serves an algorithm to calculate the value of pi. Lastly, Gou Gu similars provide algorithms of calculating heights and lengths of buildings on the mathematical basis of similar right triangles.

Completing of squares and solutions of organization of equations [edit]

The methods of completing the squares and cubes as well every bit solving simultaneous linear equations listed in The Nine Chapters on the Mathematical Art can exist regarded one of the major content of ancient Chinese mathematics. The discussion of these algorithms in The Nine Chapters on the Mathematical Art are very detailed. Through these discussions, i can understand the achievements of the evolution of ancient Chinese mathematics.^[8]

Completing the squaring and cubes can not only solve systems of two linear equations with two unknowns, but too full general quadratic and cubic equations. It is the ground for solving higher-order equations in ancient China, and it also plays an important role in the development of mathematics.^[eight]

The "equations" discussed in the Fang Cheng chapter are equivalent to today'southward simultaneous linear equations. The solution method called "Fang Cheng Shi" is best known today as Gaussian elimination. Among the eighteen issues listed in the Fang Cheng chapter, some are equivalent to simultaneous linear equations with two unknowns, some are equivalent to simultaneous linear equations with iii unknowns, and the most complex example analyzes the solution to a organization of linear equations with up to 5 unknowns.^[eight]

Significance [edit]

The word "Jiu", or "ix", means more than only a digit in ancient Chinese. In fact, since it is the largest digit, it often refers to something of a yard scale or a supreme authority. Farther, the give-and-take "Zhang", or "Chapter", too has more connotations than simply being the "affiliate". It may refer to a department, several parts of an article, or an entire treatise.^[9] Given these historical understanding of ancient Chinese, the book The 9 Chapters on the Mathematical Fine art is actually a somewhat mistranslation; information technology should really mean a g book for mathematics.

In this light, many scholars of the history of Chinese mathematics compare the significance of The Nine Capacity on the Mathematical Fine art on the development of Eastern mathematical traditions to that of Euclid's Elements on the Western mathematical traditions.^{[10] [11]} All the same, the influence of The Nine Capacity on the Mathematical Fine art stops short at the advocacy of modern mathematics due to its focus on practical issues and anterior proof methods every bit opposed to the deductive, axiomatic tradition that Euclid's Elements establishes.

However, it is dismissive to say that The Nine Chapters on the Mathematical Art has no bear upon at all on modern mathematics. The style and structure of The Nine Chapters on the Mathematical Fine art can exist best concluded every bit "problem, formula, and ciphering".^[12] This process of solving practical mathematical problems is at present pretty much the standard approach in the field of practical mathematics.

Notable translations [edit]

• Abridged English language translation: Yoshio Mikami: "Arithmetic in Nine Sections", in The Development of Mathematics in China and Japan, 1913.
• Highly Abridged English translation: Florian Cajori: "Arithmetic in Nine Sections", in A History of Mathematics, Second Edition, 1919 (maybe copied or paraphrased from Mikami).
• Abridged English translation: Lam Lay Yong: Jiu Zhang Suanshu: An Overview, Archive for History of Verbal Sciences, Springer Verlag, 1994.
• A full translation and written report of the Nine Chapters and Liu Hui'due south commentary is available in Kangshen Shen, The Nine Capacity on the Mathematical Fine art, Oxford University Press, 1999. ISBN 0-19-853936-3
• A French translation with detailed scholarly addenda and a critical edition of the Chinese text of both the book and its commentary by Karine Chemla and Shuchun Guo is Les neuf chapitres: le classique mathématique de la Chine ancienne et ses commentaires. Paris: Dunod, 2004. ISBN 978-ii-10-049589-4.
• German translation: Kurt Vogel, Neun Bücher Arithmetischer Technik, Friedrich Vieweg und Sohn Braunsweig, 1968
• Russian translation: Eastward. I Beriozkina, Математика в девяти книгах (Mathematika V Devyati Knigah), Moscow: Nauka, 1980.

See also [edit]

• Haidao Suanjing
• History of mathematics
• History of geometry

Notes [edit]

1. ^ Needham, Book 3, 24-25.
2. ^ Straffin, 164.
3. ^ Needham, Volume three, 22.
4. ^ Needham, Volume 3, 24.
5. ^ ^{a b} Dauben, Joseph Westward. (2013). "九章箅术 "Jiu zhang suan shu" (9 Chapters on the Art of Mathematics) - An Appraisement of the Text, its Editions, and Translations". Sudhoffs Archiv. 97 (ii): 199–235. ISSN 0039-4564. JSTOR 43694474.
6. ^ O'Connor.
7. ^ http://www.dam.brown.edu/people/mumford/beyond/papers/2010b--Negatives-PrfShts.pdf^{[ blank URL PDF ]}
8. ^ ^{a b c d e f} 中國文明史 第三卷 秦漢時代 中冊. 地球社编辑部. 1992. pp. 515–531.
9. ^ Dauben, Joseph W. (1992), "The "Pythagorean theorem" and Chinese Mathematics Liu Hui'south Commentary on the 勾股 (Gou-Gu) Theorem in Chapter 9 of the Jiu Zhang Suan Shu", Amphora, Birkhäuser Basel, pp. 133–155, doi:x.1007/978-3-0348-8599-7_7, ISBN978-3-0348-9696-2
10. ^ Siu, Human being-Keung (December 1993). "Proof and pedagogy in ancient China: Examples from Liu Hui's commentary on JIU ZHANG SUAN SHU". Educational Studies in Mathematics. 24 (4): 345–357. doi:10.1007/bf01273370. ISSN 0013-1954. S2CID 120420378.
11. ^ Dauben, Joseph W. (September 1998). "Ancient Chinese mathematics: the (Jiu Zhang Suan Shu) vs Euclid's Elements. Aspects of proof and the linguistic limits of cognition". International Journal of Technology Science. 36 (12–fourteen): 1339–1359. doi:ten.1016/s0020-7225(98)00036-6. ISSN 0020-7225.
12. ^ 吴, 文俊 (1982). 九章算术与刘辉. 北京: 北京师范大学出版社. p. 118.

References [edit]

• Needham, Joseph (1986). Science and Civilization in China: Book 3, Mathematics and the Sciences of the Heavens and the World. Taipei: Caves Books, Ltd.
• Straffin, Philip D. "Liu Hui and the First Golden Historic period of Chinese Mathematics", Mathematics Magazine (Volume 71, Number 3, 1998): 163–181.
• O'Connor, John J.; Robertson, Edmund F., "Liu Hui", MacTutor History of Mathematics archive, University of St Andrews

External links [edit]

• Full text of the volume (Chinese)

parkerhimathat.blogspot.com

Source: https://en.wikipedia.org/wiki/The_Nine_Chapters_on_the_Mathematical_Art

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