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Chinese mathematics

May 01, 2023

Mathematics in China emerged independently by the 11th century BCE.[1] The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system (base 2 and base 10), algebra, geometry, number theory and trigonometry.

Since the Han Dynasty, as diophantine approximation being a prominent numerical method, the Chinese made substantial progress on polynomial evaluation. Algorithms like regula falsi and expressions like continued fractions are widely used and have been well-documented ever-since. They deliberately find the principal nth root of positive numbers and the roots of equations.[2][3] The major texts from the period, The Nine Chapters on the Mathematical Art and the Book on Numbers and Computation gave detailed processes for solving various mathematical problems in daily life.[4] All procedures were computed using a counting board in both texts, and they included inverse elements as well as Euclidean divisions. The texts provide procedures similar to that of Gaussian elimination and Horner's method for linear algebra.[5] The achievement of Chinese algebra reached a zenith in the 13th century during the Yuan dynasty with the development of tiān yuán shù.

As a result of obvious linguistic and geographic barriers, as well as content, Chinese mathematics and the mathematics of the ancient Mediterranean world are presumed to have developed more or less independently up to the time when The Nine Chapters on the Mathematical Art reached its final form, while the Book on Numbers and Computation and Huainanzi are roughly contemporary with classical Greek mathematics. Some exchange of ideas across Asia through known cultural exchanges from at least Roman times is likely. Frequently, elements of the mathematics of early societies correspond to rudimentary results found later in branches of modern mathematics such as geometry or number theory. The Pythagorean theorem for example, has been attested to the time of the Duke of Zhou. Knowledge of Pascal's triangle has also been shown to have existed in China centuries before Pascal,[6] such as the Song dynasty Chinese polymath Shen Kuo.

Early Chinese mathematics

 
Visual proof for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 BC.
 
Oracle bone script numeral system
 
counting rod place value decimal

Shang Dynasty (1600–1050 BC). One of the oldest surviving mathematical works is the I Ching, which greatly influenced written literature during the Zhou Dynasty (1050–256 BC). For mathematics, the book included a sophisticated use of hexagrams. Leibniz pointed out, the I Ching (Yi Jing) contained elements of binary numbers.

Since the Shang period, the Chinese had already fully developed a decimal system. Since early times, Chinese understood basic arithmetic (which dominated far eastern history), algebra, equations, and negative numbers with counting rods.[citation needed] Although the Chinese were more focused on arithmetic and advanced algebra for astronomical uses, they were also the first to develop negative numbers, algebraic geometry (only Chinese geometry) and the usage of decimals.

Math was one of the Liù Yì (六藝) or Six Arts, students were required to master during the Zhou Dynasty (1122–256 BC). Learning them all perfectly was required to be a perfect gentleman, or in the Chinese sense, a "Renaissance Man". Six Arts have their roots in the Confucian philosophy.

The oldest existent work on geometry in China comes from the philosophical Mohist canon of c. 330 BC, compiled by the followers of Mozi (470–390 BC). The Mo Jing described various aspects of many fields associated with physical science, and provided a small wealth of information on mathematics as well. It provided an 'atomic' definition of the geometric point, stating that a line is separated into parts, and the part which has no remaining parts (i.e. cannot be divided into smaller parts) and thus forms the extreme end of a line is a point.[7] Much like Euclid's first and third definitions and Plato's 'beginning of a line', the Mo Jing stated that "a point may stand at the end (of a line) or at its beginning like a head-presentation in childbirth. (As to its invisibility) there is nothing similar to it."[8] Similar to the atomists of Democritus, the Mo Jing stated that a point is the smallest unit, and cannot be cut in half, since 'nothing' cannot be halved.[8] It stated that two lines of equal length will always finish at the same place,[8] while providing definitions for the comparison of lengths and for parallels,[9] along with principles of space and bounded space.[10] It also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch.[11] The book provided word recognition for circumference, diameter, and radius, along with the definition of volume.[12]

The history of mathematical development lacks some evidence. There are still debates about certain mathematical classics. For example, the Zhoubi Suanjing dates around 1200–1000 BC, yet many scholars believed it was written between 300 and 250 BC. The Zhoubi Suanjing contains an in-depth proof of the Gougu Theorem (a special case of the Pythagorean Theorem) but focuses more on astronomical calculations. However, the recent archaeological discovery of the Tsinghua Bamboo Slips, dated c. 305 BC, has revealed some aspects of pre-Qin mathematics, such as the first known decimal multiplication table.[13]

The abacus was first mentioned in the second century BC, alongside 'calculation with rods' (suan zi) in which small bamboo sticks are placed in successive squares of a checkerboard.[14]

Qin mathematics

Not much is known about Qin dynasty mathematics, or before, due to the burning of books and burying of scholars, circa 213–210 BC. Knowledge of this period can be determined from civil projects and historical evidence. The Qin dynasty created a standard system of weights. Civil projects of the Qin dynasty were significant feats of human engineering. Emperor Qin Shihuang (秦始皇) ordered many men to build large, lifesize statues for the palace tomb along with other temples and shrines, and the shape of the tomb was designed with geometric skills of architecture. It is certain that one of the greatest feats of human history, the Great Wall of China, required many mathematical techniques. All Qin dynasty buildings and grand projects used advanced computation formulas for volume, area and proportion.

Qin bamboo cash purchased at the antiquarian market of Hong Kong by the Yuelu Academy, according to the preliminary reports, contains the earliest epigraphic sample of a mathematical treatise.

Han mathematics

Further information: Science and technology of the Han dynasty § Mathematics and astronomy

In the Han Dynasty, numbers were developed into a place value decimal system and used on a counting board with a set of counting rods called chousuan, consisting of only nine symbols with a blank space on the counting board representing zero.[3] Negative numbers and fractions were also incorporated into solutions of the great mathematical texts of the period.[4] The mathematical texts of the time, the Suàn shù shū and the Jiuzhang suanshu solved basic arithmetic problems such as addition, subtraction, multiplication and division.[4] Furthermore, they gave the processes for square and cubed root extraction, which eventually was applied to solving quadratic equations up to the third order.[5] Both texts also made substantial progress in Linear Algebra, namely solving systems of equations with multiple unknowns.[15] The value of pi is taken to be equal to three in both texts.[16] However, the mathematicians Liu Xin (d. 23) and Zhang Heng (78–139) gave more accurate approximations for pi than Chinese of previous centuries had used.[4] Mathematics was developed to solve practical problems in the time such as division of land or problems related to division of payment.[17] The Chinese did not focus on theoretical proofs based on geometry or algebra in the modern sense of proving equations to find area or volume.[18] The Book of Computations and The Nine Chapters on the Mathematical Art provide numerous practical examples that would be used in daily life.[18]

Suan shu shu

The Suàn shù shū (Writings on Reckoning or The Book of Computations) is an ancient Chinese text on mathematics approximately seven thousand characters in length, written on 190 bamboo strips.[19] It was discovered together with other writings in 1984 when archaeologists opened a tomb at Zhangjiashan in Hubei province. From documentary evidence this tomb is known to have been closed in 186 BC, early in the Western Han dynasty.[4] While its relationship to the Nine Chapters is still under discussion by scholars, some of its contents are clearly paralleled there. The text of the Suan shu shu is however much less systematic than the Nine Chapters, and appears to consist of a number of more or less independent short sections of text drawn from a number of sources.[19]

The Book of Computations contains many perquisites to problems that would be expanded upon in The Nine Chapters on the Mathematical Art.[19] An example of the elementary mathematics in the Suàn shù shū, the square root is approximated by using false position method which says to "combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend."[19] Furthermore, The Book of Computations solves systems of two equations and two unknowns using the same false position method.[15]

The Nine Chapters on the Mathematical Art

The Nine Chapters on the Mathematical Art is a Chinese mathematics book, its oldest archeological date being 179 AD (traditionally dated 1000 BC), but perhaps as early as 300–200 BC.[20] Although the author(s) are unknown, they made a major contribution in the eastern world. Problems are set up with questions immediately followed by answers and procedure.[17] There are no formal mathematical proofs within the text, just a step-by-step procedure.[21] The commentary of Liu Hui provided geometrical and algebraic proofs to the problems given within the text.[3]

The Nine Chapters on the Mathematical Art was one of the most influential of all Chinese mathematical books and it is composed of 246 problems.[20] It was later incorporated into The Ten Computational Canons, which became the core of mathematical education in later centuries.[17] This book includes 246 problems on surveying, agriculture, partnerships, engineering, taxation, calculation, the solution of equations, and the properties of right triangles.[17] The Nine Chapters made significant additions to solving quadratic equations in a way similar to Horner's method.[5] It also made advanced contributions to "fangcheng" or what is now known as linear algebra.[15] Chapter seven solves system of linear equations with two unknowns using the false position method, similar to The Book of Computations.[15] Chapter eight deals with solving determinate and indeterminate simultaneous linear equations using positive and negative numbers, with one problem dealing with solving four equations in five unknowns.[15] The Nine Chapters solves systems of equations using methods similar to the modern Gaussian elimination and back substitution.[15]

The version of The Nine Chapters that has served as the foundation for modern renditions was a result of the efforts of the scholar Dai Zhen. Transcribing the problems directly from Yongle Encyclopedia, he then proceeded to make revisions to the original text, along with the inclusion his own notes explaining his reasoning behind the alterations.[22] His finished work would be first published in 1774, but a new revision would be published in 1776 to correct various errors as well as include a version of The Nine Chapters from the Southern Song that contained the commentaries of Lui Hui and Li Chunfeng. The final version of Dai Zhen's work would come in 1777, titled Ripple Pavilion, with this final rendition being widely distributed and coming to serve as the standard for modern versions of The Nine Chapters.[23] However, this version has come under scrutiny from Guo Shuchen, alleging that the edited version still contains numerous errors and that not all of the original amendments were done by Dai Zhen himself.[22]

Calculation of pi

Problems in The Nine Chapters on the Mathematical Art take pi to be equal to three in calculating problems related to circles and spheres, such as spherical surface area.[20] There is no explicit formula given within the text for the calculation of pi to be three, but it is used throughout the problems of both The Nine Chapters on the Mathematical Art and the Artificer's Record, which was produced in the same time period.[16] Historians believe that this figure of pi was calculated using the 3:1 relationship between the circumference and diameter of a circle.[20] Some Han mathematicians attempted to improve this number, such as Liu Xin, who is believed to have estimated pi to be 3.154.[4] Later, Liu Hui attempted to improve the calculation by calculating pi to be 3.141024 (a low estimate of the number). Liu calculated this number by using polygons inside a hexagon as a lower limit compared to a circle.[24] Zu Chongzhi later discovered the calculation of pi to be 3.1415926 < π < 3.1415927 by using polygons with 24,576 sides. This calculation would be discovered in Europe during the 16th century.[25]

There is no explicit method or record of how he calculated this estimate.[4]

Division and root extraction

Basic arithmetic processes such as addition, subtraction, multiplication and division were present before the Han Dynasty.[4] The Nine Chapters on the Mathematical Art take these basic operations for granted and simply instruct the reader to perform them.[15] Han mathematicians calculated square and cube roots in a similar manner as division, and problems on division and root extraction both occur in Chapter Four of The Nine Chapters on the Mathematical Art.[26] Calculating the square and cube roots of numbers is done through successive approximation, the same as division, and often uses similar terms such as dividend (shi) and divisor (fa) throughout the process.[5] This process of successive approximation was then extended to solving quadratics of the second and third order, such as  , using a method similar to Horner's method.[5] The method was not extended to solve quadratics of the nth order during the Han Dynasty; however, this method was eventually used to solve these equations.[5]

 
Fangcheng on a counting board

Linear algebra

The Book of Computations is the first known text to solve systems of equations with two unknowns.[15] There are a total of three sets of problems within The Book of Computations involving solving systems of equations with the false position method, which again are put into practical terms.[15] Chapter Seven of The Nine Chapters on the Mathematical Art also deals with solving a system of two equations with two unknowns with the false position method.[15] To solve for the greater of the two unknowns, the false position method instructs the reader to cross-multiply the minor terms or zi (which are the values given for the excess and deficit) with the major terms mu.[15] To solve for the lesser of the two unknowns, simply add the minor terms together.[15]

Chapter Eight of The Nine Chapters on the Mathematical Art deals with solving infinite equations with infinite unknowns.[15] This process is referred to as the "fangcheng procedure" throughout the chapter.[15] Many historians chose to leave the term fangcheng untranslated due to conflicting evidence of what the term means. Many historians translate the word to linear algebra today. In this chapter, the process of Gaussian elimination and back-substitution are used to solve systems of equations with many unknowns.[15] Problems were done on a counting board and included the use of negative numbers as well as fractions.[15] The counting board was effectively a matrix, where the top line is the first variable of one equation and the bottom was the last.[15]

Liu Hui's commentary on The Nine Chapters on the Mathematical Art

 
Liu Hui's exhaustion method

Liu Hui's commentary on The Nine Chapters on the Mathematical Art is the earliest edition of the original text available.[20] Hui is believed by most to be a mathematician shortly after the Han dynasty. Within his commentary, Hui qualified and proved some of the problems from either an algebraic or geometrical standpoint.[18] For instance, throughout The Nine Chapters on the Mathematical Art, the value of pi is taken to be equal to three in problems regarding circles or spheres.[16] In his commentary, Liu Hui finds a more accurate estimation of pi using the method of exhaustion.[16] The method involves creating successive polynomials within a circle so that eventually the area of a higher-order polygon will be identical to that of the circle.[16] From this method, Liu Hui asserted that the value of pi is about 3.14.[4] Liu Hui also presented a geometric proof of square and cubed root extraction similar to the Greek method, which involved cutting a square or cube in any line or section and determining the square root through symmetry of the remaining rectangles.[26]

Mathematics in the period of disunity

 
Liu Hui's Survey of sea island
 
Sunzi algorithm for division 400 AD
 
al Khwarizmi division in the 9th century
 
Statue of Zu Chongzhi.

In the third century Liu Hui wrote his commentary on the Nine Chapters and also wrote Haidao Suanjing which dealt with using Pythagorean theorem (already known by the 9 chapters), and triple, quadruple triangulation for surveying; his accomplishment in the mathematical surveying exceeded those accomplished in the west by a millennium.[27] He was the first Chinese mathematician to calculate π=3.1416 with his π algorithm. He discovered the usage of Cavalieri's principle to find an accurate formula for the volume of a cylinder, and also developed elements of the infinitesimal calculus during the 3rd century CE.

 
fraction interpolation for pi

In the fourth century, another influential mathematician named Zu Chongzhi, introduced the Da Ming Li. This calendar was specifically calculated to predict many cosmological cycles that will occur in a period of time. Very little is really known about his life. Today, the only sources are found in Book of Sui, we now know that Zu Chongzhi was one of the generations of mathematicians. He used Liu Hui's pi-algorithm applied to a 12288-gon and obtained a value of pi to 7 accurate decimal places (between 3.1415926 and 3.1415927), which would remain the most accurate approximation of π available for the next 900 years. He also applied He Chengtian's interpolation for approximating irrational number with fraction in his astronomy and mathematical works, he obtained   as a good fraction approximate for pi; Yoshio Mikami commented that neither the Greeks, nor the Hindus nor Arabs knew about this fraction approximation to pi, not until the Dutch mathematician Adrian Anthoniszoom rediscovered it in 1585, "the Chinese had therefore been possessed of this the most extraordinary of all fractional values over a whole millennium earlier than Europe"[28]

Along with his son, Zu Geng, Zu Chongzhi applied the Cavalieri's principle to find an accurate solution for calculating the volume of the sphere. Besides containing formulas for the volume of the sphere, his book also included formulas of cubic equations and the accurate value of pi. His work, Zhui Shu was discarded out of the syllabus of mathematics during the Song dynasty and lost. Many believed that Zhui Shu contains the formulas and methods for linear, matrix algebra, algorithm for calculating the value of π, formula for the volume of the sphere. The text should also associate with his astronomical methods of interpolation, which would contain knowledge, similar to our modern mathematics.

A mathematical manual called Sunzi mathematical classic dated between 200 and 400 CE contained the most detailed step by step description of multiplication and division algorithm with counting rods. Intriguingly, Sunzi may have influenced the development of place-value systems and place-value systems and the associated Galley division in the West. European sources learned place-value techniques in the 13th century, from a Latin translation an early-9th-century work by Al-Khwarizmi. Khwarizmi's presentation is almost identical to the division algorithm in Sunzi, even regarding stylistic matters (for example, using blank spaces to represent trailing zeros); the similarity suggests that the results may not have been an independent discovery. Islamic commentators on Al-Khwarizmi's work believed that it primarily summarized Hindu knowledge; Al-Khwarizmi's failure to cite his sources makes it difficult to determine whether those sources had in turn learned the procedure from China.[29]

In the fifth century the manual called "Zhang Qiujian suanjing" discussed linear and quadratic equations. By this point the Chinese had the concept of negative numbers.

Tang mathematics

By the Tang Dynasty study of mathematics was fairly standard in the great schools. The Ten Computational Canons was a collection of ten Chinese mathematical works, compiled by early Tang dynasty mathematician Li Chunfeng (李淳風 602–670), as the official mathematical texts for imperial examinations in mathematics. The Sui dynasty and Tang dynasty ran the "School of Computations".[30]

Wang Xiaotong was a great mathematician in the beginning of the Tang Dynasty, and he wrote a book: Jigu Suanjing (Continuation of Ancient Mathematics), where numerical solutions which general cubic equations appear for the first time[31]

The Tibetans obtained their first knowledge of mathematics (arithmetic) from China during the reign of Nam-ri srong btsan, who died in 630.[32][33]

The table of sines by the Indian mathematician, Aryabhata, were translated into the Chinese mathematical book of the Kaiyuan Zhanjing, compiled in 718 AD during the Tang Dynasty.[34] Although the Chinese excelled in other fields of mathematics such as solid geometry, binomial theorem, and complex algebraic formulas, early forms of trigonometry were not as widely appreciated as in the contemporary Indian and Islamic mathematics.[35]

Yi Xing, the mathematician and Buddhist monk was credited for calculating the tangent table. Instead, the early Chinese used an empirical substitute known as chong cha, while practical use of plane trigonometry in using the sine, the tangent, and the secant were known.[34] Yi Xing was famed for his genius, and was known to have calculated the number of possible positions on a go board game (though without a symbol for zero he had difficulties expressing the number).

Song and Yuan mathematics

Northern Song Dynasty mathematician Jia Xian developed an additive multiplicative method for extraction of square root and cubic root which implemented the "Horner" rule.[36]

 
Yang Hui triangle (Pascal's triangle) using rod numerals, as depicted in a publication of Zhu Shijie in 1303 AD

Four outstanding mathematicians arose during the Song Dynasty and Yuan Dynasty, particularly in the twelfth and thirteenth centuries: Yang Hui, Qin Jiushao, Li Zhi (Li Ye), and Zhu Shijie. Yang Hui, Qin Jiushao, Zhu Shijie all used the Horner-Ruffini method six hundred years earlier to solve certain types of simultaneous equations, roots, quadratic, cubic, and quartic equations. Yang Hui was also the first person in history to discover and prove "Pascal's Triangle", along with its binomial proof (although the earliest mention of the Pascal's triangle in China exists before the eleventh century AD). Li Zhi on the other hand, investigated on a form of algebraic geometry based on tiān yuán shù. His book; Ceyuan haijing revolutionized the idea of inscribing a circle into triangles, by turning this geometry problem by algebra instead of the traditional method of using Pythagorean theorem. Guo Shoujing of this era also worked on spherical trigonometry for precise astronomical calculations. At this point of mathematical history, a lot of modern western mathematics were already discovered by Chinese mathematicians. Things grew quiet for a time until the thirteenth century Renaissance of Chinese math. This saw Chinese mathematicians solving equations with methods Europe would not know until the eighteenth century. The high point of this era came with Zhu Shijie's two books Suanxue qimeng and the Siyuan yujian. In one case he reportedly gave a method equivalent to Gauss's pivotal condensation.

Qin Jiushao (c. 1202–1261) was the first to introduce the zero symbol into Chinese mathematics.[37] Before this innovation, blank spaces were used instead of zeros in the system of counting rods.[38] One of the most important contribution of Qin Jiushao was his method of solving high order numerical equations. Referring to Qin's solution of a 4th order equation, Yoshio Mikami put it: "Who can deny the fact of Horner's illustrious process being used in China at least nearly six long centuries earlier than in Europe?"[39] Qin also solved a 10th order equation.[40]

Pascal's triangle was first illustrated in China by Yang Hui in his book Xiangjie Jiuzhang Suanfa (詳解九章算法), although it was described earlier around 1100 by Jia Xian.[41] Although the Introduction to Computational Studies (算學啓蒙) written by Zhu Shijie (fl. 13th century) in 1299 contained nothing new in Chinese algebra, it had a great impact on the development of Japanese mathematics.[42]

Algebra

Ceyuan haijing

Main article: Ceyuan haijing
 
Li Ye's inscribed circle in triangle:Diagram of a round town
 
Yang Hui's magic concentric circles – numbers on each circle and diameter (ignoring the middle 9) sum to 138

Ceyuan haijing (Chinese: 測圓海鏡; pinyin: Cèyuán Hǎijìng), or Sea-Mirror of the Circle Measurements, is a collection of 692 formula and 170 problems related to inscribed circle in a triangle, written by Li Zhi (or Li Ye) (1192–1272 AD). He used Tian yuan shu to convert intricated geometry problems into pure algebra problems. He then used fan fa, or Horner's method, to solve equations of degree as high as six, although he did not describe his method of solving equations.[43] "Li Chih (or Li Yeh, 1192–1279), a mathematician of Peking who was offered a government post by Khublai Khan in 1206, but politely found an excuse to decline it. His Ts'e-yuan hai-ching (Sea-Mirror of the Circle Measurements) includes 170 problems dealing with[...]some of the problems leading to polynomial equations of sixth degree. Although he did not describe his method of solution of equations, it appears that it was not very different from that used by Chu Shih-chieh and Horner. Others who used the Horner method were Ch'in Chiu-shao (ca. 1202 – ca.1261) and Yang Hui (fl. ca. 1261–1275).

Jade Mirror of the Four Unknowns

 
Facsimile of Zhu Shijie's Jade Mirror of Four Unknowns

Si-yüan yü-jian (四元玉鑒), or Jade Mirror of the Four Unknowns, was written by Zhu Shijie in 1303 AD and marks the peak in the development of Chinese algebra. The four elements, called heaven, earth, man and matter, represented the four unknown quantities in his algebraic equations. It deals with simultaneous equations and with equations of degrees as high as fourteen. The author uses the method of fan fa, today called Horner's method, to solve these equations.[44]

There are many summation series equations given without proof in the Mirror. A few of the summation series are:[45]

 
 

Mathematical Treatise in Nine Sections

Shu-shu chiu-chang, or Mathematical Treatise in Nine Sections, was written by the wealthy governor and minister Ch'in Chiu-shao (ca. 1202 – ca. 1261 AD) and with the invention of a method of solving simultaneous congruences, it marks the high point in Chinese indeterminate analysis.[43]

Magic squares and magic circles

The earliest known magic squares of order greater than three are attributed to Yang Hui (fl. ca. 1261–1275), who worked with magic squares of order as high as ten.[46] He also worked with magic circle.

Trigonometry

The embryonic state of trigonometry in China slowly began to change and advance during the Song Dynasty (960–1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendarical science and astronomical calculations.[34] The polymath Chinese scientist, mathematician and official Shen Kuo (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs.[34] Victor J. Katz writes that in Shen's formula "technique of intersecting circles", he created an approximation of the arc of a circle s by s = c + 2v2/d, where d is the diameter, v is the versine, c is the length of the chord c subtending the arc.[47] Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for spherical trigonometry developed in the 13th century by the mathematician and astronomer Guo Shoujing (1231–1316).[48] As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve the calendar system and Chinese astronomy.[34][49] Along with a later 17th-century Chinese illustration of Guo's mathematical proofs, Needham states that:

Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with two meridian arcs, one of which passed through the summer solstice point...By such methods he was able to obtain the du lü (degrees of equator corresponding to degrees of ecliptic), the ji cha (values of chords for given ecliptic arcs), and the cha lü (difference between chords of arcs differing by 1 degree).[50]

Despite the achievements of Shen and Guo's work in trigonometry, another substantial work in Chinese trigonometry would not be published again until 1607, with the dual publication of Euclid's Elements by Chinese official and astronomer Xu Guangqi (1562–1633) and the Italian Jesuit Matteo Ricci (1552–1610).[51]

Ming mathematics

After the overthrow of the Yuan Dynasty, China became suspicious of Mongol-favored knowledge. The court turned away from math and physics in favor of botany and pharmacology. Imperial examinations included little mathematics, and what little they included ignored recent developments. Martzloff writes:

At the end of the 16th century, Chinese autochthonous mathematics known by the Chinese themselves amounted to almost nothing, little more than calculation on the abacus, whilst in the 17th and 18th centuries nothing could be paralleled with the revolutionary progress in the theatre of European science. Moreover, at this same period, no one could report what had taken place in the more distant past, since the Chinese themselves only had a fragmentary knowledge of that. One should not forget that, in China itself, autochthonous mathematics was not rediscovered on a large scale prior to the last quarter of the 18th century.[52]

Correspondingly, scholars paid less attention to mathematics; pre-eminent mathematicians such as Gu Yingxiang and Tang Shunzhi appear to have been ignorant of the Tian yuan shu (Increase multiply) method.[53] Without oral interlocutors to explicate them, the texts rapidly became incomprehensible; worse yet, most problems could be solved with more elementary methods. To the average scholar, then, tianyuan seemed numerology. When Wu Jing collated all the mathematical works of previous dynasties into The Annotations of Calculations in the Nine Chapters on the Mathematical Art, he omitted Tian yuan shu and the increase multiply method.[54][failed verification]

 
An abacus.

Instead, mathematical progress became focused on computational tools. In 15 century, abacus came into its suan pan form. Easy to use and carry, both fast and accurate, it rapidly overtook rod calculus as the preferred form of computation. Zhusuan, the arithmetic calculation through abacus, inspired multiple new works. Suanfa Tongzong (General Source of Computational Methods), a 17-volume work published in 1592 by Cheng Dawei, remained in use for over 300 years.[55] Zhu Zaiyu, Prince of Zheng used 81 position abacus to calculate the square root and cubic root of 2 to 25 figure accuracy, a precision that enabled his development of the equal-temperament system.

Although this switch from counting rods to the abacus allowed for reduced computation times, it may have also led to the stagnation and decline of Chinese mathematics. The pattern rich layout of counting rod numerals on counting boards inspired many Chinese inventions in mathematics, such as the cross multiplication principle of fractions and methods for solving linear equations. Similarly, Japanese mathematicians were influenced by the counting rod numeral layout in their definition of the concept of a matrix. Algorithms for the abacus did not lead to similar conceptual advances. (This distinction, of course, is a modern one: until the 20th century, Chinese mathematics was exclusively a computational science.[56])

In the late 16th century, Matteo Ricci decided to published Western scientific works in order to establish a position at the Imperial Court. With the assistance of Xu Guangqi, he was able to translate Euclid's Elements using the same techniques used to teach classical Buddhist texts.[57] Other missionaries followed in his example, translating Western works on special functions (trigonometry and logarithms) that were neglected in the Chinese tradition.[58] However, contemporary scholars found the emphasis on proofs — as opposed to solved problems — baffling, and most continued to work from classical texts alone.[59]

Qing dynasty

Under the Kangxi Emperor, who learned Western mathematics from the Jesuits and was open to outside knowledge and ideas, Chinese mathematics enjoyed a brief period of official support.[60] At Kangxi's direction, Mei Goucheng and three other outstanding mathematicians compiled a 53-volume Shuli Jingyun [The Essence of Mathematical Study] (printed 1723) which gave a systematic introduction to western mathematical knowledge.[61] At the same time, Mei Goucheng also developed to Meishi Congshu Jiyang [The Compiled works of Mei]. Meishi Congshu Jiyang was an encyclopedic summary of nearly all schools of Chinese mathematics at that time, but it also included the cross-cultural works of Mei Wending (1633-1721), Goucheng's grandfather.[62][63] The enterprise sought to alleviate the difficulties for Chinese mathematicians working on Western mathematics in tracking down citations.[64]

However, no sooner were the encyclopedias published than the Yongzheng Emperor acceded to the throne. Yongzheng introduced a sharply anti-Western turn to Chinese policy, and banished most missionaries from the Court. With access to neither Western texts nor intelligible Chinese ones, Chinese mathematics stagnated.

In 1773, the Qianlong Emperor decided to compile Siku Quanshu (The Complete Library of the Four Treasuries). Dai Zhen (1724-1777) selected and proofread The Nine Chapters on the Mathematical Art from Yongle Encyclopedia and several other mathematical works from Han and Tang dynasties.[65] The long-missing mathematical works from Song and Yuan dynasties such as Si-yüan yü-jian and Ceyuan haijing were also found and printed, which directly led to a wave of new research.[66] The most annotated work were Jiuzhang suanshu xicaotushuo (The Illustrations of Calculation Process for The Nine Chapters on the Mathematical Art ) contributed by Li Huang and Siyuan yujian xicao (The Detailed Explanation of Si-yuan yu-jian) by Luo Shilin.[67]

Western influences

In 1840, the First Opium War forced China to open its door and look at the outside world, which also led to an influx of western mathematical studies at a rate unrivaled in the previous centuries. In 1852, the Chinese mathematician Li Shanlan and the British missionary Alexander Wylie co-translated the later nine volumes of Elements and 13 volumes on Algebra.[68][69] With the assistance of Joseph Edkins, more works on astronomy and calculus soon followed. Chinese scholars were initially unsure whether to approach the new works: was study of Western knowledge a form of submission to foreign invaders? But by the end of the century, it became clear that China could only begin to recover its sovereignty by incorporating Western works. Chinese scholars, taught in Western missionary schools, from (translated) Western texts, rapidly lost touch with the indigenous tradition. Those who were self-trained or in traditionalist circles nevertheless continued to work within the traditional framework of algorithmic mathematics without resorting to Western symbolism.[70] Yet, as Martzloff notes, "from 1911 onwards, solely Western mathematics has been practised in China."[71]

Western mathematics in modern China

Chinese mathematics experienced a great surge of revival following the establishment of a modern Chinese republic in 1912. Ever since then, modern Chinese mathematicians have made numerous achievements in various mathematical fields.

Some famous modern ethnic Chinese mathematicians include:

  • Shiing-Shen Chern was widely regarded as a leader in geometry and one of the greatest mathematicians of the twentieth century and was awarded the Wolf prize for his immense number of mathematical contributions.[72][73]
  • Ky Fan, made a tremendous number of fundamental contributions to many different fields of mathematics. His work in fixed point theory, in addition to influencing nonlinear functional analysis, has found wide application in mathematical economics and game theory, potential theory, calculus of variations, and differential equations.
  • Shing-Tung Yau, his contributions have influenced both physics and mathematics, and he has been active at the interface between geometry and theoretical physics and subsequently awarded the Fields medal for his contributions.
  • Terence Tao, an ethnic Chinese child prodigy who received his master's degree at age 16, was the youngest participant in the International Mathematical Olympiad's entire history, first competing at the age of ten, winning a bronze, silver, and gold medal. He remains the youngest winner of each of the three medals in the Olympiad's history. He went on to receive the Fields medal.
  • Yitang Zhang, a number theorist who established the first finite bound on gaps between prime numbers.
  • Chen Jingrun, a number theorist who proved that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes) which is now called Chen's theorem .[74] His work was known as a milestone in the research of Goldbach's conjecture.

Mathematics in the People's Republic of China

In 1949, at the beginning of the founding of the People's Republic of China, the government paid great attention to the cause of science although the country was in a predicament of lack of funds. The Chinese Academy of Sciences was established in November 1949. The Institute of Mathematics was formally established in July 1952. Then, the Chinese Mathematical Society and its founding journals restored and added other special journals. In the 18 years after 1949, the number of published papers accounted for more than three times the total number of articles before 1949. Many of them not only filled the gaps in China's past, but also reached the world's advanced level.[75]

During the chaos of the Cultural Revolution, the sciences declined. In the field of mathematics, in addition to Chen Jingrun, Hua Luogeng, Zhang Guanghou and other mathematicians struggling to continue their work. After the catastrophe, with the publication of Guo Moruo's literary "Spring of Science", Chinese sciences and mathematics experienced a revival. In 1977, a new mathematical development plan was formulated in Beijing, the work of the mathematics society was resumed, the journal was re-published, the academic journal was published, the mathematics education was strengthened, and basic theoretical research was strengthened.[75]

An important mathematical achievement of the Chinese mathematician in the direction of the power system is how Xia Zhihong proved the Painleve conjecture in 1988. When there are some initial states of N celestial bodies, one of the celestial bodies ran to infinity or speed in a limited time. Infinity is reached, that is, there are non-collision singularities. The Painleve conjecture is an important conjecture in the field of power systems proposed in 1895. A very important recent development for the 4-body problem is that Xue Jinxin and Dolgopyat proved a non-collision singularity in a simplified version of the 4-body system around 2013.[76]

In addition, in 2007, Shen Weixiao and Kozlovski, Van-Strien proved the Real Fatou conjecture: Real hyperbolic polynomials are dense in the space of real polynomials with fixed degree. This conjecture can be traced back to Fatou in the 1920s, and later Smale posed it in the 1960s. The proof of Real Fatou conjecture is one of the most important developments in conformal dynamics in the past decade.[76]

Performance at the IMO

In comparison to other participating countries at the International Mathematical Olympiad, China has highest team scores and has won the all-members-gold IMO with a full team the most number of times.[77]

Mathematical texts

Zhou Dynasty

Zhoubi Suanjing c. 1000 BCE-100 CE

  • Astronomical theories, and computation techniques
  • Proof of the Pythagorean theorem (Shang Gao Theorem)
  • Fractional computations
  • Pythagorean theorem for astronomical purposes

Nine Chapters on the Mathematical Art 1000 BCE? – 50 CE

  • ch.1, computational algorithm, area of plane figures, GCF, LCD
  • ch.2, proportions
  • ch.3, proportions
  • ch.4, square, cube roots, finding unknowns
  • ch.5, volume and usage of pi as 3
  • ch.6, proportions
  • ch,7, interdeterminate equations
  • ch.8, Gaussian elimination and matrices
  • ch.9, Pythagorean theorem (Gougu Theorem)

Han Dynasty

Book on Numbers and Computation 202 BC-186 BC

  • Calculation of the volume of various 3-dimensional shapes
  • Calculation of unknown side of rectangle, given area and one side
  • Using the false position method for finding roots and the extraction of approximate square roots
  • Conversion between different units

Mathematics in education

The first reference to a book being used in learning mathematics in China is dated to the second century CE (Hou Hanshu: 24, 862; 35,1207). We are told that Ma Xu (a youth ca 110) and Zheng Xuan (127-200) both studied the Nine Chapters on Mathematical procedures. C.Cullen claims that mathematics, in a manner akin to medicine, was taught orally. The stylistics of the Suàn shù shū from Zhangjiashan suggest that the text was assembled from various sources and then underwent codification.[78]

See also

References

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Citations


Sources

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Public domain
  •   This article incorporates text from The Encyclopædia Britannica: a dictionary of arts, sciences, literature and general information, Volume 26, by Hugh Chisholm, a publication from 1911, now in the public domain in the United States.
  •   This article incorporates text from The Life of the Buddha and the early history of his order: derived from Tibetan works in the Bkah-hgyur and Bstan-hgyur followed by notices on the early history of Tibet and Khoten, by Translated by William Woodville Rockhill, Ernst Leumann, Bunyiu Nanjio, a publication from 1907, now in the public domain in the United States.

External links

May 01, 2023
chinese, mathematics, mathematics, china, emerged, independently, 11th, century, chinese, independently, developed, real, number, system, that, includes, significantly, large, negative, numbers, more, than, numeral, system, base, base, algebra, geometry, numbe. Mathematics in China emerged independently by the 11th century BCE 1 The Chinese independently developed a real number system that includes significantly large and negative numbers more than one numeral system base 2 and base 10 algebra geometry number theory and trigonometry Since the Han Dynasty as diophantine approximation being a prominent numerical method the Chinese made substantial progress on polynomial evaluation Algorithms like regula falsi and expressions like continued fractions are widely used and have been well documented ever since They deliberately find the principal nth root of positive numbers and the roots of equations 2 3 The major texts from the period The Nine Chapters on the Mathematical Art and the Book on Numbers and Computation gave detailed processes for solving various mathematical problems in daily life 4 All procedures were computed using a counting board in both texts and they included inverse elements as well as Euclidean divisions The texts provide procedures similar to that of Gaussian elimination and Horner s method for linear algebra 5 The achievement of Chinese algebra reached a zenith in the 13th century during the Yuan dynasty with the development of tian yuan shu As a result of obvious linguistic and geographic barriers as well as content Chinese mathematics and the mathematics of the ancient Mediterranean world are presumed to have developed more or less independently up to the time when The Nine Chapters on the Mathematical Art reached its final form while the Book on Numbers and Computation and Huainanzi are roughly contemporary with classical Greek mathematics Some exchange of ideas across Asia through known cultural exchanges from at least Roman times is likely Frequently elements of the mathematics of early societies correspond to rudimentary results found later in branches of modern mathematics such as geometry or number theory The Pythagorean theorem for example has been attested to the time of the Duke of Zhou Knowledge of Pascal s triangle has also been shown to have existed in China centuries before Pascal 6 such as the Song dynasty Chinese polymath Shen Kuo Contents 1 Early Chinese mathematics 2 Qin mathematics 3 Han mathematics 3 1 Suan shu shu 3 2 The Nine Chapters on the Mathematical Art 3 3 Calculation of pi 3 4 Division and root extraction 3 5 Linear algebra 3 6 Liu Hui s commentary on The Nine Chapters on the Mathematical Art 4 Mathematics in the period of disunity 5 Tang mathematics 6 Song and Yuan mathematics 6 1 Algebra 6 1 1 Ceyuan haijing 6 1 2 Jade Mirror of the Four Unknowns 6 1 3 Mathematical Treatise in Nine Sections 6 1 4 Magic squares and magic circles 6 2 Trigonometry 7 Ming mathematics 8 Qing dynasty 9 Western influences 9 1 Western mathematics in modern China 10 Mathematics in the People s Republic of China 10 1 Performance at the IMO 11 Mathematical texts 12 Mathematics in education 13 See also 14 References 14 1 Citations 14 2 Sources 15 External linksEarly Chinese mathematics Edit Visual proof for the 3 4 5 triangle as in the Zhoubi Suanjing 500 200 BC Oracle bone script numeral system counting rod place value decimal Shang Dynasty 1600 1050 BC One of the oldest surviving mathematical works is the I Ching which greatly influenced written literature during the Zhou Dynasty 1050 256 BC For mathematics the book included a sophisticated use of hexagrams Leibniz pointed out the I Ching Yi Jing contained elements of binary numbers Since the Shang period the Chinese had already fully developed a decimal system Since early times Chinese understood basic arithmetic which dominated far eastern history algebra equations and negative numbers with counting rods citation needed Although the Chinese were more focused on arithmetic and advanced algebra for astronomical uses they were also the first to develop negative numbers algebraic geometry only Chinese geometry and the usage of decimals Math was one of the Liu Yi 六藝 or Six Arts students were required to master during the Zhou Dynasty 1122 256 BC Learning them all perfectly was required to be a perfect gentleman or in the Chinese sense a Renaissance Man Six Arts have their roots in the Confucian philosophy The oldest existent work on geometry in China comes from the philosophical Mohist canon of c 330 BC compiled by the followers of Mozi 470 390 BC The Mo Jing described various aspects of many fields associated with physical science and provided a small wealth of information on mathematics as well It provided an atomic definition of the geometric point stating that a line is separated into parts and the part which has no remaining parts i e cannot be divided into smaller parts and thus forms the extreme end of a line is a point 7 Much like Euclid s first and third definitions and Plato s beginning of a line the Mo Jing stated that a point may stand at the end of a line or at its beginning like a head presentation in childbirth As to its invisibility there is nothing similar to it 8 Similar to the atomists of Democritus the Mo Jing stated that a point is the smallest unit and cannot be cut in half since nothing cannot be halved 8 It stated that two lines of equal length will always finish at the same place 8 while providing definitions for the comparison of lengths and for parallels 9 along with principles of space and bounded space 10 It also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch 11 The book provided word recognition for circumference diameter and radius along with the definition of volume 12 The history of mathematical development lacks some evidence There are still debates about certain mathematical classics For example the Zhoubi Suanjing dates around 1200 1000 BC yet many scholars believed it was written between 300 and 250 BC The Zhoubi Suanjing contains an in depth proof of the Gougu Theorem a special case of the Pythagorean Theorem but focuses more on astronomical calculations However the recent archaeological discovery of the Tsinghua Bamboo Slips dated c 305 BC has revealed some aspects of pre Qin mathematics such as the first known decimal multiplication table 13 The abacus was first mentioned in the second century BC alongside calculation with rods suan zi in which small bamboo sticks are placed in successive squares of a checkerboard 14 Qin mathematics EditNot much is known about Qin dynasty mathematics or before due to the burning of books and burying of scholars circa 213 210 BC Knowledge of this period can be determined from civil projects and historical evidence The Qin dynasty created a standard system of weights Civil projects of the Qin dynasty were significant feats of human engineering Emperor Qin Shihuang 秦始皇 ordered many men to build large lifesize statues for the palace tomb along with other temples and shrines and the shape of the tomb was designed with geometric skills of architecture It is certain that one of the greatest feats of human history the Great Wall of China required many mathematical techniques All Qin dynasty buildings and grand projects used advanced computation formulas for volume area and proportion Qin bamboo cash purchased at the antiquarian market of Hong Kong by the Yuelu Academy according to the preliminary reports contains the earliest epigraphic sample of a mathematical treatise Han mathematics EditFurther information Science and technology of the Han dynasty Mathematics and astronomy The Nine Chapters on the Mathematical Art In the Han Dynasty numbers were developed into a place value decimal system and used on a counting board with a set of counting rods called chousuan consisting of only nine symbols with a blank space on the counting board representing zero 3 Negative numbers and fractions were also incorporated into solutions of the great mathematical texts of the period 4 The mathematical texts of the time the Suan shu shu and the Jiuzhang suanshu solved basic arithmetic problems such as addition subtraction multiplication and division 4 Furthermore they gave the processes for square and cubed root extraction which eventually was applied to solving quadratic equations up to the third order 5 Both texts also made substantial progress in Linear Algebra namely solving systems of equations with multiple unknowns 15 The value of pi is taken to be equal to three in both texts 16 However the mathematicians Liu Xin d 23 and Zhang Heng 78 139 gave more accurate approximations for pi than Chinese of previous centuries had used 4 Mathematics was developed to solve practical problems in the time such as division of land or problems related to division of payment 17 The Chinese did not focus on theoretical proofs based on geometry or algebra in the modern sense of proving equations to find area or volume 18 The Book of Computations and The Nine Chapters on the Mathematical Art provide numerous practical examples that would be used in daily life 18 Suan shu shu Edit The Suan shu shu Writings on Reckoning or The Book of Computations is an ancient Chinese text on mathematics approximately seven thousand characters in length written on 190 bamboo strips 19 It was discovered together with other writings in 1984 when archaeologists opened a tomb at Zhangjiashan in Hubei province From documentary evidence this tomb is known to have been closed in 186 BC early in the Western Han dynasty 4 While its relationship to the Nine Chapters is still under discussion by scholars some of its contents are clearly paralleled there The text of the Suan shu shu is however much less systematic than the Nine Chapters and appears to consist of a number of more or less independent short sections of text drawn from a number of sources 19 The Book of Computations contains many perquisites to problems that would be expanded upon in The Nine Chapters on the Mathematical Art 19 An example of the elementary mathematics in the Suan shu shu the square root is approximated by using false position method which says to combine the excess and deficiency as the divisor taking the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator combine them as the dividend 19 Furthermore The Book of Computations solves systems of two equations and two unknowns using the same false position method 15 The Nine Chapters on the Mathematical Art Edit The Nine Chapters on the Mathematical Art is a Chinese mathematics book its oldest archeological date being 179 AD traditionally dated 1000 BC but perhaps as early as 300 200 BC 20 Although the author s are unknown they made a major contribution in the eastern world Problems are set up with questions immediately followed by answers and procedure 17 There are no formal mathematical proofs within the text just a step by step procedure 21 The commentary of Liu Hui provided geometrical and algebraic proofs to the problems given within the text 3 The Nine Chapters on the Mathematical Art was one of the most influential of all Chinese mathematical books and it is composed of 246 problems 20 It was later incorporated into The Ten Computational Canons which became the core of mathematical education in later centuries 17 This book includes 246 problems on surveying agriculture partnerships engineering taxation calculation the solution of equations and the properties of right triangles 17 The Nine Chapters made significant additions to solving quadratic equations in a way similar to Horner s method 5 It also made advanced contributions to fangcheng or what is now known as linear algebra 15 Chapter seven solves system of linear equations with two unknowns using the false position method similar to The Book of Computations 15 Chapter eight deals with solving determinate and indeterminate simultaneous linear equations using positive and negative numbers with one problem dealing with solving four equations in five unknowns 15 The Nine Chapters solves systems of equations using methods similar to the modern Gaussian elimination and back substitution 15 The version of The Nine Chapters that has served as the foundation for modern renditions was a result of the efforts of the scholar Dai Zhen Transcribing the problems directly from Yongle Encyclopedia he then proceeded to make revisions to the original text along with the inclusion his own notes explaining his reasoning behind the alterations 22 His finished work would be first published in 1774 but a new revision would be published in 1776 to correct various errors as well as include a version of The Nine Chapters from the Southern Song that contained the commentaries of Lui Hui and Li Chunfeng The final version of Dai Zhen s work would come in 1777 titled Ripple Pavilion with this final rendition being widely distributed and coming to serve as the standard for modern versions of The Nine Chapters 23 However this version has come under scrutiny from Guo Shuchen alleging that the edited version still contains numerous errors and that not all of the original amendments were done by Dai Zhen himself 22 Calculation of pi Edit Problems in The Nine Chapters on the Mathematical Art take pi to be equal to three in calculating problems related to circles and spheres such as spherical surface area 20 There is no explicit formula given within the text for the calculation of pi to be three but it is used throughout the problems of both The Nine Chapters on the Mathematical Art and the Artificer s Record which was produced in the same time period 16 Historians believe that this figure of pi was calculated using the 3 1 relationship between the circumference and diameter of a circle 20 Some Han mathematicians attempted to improve this number such as Liu Xin who is believed to have estimated pi to be 3 154 4 Later Liu Hui attempted to improve the calculation by calculating pi to be 3 141024 a low estimate of the number Liu calculated this number by using polygons inside a hexagon as a lower limit compared to a circle 24 Zu Chongzhi later discovered the calculation of pi to be 3 1415926 lt p lt 3 1415927 by using polygons with 24 576 sides This calculation would be discovered in Europe during the 16th century 25 There is no explicit method or record of how he calculated this estimate 4 Division and root extraction Edit Basic arithmetic processes such as addition subtraction multiplication and division were present before the Han Dynasty 4 The Nine Chapters on the Mathematical Art take these basic operations for granted and simply instruct the reader to perform them 15 Han mathematicians calculated square and cube roots in a similar manner as division and problems on division and root extraction both occur in Chapter Four of The Nine Chapters on the Mathematical Art 26 Calculating the square and cube roots of numbers is done through successive approximation the same as division and often uses similar terms such as dividend shi and divisor fa throughout the process 5 This process of successive approximation was then extended to solving quadratics of the second and third order such as x 2 a b displaystyle x 2 a b using a method similar to Horner s method 5 The method was not extended to solve quadratics of the nth order during the Han Dynasty however this method was eventually used to solve these equations 5 Fangcheng on a counting board Linear algebra Edit The Book of Computations is the first known text to solve systems of equations with two unknowns 15 There are a total of three sets of problems within The Book of Computations involving solving systems of equations with the false position method which again are put into practical terms 15 Chapter Seven of The Nine Chapters on the Mathematical Art also deals with solving a system of two equations with two unknowns with the false position method 15 To solve for the greater of the two unknowns the false position method instructs the reader to cross multiply the minor terms or zi which are the values given for the excess and deficit with the major terms mu 15 To solve for the lesser of the two unknowns simply add the minor terms together 15 Chapter Eight of The Nine Chapters on the Mathematical Art deals with solving infinite equations with infinite unknowns 15 This process is referred to as the fangcheng procedure throughout the chapter 15 Many historians chose to leave the term fangcheng untranslated due to conflicting evidence of what the term means Many historians translate the word to linear algebra today In this chapter the process of Gaussian elimination and back substitution are used to solve systems of equations with many unknowns 15 Problems were done on a counting board and included the use of negative numbers as well as fractions 15 The counting board was effectively a matrix where the top line is the first variable of one equation and the bottom was the last 15 Liu Hui s commentary on The Nine Chapters on the Mathematical Art Edit Liu Hui s exhaustion method Liu Hui s commentary on The Nine Chapters on the Mathematical Art is the earliest edition of the original text available 20 Hui is believed by most to be a mathematician shortly after the Han dynasty Within his commentary Hui qualified and proved some of the problems from either an algebraic or geometrical standpoint 18 For instance throughout The Nine Chapters on the Mathematical Art the value of pi is taken to be equal to three in problems regarding circles or spheres 16 In his commentary Liu Hui finds a more accurate estimation of pi using the method of exhaustion 16 The method involves creating successive polynomials within a circle so that eventually the area of a higher order polygon will be identical to that of the circle 16 From this method Liu Hui asserted that the value of pi is about 3 14 4 Liu Hui also presented a geometric proof of square and cubed root extraction similar to the Greek method which involved cutting a square or cube in any line or section and determining the square root through symmetry of the remaining rectangles 26 Mathematics in the period of disunity Edit Liu Hui s Survey of sea island Sunzi algorithm for division 400 AD al Khwarizmi division in the 9th century Statue of Zu Chongzhi In the third century Liu Hui wrote his commentary on the Nine Chapters and also wrote Haidao Suanjing which dealt with using Pythagorean theorem already known by the 9 chapters and triple quadruple triangulation for surveying his accomplishment in the mathematical surveying exceeded those accomplished in the west by a millennium 27 He was the first Chinese mathematician to calculate p 3 1416 with his p algorithm He discovered the usage of Cavalieri s principle to find an accurate formula for the volume of a cylinder and also developed elements of the infinitesimal calculus during the 3rd century CE fraction interpolation for pi In the fourth century another influential mathematician named Zu Chongzhi introduced the Da Ming Li This calendar was specifically calculated to predict many cosmological cycles that will occur in a period of time Very little is really known about his life Today the only sources are found in Book of Sui we now know that Zu Chongzhi was one of the generations of mathematicians He used Liu Hui s pi algorithm applied to a 12288 gon and obtained a value of pi to 7 accurate decimal places between 3 1415926 and 3 1415927 which would remain the most accurate approximation of p available for the next 900 years He also applied He Chengtian s interpolation for approximating irrational number with fraction in his astronomy and mathematical works he obtained 355 113 displaystyle tfrac 355 113 as a good fraction approximate for pi Yoshio Mikami commented that neither the Greeks nor the Hindus nor Arabs knew about this fraction approximation to pi not until the Dutch mathematician Adrian Anthoniszoom rediscovered it in 1585 the Chinese had therefore been possessed of this the most extraordinary of all fractional values over a whole millennium earlier than Europe 28 Along with his son Zu Geng Zu Chongzhi applied the Cavalieri s principle to find an accurate solution for calculating the volume of the sphere Besides containing formulas for the volume of the sphere his book also included formulas of cubic equations and the accurate value of pi His work Zhui Shu was discarded out of the syllabus of mathematics during the Song dynasty and lost Many believed that Zhui Shu contains the formulas and methods for linear matrix algebra algorithm for calculating the value of p formula for the volume of the sphere The text should also associate with his astronomical methods of interpolation which would contain knowledge similar to our modern mathematics A mathematical manual called Sunzi mathematical classic dated between 200 and 400 CE contained the most detailed step by step description of multiplication and division algorithm with counting rods Intriguingly Sunzi may have influenced the development of place value systems and place value systems and the associated Galley division in the West European sources learned place value techniques in the 13th century from a Latin translation an early 9th century work by Al Khwarizmi Khwarizmi s presentation is almost identical to the division algorithm in Sunzi even regarding stylistic matters for example using blank spaces to represent trailing zeros the similarity suggests that the results may not have been an independent discovery Islamic commentators on Al Khwarizmi s work believed that it primarily summarized Hindu knowledge Al Khwarizmi s failure to cite his sources makes it difficult to determine whether those sources had in turn learned the procedure from China 29 In the fifth century the manual called Zhang Qiujian suanjing discussed linear and quadratic equations By this point the Chinese had the concept of negative numbers Tang mathematics EditBy the Tang Dynasty study of mathematics was fairly standard in the great schools The Ten Computational Canons was a collection of ten Chinese mathematical works compiled by early Tang dynasty mathematician Li Chunfeng 李淳風 602 670 as the official mathematical texts for imperial examinations in mathematics The Sui dynasty and Tang dynasty ran the School of Computations 30 Wang Xiaotong was a great mathematician in the beginning of the Tang Dynasty and he wrote a book Jigu Suanjing Continuation of Ancient Mathematics where numerical solutions which general cubic equations appear for the first time 31 The Tibetans obtained their first knowledge of mathematics arithmetic from China during the reign of Nam ri srong btsan who died in 630 32 33 The table of sines by the Indian mathematician Aryabhata were translated into the Chinese mathematical book of the Kaiyuan Zhanjing compiled in 718 AD during the Tang Dynasty 34 Although the Chinese excelled in other fields of mathematics such as solid geometry binomial theorem and complex algebraic formulas early forms of trigonometry were not as widely appreciated as in the contemporary Indian and Islamic mathematics 35 Yi Xing the mathematician and Buddhist monk was credited for calculating the tangent table Instead the early Chinese used an empirical substitute known as chong cha while practical use of plane trigonometry in using the sine the tangent and the secant were known 34 Yi Xing was famed for his genius and was known to have calculated the number of possible positions on a go board game though without a symbol for zero he had difficulties expressing the number Song and Yuan mathematics EditNorthern Song Dynasty mathematician Jia Xian developed an additive multiplicative method for extraction of square root and cubic root which implemented the Horner rule 36 Yang Hui triangle Pascal s triangle using rod numerals as depicted in a publication of Zhu Shijie in 1303 AD Four outstanding mathematicians arose during the Song Dynasty and Yuan Dynasty particularly in the twelfth and thirteenth centuries Yang Hui Qin Jiushao Li Zhi Li Ye and Zhu Shijie Yang Hui Qin Jiushao Zhu Shijie all used the Horner Ruffini method six hundred years earlier to solve certain types of simultaneous equations roots quadratic cubic and quartic equations Yang Hui was also the first person in history to discover and prove Pascal s Triangle along with its binomial proof although the earliest mention of the Pascal s triangle in China exists before the eleventh century AD Li Zhi on the other hand investigated on a form of algebraic geometry based on tian yuan shu His book Ceyuan haijing revolutionized the idea of inscribing a circle into triangles by turning this geometry problem by algebra instead of the traditional method of using Pythagorean theorem Guo Shoujing of this era also worked on spherical trigonometry for precise astronomical calculations At this point of mathematical history a lot of modern western mathematics were already discovered by Chinese mathematicians Things grew quiet for a time until the thirteenth century Renaissance of Chinese math This saw Chinese mathematicians solving equations with methods Europe would not know until the eighteenth century The high point of this era came with Zhu Shijie s two books Suanxue qimeng and the Siyuan yujian In one case he reportedly gave a method equivalent to Gauss s pivotal condensation Qin Jiushao c 1202 1261 was the first to introduce the zero symbol into Chinese mathematics 37 Before this innovation blank spaces were used instead of zeros in the system of counting rods 38 One of the most important contribution of Qin Jiushao was his method of solving high order numerical equations Referring to Qin s solution of a 4th order equation Yoshio Mikami put it Who can deny the fact of Horner s illustrious process being used in China at least nearly six long centuries earlier than in Europe 39 Qin also solved a 10th order equation 40 Pascal s triangle was first illustrated in China by Yang Hui in his book Xiangjie Jiuzhang Suanfa 詳解九章算法 although it was described earlier around 1100 by Jia Xian 41 Although the Introduction to Computational Studies 算學啓蒙 written by Zhu Shijie fl 13th century in 1299 contained nothing new in Chinese algebra it had a great impact on the development of Japanese mathematics 42 Algebra Edit Ceyuan haijing Edit Main article Ceyuan haijing Li Ye s inscribed circle in triangle Diagram of a round town Yang Hui s magic concentric circles numbers on each circle and diameter ignoring the middle 9 sum to 138 Ceyuan haijing Chinese 測圓海鏡 pinyin Ceyuan Hǎijing or Sea Mirror of the Circle Measurements is a collection of 692 formula and 170 problems related to inscribed circle in a triangle written by Li Zhi or Li Ye 1192 1272 AD He used Tian yuan shu to convert intricated geometry problems into pure algebra problems He then used fan fa or Horner s method to solve equations of degree as high as six although he did not describe his method of solving equations 43 Li Chih or Li Yeh 1192 1279 a mathematician of Peking who was offered a government post by Khublai Khan in 1206 but politely found an excuse to decline it His Ts e yuan hai ching Sea Mirror of the Circle Measurements includes 170 problems dealing with some of the problems leading to polynomial equations of sixth degree Although he did not describe his method of solution of equations it appears that it was not very different from that used by Chu Shih chieh and Horner Others who used the Horner method were Ch in Chiu shao ca 1202 ca 1261 and Yang Hui fl ca 1261 1275 Jade Mirror of the Four Unknowns Edit Facsimile of Zhu Shijie s Jade Mirror of Four Unknowns Si yuan yu jian 四元玉鑒 or Jade Mirror of the Four Unknowns was written by Zhu Shijie in 1303 AD and marks the peak in the development of Chinese algebra The four elements called heaven earth man and matter represented the four unknown quantities in his algebraic equations It deals with simultaneous equations and with equations of degrees as high as fourteen The author uses the method of fan fa today called Horner s method to solve these equations 44 There are many summation series equations given without proof in the Mirror A few of the summation series are 45 1 2 2 2 3 2 n 2 n n 1 2 n 1 3 displaystyle 1 2 2 2 3 2 cdots n 2 n n 1 2n 1 over 3 1 8 30 80 n 2 n 1 n 2 3 n n 1 n 2 n 3 4 n 1 5 displaystyle 1 8 30 80 cdots n 2 n 1 n 2 over 3 n n 1 n 2 n 3 4n 1 over 5 Mathematical Treatise in Nine Sections Edit Shu shu chiu chang or Mathematical Treatise in Nine Sections was written by the wealthy governor and minister Ch in Chiu shao ca 1202 ca 1261 AD and with the invention of a method of solving simultaneous congruences it marks the high point in Chinese indeterminate analysis 43 Magic squares and magic circles Edit The earliest known magic squares of order greater than three are attributed to Yang Hui fl ca 1261 1275 who worked with magic squares of order as high as ten 46 He also worked with magic circle Trigonometry Edit The embryonic state of trigonometry in China slowly began to change and advance during the Song Dynasty 960 1279 where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendarical science and astronomical calculations 34 The polymath Chinese scientist mathematician and official Shen Kuo 1031 1095 used trigonometric functions to solve mathematical problems of chords and arcs 34 Victor J Katz writes that in Shen s formula technique of intersecting circles he created an approximation of the arc of a circle s by s c 2v2 d where d is the diameter v is the versine c is the length of the chord c subtending the arc 47 Sal Restivo writes that Shen s work in the lengths of arcs of circles provided the basis for spherical trigonometry developed in the 13th century by the mathematician and astronomer Guo Shoujing 1231 1316 48 As the historians L Gauchet and Joseph Needham state Guo Shoujing used spherical trigonometry in his calculations to improve the calendar system and Chinese astronomy 34 49 Along with a later 17th century Chinese illustration of Guo s mathematical proofs Needham states that Guo used a quadrangular spherical pyramid the basal quadrilateral of which consisted of one equatorial and one ecliptic arc together with two meridian arcs one of which passed through the summer solstice point By such methods he was able to obtain the du lu degrees of equator corresponding to degrees of ecliptic the ji cha values of chords for given ecliptic arcs and the cha lu difference between chords of arcs differing by 1 degree 50 Despite the achievements of Shen and Guo s work in trigonometry another substantial work in Chinese trigonometry would not be published again until 1607 with the dual publication of Euclid s Elements by Chinese official and astronomer Xu Guangqi 1562 1633 and the Italian Jesuit Matteo Ricci 1552 1610 51 Ming mathematics EditAfter the overthrow of the Yuan Dynasty China became suspicious of Mongol favored knowledge The court turned away from math and physics in favor of botany and pharmacology Imperial examinations included little mathematics and what little they included ignored recent developments Martzloff writes At the end of the 16th century Chinese autochthonous mathematics known by the Chinese themselves amounted to almost nothing little more than calculation on the abacus whilst in the 17th and 18th centuries nothing could be paralleled with the revolutionary progress in the theatre of European science Moreover at this same period no one could report what had taken place in the more distant past since the Chinese themselves only had a fragmentary knowledge of that One should not forget that in China itself autochthonous mathematics was not rediscovered on a large scale prior to the last quarter of the 18th century 52 Correspondingly scholars paid less attention to mathematics pre eminent mathematicians such as Gu Yingxiang and Tang Shunzhi appear to have been ignorant of the Tian yuan shu Increase multiply method 53 Without oral interlocutors to explicate them the texts rapidly became incomprehensible worse yet most problems could be solved with more elementary methods To the average scholar then tianyuan seemed numerology When Wu Jing collated all the mathematical works of previous dynasties into The Annotations of Calculations in the Nine Chapters on the Mathematical Art he omitted Tian yuan shu and the increase multiply method 54 failed verification An abacus Instead mathematical progress became focused on computational tools In 15 century abacus came into its suan pan form Easy to use and carry both fast and accurate it rapidly overtook rod calculus as the preferred form of computation Zhusuan the arithmetic calculation through abacus inspired multiple new works Suanfa Tongzong General Source of Computational Methods a 17 volume work published in 1592 by Cheng Dawei remained in use for over 300 years 55 Zhu Zaiyu Prince of Zheng used 81 position abacus to calculate the square root and cubic root of 2 to 25 figure accuracy a precision that enabled his development of the equal temperament system Although this switch from counting rods to the abacus allowed for reduced computation times it may have also led to the stagnation and decline of Chinese mathematics The pattern rich layout of counting rod numerals on counting boards inspired many Chinese inventions in mathematics such as the cross multiplication principle of fractions and methods for solving linear equations Similarly Japanese mathematicians were influenced by the counting rod numeral layout in their definition of the concept of a matrix Algorithms for the abacus did not lead to similar conceptual advances This distinction of course is a modern one until the 20th century Chinese mathematics was exclusively a computational science 56 In the late 16th century Matteo Ricci decided to published Western scientific works in order to establish a position at the Imperial Court With the assistance of Xu Guangqi he was able to translate Euclid s Elements using the same techniques used to teach classical Buddhist texts 57 Other missionaries followed in his example translating Western works on special functions trigonometry and logarithms that were neglected in the Chinese tradition 58 However contemporary scholars found the emphasis on proofs as opposed to solved problems baffling and most continued to work from classical texts alone 59 Qing dynasty EditUnder the Kangxi Emperor who learned Western mathematics from the Jesuits and was open to outside knowledge and ideas Chinese mathematics enjoyed a brief period of official support 60 At Kangxi s direction Mei Goucheng and three other outstanding mathematicians compiled a 53 volume Shuli Jingyun The Essence of Mathematical Study printed 1723 which gave a systematic introduction to western mathematical knowledge 61 At the same time Mei Goucheng also developed to Meishi Congshu Jiyang The Compiled works of Mei Meishi Congshu Jiyang was an encyclopedic summary of nearly all schools of Chinese mathematics at that time but it also included the cross cultural works of Mei Wending 1633 1721 Goucheng s grandfather 62 63 The enterprise sought to alleviate the difficulties for Chinese mathematicians working on Western mathematics in tracking down citations 64 However no sooner were the encyclopedias published than the Yongzheng Emperor acceded to the throne Yongzheng introduced a sharply anti Western turn to Chinese policy and banished most missionaries from the Court With access to neither Western texts nor intelligible Chinese ones Chinese mathematics stagnated In 1773 the Qianlong Emperor decided to compile Siku Quanshu The Complete Library of the Four Treasuries Dai Zhen 1724 1777 selected and proofread The Nine Chapters on the Mathematical Art from Yongle Encyclopedia and several other mathematical works from Han and Tang dynasties 65 The long missing mathematical works from Song and Yuan dynasties such as Si yuan yu jian and Ceyuan haijing were also found and printed which directly led to a wave of new research 66 The most annotated work were Jiuzhang suanshu xicaotushuo The Illustrations of Calculation Process for The Nine Chapters on the Mathematical Art contributed by Li Huang and Siyuan yujian xicao The Detailed Explanation of Si yuan yu jian by Luo Shilin 67 Western influences EditIn 1840 the First Opium War forced China to open its door and look at the outside world which also led to an influx of western mathematical studies at a rate unrivaled in the previous centuries In 1852 the Chinese mathematician Li Shanlan and the British missionary Alexander Wylie co translated the later nine volumes of Elements and 13 volumes on Algebra 68 69 With the assistance of Joseph Edkins more works on astronomy and calculus soon followed Chinese scholars were initially unsure whether to approach the new works was study of Western knowledge a form of submission to foreign invaders But by the end of the century it became clear that China could only begin to recover its sovereignty by incorporating Western works Chinese scholars taught in Western missionary schools from translated Western texts rapidly lost touch with the indigenous tradition Those who were self trained or in traditionalist circles nevertheless continued to work within the traditional framework of algorithmic mathematics without resorting to Western symbolism 70 Yet as Martzloff notes from 1911 onwards solely Western mathematics has been practised in China 71 Western mathematics in modern China Edit Chinese mathematics experienced a great surge of revival following the establishment of a modern Chinese republic in 1912 Ever since then modern Chinese mathematicians have made numerous achievements in various mathematical fields Some famous modern ethnic Chinese mathematicians include Shiing Shen Chern was widely regarded as a leader in geometry and one of the greatest mathematicians of the twentieth century and was awarded the Wolf prize for his immense number of mathematical contributions 72 73 Ky Fan made a tremendous number of fundamental contributions to many different fields of mathematics His work in fixed point theory in addition to influencing nonlinear functional analysis has found wide application in mathematical economics and game theory potential theory calculus of variations and differential equations Shing Tung Yau his contributions have influenced both physics and mathematics and he has been active at the interface between geometry and theoretical physics and subsequently awarded the Fields medal for his contributions Terence Tao an ethnic Chinese child prodigy who received his master s degree at age 16 was the youngest participant in the International Mathematical Olympiad s entire history first competing at the age of ten winning a bronze silver and gold medal He remains the youngest winner of each of the three medals in the Olympiad s history He went on to receive the Fields medal Yitang Zhang a number theorist who established the first finite bound on gaps between prime numbers Chen Jingrun a number theorist who proved that every sufficiently large even number can be written as the sum of either two primes or a prime and a semiprime the product of two primes which is now called Chen s theorem 74 His work was known as a milestone in the research of Goldbach s conjecture Mathematics in the People s Republic of China EditIn 1949 at the beginning of the founding of the People s Republic of China the government paid great attention to the cause of science although the country was in a predicament of lack of funds The Chinese Academy of Sciences was established in November 1949 The Institute of Mathematics was formally established in July 1952 Then the Chinese Mathematical Society and its founding journals restored and added other special journals In the 18 years after 1949 the number of published papers accounted for more than three times the total number of articles before 1949 Many of them not only filled the gaps in China s past but also reached the world s advanced level 75 During the chaos of the Cultural Revolution the sciences declined In the field of mathematics in addition to Chen Jingrun Hua Luogeng Zhang Guanghou and other mathematicians struggling to continue their work After the catastrophe with the publication of Guo Moruo s literary Spring of Science Chinese sciences and mathematics experienced a revival In 1977 a new mathematical development plan was formulated in Beijing the work of the mathematics society was resumed the journal was re published the academic journal was published the mathematics education was strengthened and basic theoretical research was strengthened 75 An important mathematical achievement of the Chinese mathematician in the direction of the power system is how Xia Zhihong proved the Painleve conjecture in 1988 When there are some initial states of N celestial bodies one of the celestial bodies ran to infinity or speed in a limited time Infinity is reached that is there are non collision singularities The Painleve conjecture is an important conjecture in the field of power systems proposed in 1895 A very important recent development for the 4 body problem is that Xue Jinxin and Dolgopyat proved a non collision singularity in a simplified version of the 4 body system around 2013 76 In addition in 2007 Shen Weixiao and Kozlovski Van Strien proved the Real Fatou conjecture Real hyperbolic polynomials are dense in the space of real polynomials with fixed degree This conjecture can be traced back to Fatou in the 1920s and later Smale posed it in the 1960s The proof of Real Fatou conjecture is one of the most important developments in conformal dynamics in the past decade 76 Performance at the IMO Edit In comparison to other participating countries at the International Mathematical Olympiad China has highest team scores and has won the all members gold IMO with a full team the most number of times 77 Mathematical texts EditZhou DynastyZhoubi Suanjing c 1000 BCE 100 CE Astronomical theories and computation techniques Proof of the Pythagorean theorem Shang Gao Theorem Fractional computations Pythagorean theorem for astronomical purposesNine Chapters on the Mathematical Art 1000 BCE 50 CE ch 1 computational algorithm area of plane figures GCF LCD ch 2 proportions ch 3 proportions ch 4 square cube roots finding unknowns ch 5 volume and usage of pi as 3 ch 6 proportions ch 7 interdeterminate equations ch 8 Gaussian elimination and matrices ch 9 Pythagorean theorem Gougu Theorem Han DynastyBook on Numbers and Computation 202 BC 186 BC Calculation of the volume of various 3 dimensional shapes Calculation of unknown side of rectangle given area and one side Using the false position method for finding roots and the extraction of approximate square roots Conversion between different unitsMathematics in education EditThe first reference to a book being used in learning mathematics in China is dated to the second century CE Hou Hanshu 24 862 35 1207 We are told that Ma Xu a youth ca 110 and Zheng Xuan 127 200 both studied the Nine Chapters on Mathematical procedures C Cullen claims that mathematics in a manner akin to medicine was taught orally The stylistics of the Suan shu shu from Zhangjiashan suggest that the text was assembled from various sources and then underwent codification 78 See also EditChinese astronomy History of mathematics Indian mathematics Islamic mathematics Japanese mathematics List of Chinese discoveries List of Chinese mathematicians Numbers in Chinese cultureReferences Edit Chinese overview Needham 1986 pp 65 66 a b c Chemla Karine East Asian Mathematics Britannica Online Encyclopedia a b c d e f g h i Needham Joseph 1959 Science and Civilization in China England Cambridge University Press pp 1 886 ISBN 0 521 05801 5 a b c d e f Needham Joseph 1955 Horner s Method in Chinese Mathematics T oung Pao Second Series 43 5 345 401 JSTOR 4527405 Frank J Swetz and T I Kao Was Pythagoras Chinese Needham Volume 3 91 a b c Needham Volume 3 92 Needham Volume 3 92 93 Needham Volume 3 93 Needham Volume 3 93 94 Needham Volume 3 94 Qiu Jane 7 January 2014 Ancient times table hidden in Chinese bamboo strips Nature doi 10 1038 nature 2014 14482 S2CID 130132289 Retrieved 15 September 2016 Ifrah Georges 2001 The Universal History of Computing From the Abacus to the Quantum Computer New York NY John Wiley amp Sons Inc ISBN 978 0471396710 a b c d e f g h i j k l m n o p q Hart Roger The Chinese Roots of Linear Algebra Johns Hopkins University pp 11 85 ISBN 978 0801897559 a b c d e Lennart Bergren 1997 Pi A Source Book New York ISBN 978 1 4757 2738 8 a b c d Lay Yong Lam June 1994 Nine Chapters on the Mathematical Art An Overview Archive for History of Exact Sciences 47 1 1 51 doi 10 1007 BF01881700 JSTOR 41133972 S2CID 123502226 a b c Siu Man Keung 1993 Proof and Pedagogy in Ancient China Educational Studies in Mathematics 24 4 345 357 doi 10 1007 BF01273370 JSTOR 3482649 S2CID 120420378 a b c d Dauben Joseph W 2008 算数書 Suan Shu Shu A Book on Numbers and Computations English Translation with Commentary Archive for History of Exact Sciences 62 2 91 178 doi 10 1007 s00407 007 0124 1 JSTOR 41134274 S2CID 125757029 a b c d e Dauben Joseph 2013 九章箅术 Jiu zhang suan shu Nine Chapters on the Art of Mathematics An Appraisal of the Text its Editions and Translations Sudhoffs Archiv 97 2 199 235 doi 10 25162 sudhoff 2013 0017 JSTOR 43694474 PMID 24707775 S2CID 1159700 Straffin Philip D 1998 Liu Hui and the First Golden Age of Chinese Mathematics Mathematics Magazine 71 3 163 181 doi 10 2307 2691200 JSTOR 2691200 a b Hart Roger 2011 The Chinese roots of linear algebra Baltimore MD Johns Hopkins University Press pp 32 33 ISBN 978 0 8018 9958 4 Dauben Joseph W 2013 九章箅术 Jiu zhang suan shu Nine Chapters on the Art of Mathematics An Appraisal of the Text its Editions and Translations Sudhoffs Archiv 97 2 18 19 doi 10 25162 sudhoff 2013 0017 ISSN 0039 4564 JSTOR 43694474 S2CID 1159700 Hart Robert 2011 The Chinese Roots of Linear Algebra Baltimore MD Johns Hopkins University Press p 39 ISBN 9780801899584 Robin Wilson 2013 Early Chinese Mathematics Math Intelligencer 35 2 80 doi 10 1007 s00283 013 9364 x S2CID 122920358 a b Yong Lam Lay 1970 The Geometrical Basis of the Ancient Chinese Square Root Method Isis 61 1 92 102 doi 10 1086 350581 JSTOR 229151 S2CID 145059170 Frank J Swetz The Sea Island Mathematical Manual Surveying and Mathematics in Ancient China 4 2 Chinese Surveying Accomplishments A Comparative Retrospection p63 The Pennsylvania State University Press 1992 ISBN 0 271 00799 0 Yoshio Mikami The Development of Mathematics in China and Japan chap 7 p 50 reprint of 1913 edition Chelsea NY Library of Congress catalog 61 13497 Lam Lay Yong 1996 The Development of Hindu Arabic and Traditional Chinese Arithmetic PDF Chinese Science 13 35 54 Archived from the original PDF on 2012 03 21 Retrieved 2015 12 31 Alexander Karp Gert Schubring 25 January 2014 Handbook on the History of Mathematics Education Springer Science amp Business Media pp 59 ISBN 978 1 4614 9155 2 Yoshio Mikami Mathematics in China and Japan p53 Chisholm Hugh ed 1911 Tibet s v History Encyclopaedia Britannica Vol 26 11th ed Cambridge University Press p 926 gNam ri srong btsan who died in 630 During his reign the Tibetans obtained their first knowledge of arithmetic and medicine from China Translated by William Woodville Rockhill Ernst Leumann Bunyiu Nanjio 1907 The Life of the Buddha and the early history of his order derived from Tibetan works in the Bkah hgyur and Bstan hgyur followed by notices on the early history of Tibet and Khoten K Paul Trench Trubner p 211 Retrieved 2011 07 01 sixth century the tibetans obtained their first knowledge of arithmetic and medicine from the chinese a href Template Cite book html title Template Cite book cite book a CS1 maint multiple names authors list link a b c d e Needham Volume 3 109 Needham Volume 3 108 109 Martzloff 1987 p 142 Needham Volume 3 43 Needham Volume 3 62 63 Yoshio Mikami The development of Mathematics in China and Japan p77 Leipzig 1912 Ulrich Librecht Chinese Mathematics in the Thirteenth Century p 211 Dover 1973 Needham Volume 3 134 137 Needham Volume 3 46 a b Boyer 1991 China and India p 204 Boyer 1991 China and India p 203 Boyer 1991 China and India p 205 Boyer 1991 China and India pp 204 205 The same Horner device was used by Yang Hui about whose life almost nothing is known and who work has survived only in part Among his contributions that are extant are the earliest Chinese magic squares of order greater than three including two each of orders four through eight and one each of orders nine and ten Katz 308 Restivo Sal 1992 Mathematics in Society and History Sociological Inquiries Dordrecht Kluwer Academic Publishers p 32 ISBN 1 4020 0039 1 Gauchet 151 Needham Volume 3 109 110 Needham Volume 3 110 Martzloff 1987 p 4 He Ji Huan May 2004 Some interpolation formulas in Chinese ancient mathematics Applied Mathematics and Computation 152 2 367 371 doi 10 1016 s0096 3003 03 00559 9 ISSN 0096 3003 S2CID 28478650 Martzloff 1987 p 20 East Asian Journal on Applied Mathematics East Asian Journal on Applied Mathematics doi 10 4208 eajam Martzloff 1987 Martzloff 1987 p 21 Brucker Joseph 1912 Matteo Ricci The Catholic Encyclopedia New York Robert Appleton Company OCLC 174525342 Retrieved 17 August 2017 Martzloff 1987 p 29 Martzloff 1987 pp 25 8 Jami Catherine Qi Han 2003 01 01 The Reconstruction of Imperial Mathematics in China During the Kangxi Reign 1662 1722 Early Science and Medicine 8 2 88 110 doi 10 1163 157338203X00026 ISSN 1573 3823 Jami Catherine 2011 12 01 A mathematical scholar in Jiangnan The first half life of Mei Wending The Emperor s New Mathematics Western Learning and Imperial Authority During the Kangxi Reign 1662 1722 Oxford University Press pp 82 101 doi 10 1093 acprof oso 9780199601400 003 0005 ISBN 9780199601400 Retrieved 2018 07 28 Elman Benjamin A 2005 On their own terms science in China 1550 1900 Cambridge Mass Harvard University Press ISBN 9780674036475 OCLC 443109938 Martzloff 1987 p 28 Minghui Hu 2017 02 14 China s transition to modernity the new classical vision of Dai Zhen Seattle ISBN 978 0295741802 OCLC 963736201 Jean Claude Martzloff A History of Chinese Mathematics Springer 1997 ISBN 3 540 33782 2 Catherine Jami 2012 The emperor s new mathematics Western learning and imperial authority during the Kangxi Reign 1662 1722 Oxford Oxford University Press ISBN 9780191729218 OCLC 774104121 Carlyle Edward Irving 1900 Wylie Alexander In Lee Sidney Dictionary of National Biography 63 London Smith Elder amp Co Li Shanlan s Summation Formulae A History of Chinese Mathematics 341 351 doi 10 1007 978 3 540 33783 6 18 Breard 2019 Martzloff 1987 pp 34 9 Chern biography www history mcs st and ac uk Retrieved 2017 01 16 12 06 2004 Renowned mathematician Shiing Shen Chern who revitalized the study of geometry has died at 93 in Tianjin China www berkeley edu Retrieved 2017 01 16 J R Chen 1973 On the representation of a larger even integer as the sum of a prime and the product of at most two primes Sci Sinica a b 孔国平 著 2015 中国数学思想史 中国学术思想史 南京大学出版社 ISBN 9787305147050 a b 孔国平 October 2012 中国数学史上最光辉的篇章 吉林科学技术出版社 ISBN 9787538461541 Team Results China at International Mathematical Olympiad Christopher Cullen Numbers numeracy and the cosmos in Loewe Nylan China s Early Empires 2010 337 8 Citations Edit Sources Edit Boyer C B 1991 A History of Mathematics rev by Uta C Merzbach paperback ed Wiley ISBN 0 471 54397 7 Breard Andrea 2019 Nine Chapters on Mathematical Modernity Essays on the Global Historical Entanglements of the Science of Numbers in China eBook ed Springer ISBN 978 3 319 93695 6 Dauben Joseph W 2007 Chinese Mathematics In Victor J Katz ed The Mathematics of Egypt Mesopotamia China India and Islam A Sourcebook Princeton University Press ISBN 978 0 691 11485 9 Lander Brian State Management of River Dikes in Early China New Sources on the Environmental History of the Central Yangzi Region T oung Pao 100 4 5 2014 325 62 Martzloff Jean Claude 1987 A history of chinese mathematics PDF Translated by Wilson Stephen S Berlin Springer p 4 doi 10 1007 978 3 540 33783 6 ISBN 9783540337836 OCLC 262687287 Retrieved 1 December 2018 Needham Joseph 1986 Science and Civilization in China Volume 3 Mathematics and the Sciences of the Heavens and the Earth Taipei Caves Books Ltd Public domain This article incorporates text fromThe Encyclopaedia Britannica a dictionary of arts sciences literature and general information Volume 26 by Hugh Chisholm a publication from 1911 now in the public domain in the United States This article incorporates text fromThe Life of the Buddha and the early history of his order derived from Tibetan works in the Bkah hgyur and Bstan hgyur followed by notices on the early history of Tibet and Khoten by Translated by William Woodville Rockhill Ernst Leumann Bunyiu Nanjio a publication from 1907 now in the public domain in the United States External links EditEarly mathematics texts Chinese Chinese Text Project Overview of Chinese mathematics Chinese Mathematics Through the Han Dynasty Primer of Mathematics by Zhu Shijie Retrieved from https en wikipedia org w index php title Chinese mathematics amp oldid 1140189511, wikipedia, wiki, book, books, library,

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