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Mathematics in China emerged independently by the 11th century BCE. The Chinese independently developed a
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
system that includes significantly large and
negative number In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed m ...
s, more than one
numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbo ...
( base 2 and base 10),
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
,
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
,
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
and
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
. Since the
Han Dynasty The Han dynasty (, ; ) was an Dynasties in Chinese history, imperial dynasty of China (202 BC – 9 AD, 25–220 AD), established by Emperor Gaozu of Han, Liu Bang (Emperor Gao) and ruled by the House of Liu. The dynasty was preceded by th ...
, as
diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated ...
being a prominent
numerical method In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. Mathem ...
, 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 ''n''th root of positive numbers and the roots of equations. 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. All procedures were computed using a counting board in both texts, and they included
inverse element In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
s as well as
Euclidean division In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than ...
s. The texts provide procedures similar to that of Gaussian elimination and Horner's method for
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...
. The achievement of Chinese algebra reached a zenith in the 13th century during the
Yuan dynasty The Yuan dynasty (), officially the Great Yuan (; xng, , , literally "Great Yuan State"), was a Mongol-led imperial dynasty of China and a successor state to the Mongol Empire after its division. It was established by Kublai, the fifth ...
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 The ''Huainanzi'' is an ancient Chinese text that consists of a collection of essays that resulted from a series of scholarly debates held at the court of Liu An, Prince of Huainan, sometime before 139. The ''Huainanzi'' blends Daoist, Confu ...
'' 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, such as the Song dynasty Chinese
polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a substantial number of subjects, known to draw on complex bodies of knowledge to solve specific pro ...
Shen Kuo Shen Kuo (; 1031–1095) or Shen Gua, courtesy name Cunzhong (存中) and pseudonym Mengqi (now usually given as Mengxi) Weng (夢溪翁),Yao (2003), 544. was a Chinese polymathic scientist and statesman of the Song dynasty (960–1279). Shen wa ...
.

# Early Chinese mathematics

Shang Dynasty The Shang dynasty (), also known as the Yin dynasty (), was a Chinese royal dynasty founded by Tang of Shang (Cheng Tang) that ruled in the Yellow River valley in the second millennium BC, traditionally succeeding the Xia dynasty and ...
(1600–1050 BC). One of the oldest surviving mathematical works is the '' I Ching'', which greatly influenced written literature during the
Zhou Dynasty The Zhou dynasty ( ; Old Chinese ( B&S): *''tiw'') was a royal dynasty of China that followed the Shang dynasty. Having lasted 789 years, the Zhou dynasty was the longest dynastic regime in Chinese history. The military control of China by t ...
(1050–256 BC). For mathematics, the book included a sophisticated use of hexagrams.
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
pointed out, the I Ching (Yi Jing) contained elements of
binary number A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one). The base-2 numeral system is a positional notation ...
s. Since the Shang period, the Chinese had already fully developed a
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numer ...
system. Since early times, Chinese understood basic
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19t ...
(which dominated far eastern history),
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
,
equations In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
, and negative numbers with
counting rods Counting rods () are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient East Asia. They are placed either horizontally or vertically to represent any integer or rational number. The written ...
. Although the Chinese were more focused on arithmetic and advanced algebra for
astronomical Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galax ...
uses, they were also the first to develop negative numbers,
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
(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 The Zhou dynasty ( ; Old Chinese ( B&S): *''tiw'') was a royal dynasty of China that followed the Shang dynasty. Having lasted 789 years, the Zhou dynasty was the longest dynastic regime in Chinese history. The military control of China by t ...
(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.Needham, Volume 3, 91. Much like
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
's first and third definitions and
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institutio ...
'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."Needham, Volume 3, 92. 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. It stated that two lines of equal length will always finish at the same place, while providing definitions for the ''comparison of lengths'' and for ''parallels'',Needham, Volume 3, 92-93. along with principles of space and bounded space.Needham, Volume 3, 93. It also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch.Needham, Volume 3, 93-94. The book provided word recognition for circumference, diameter, and radius, along with the definition of volume.Needham, Volume 3, 94. The history of mathematical development lacks some evidence. There are still debates about certain mathematical classics. For example, the ''
Zhoubi Suanjing The ''Zhoubi Suanjing'' () is one of the oldest Chinese mathematical texts. "Zhou" refers to the ancient Zhou dynasty (1046–256 BCE); "Bì" literally means " thigh", but in the book refers to the gnomon of a sundial. The book is dedicated to ...
'' 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 The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numer ...
multiplication table In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system. The decimal multiplication table was traditionally taught as an essent ...
. 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.

# Qin mathematics

Qin dynasty The Qin dynasty ( ; zh, c=秦朝, p=Qín cháo, w=), or Ch'in dynasty in Wade–Giles romanization ( zh, c=, p=, w=Ch'in ch'ao), was the first dynasty of Imperial China. Named for its heartland in Qin state (modern Gansu and Shaanxi), ...
mathematics, or before, due to the
burning of books and burying of scholars The burning of books and burying of scholars (), also known as burning the books and executing the ru scholars, refers to the purported burning of texts in 213 BCE and live burial of 460 Confucian scholars in 212 BCE by the Chinese emperor Q ...
, 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 Hong Kong ( (US) or (UK); , ), officially the Hong Kong Special Administrative Region of the People's Republic of China (abbr. Hong Kong SAR or HKSAR), is a city and special administrative region of China on the eastern Pearl River Delta i ...
by the
Yuelu Academy The Yuelu Academy (also as known as the ''Yuelu Academy of Classical Learning'', ) is on the east side of Yuelu Mountain in Changsha, Hunan province, on the west bank of the Xiang River. As one of the four most prestigious academies over the la ...
, according to the preliminary reports, contains the earliest epigraphic sample of a mathematical treatise.

# Han mathematics

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 Counting rods () are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient East Asia. They are placed either horizontally or vertically to represent any integer or rational number. The written ...
called chousuan, consisting of only nine symbols with a blank space on the counting board representing zero. Negative numbers and fractions were also incorporated into solutions of the great mathematical texts of the period. 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. Furthermore, they gave the processes for square and cubed root extraction, which eventually was applied to solving quadratic equations up to the third order. Both texts also made substantial progress in Linear Algebra, namely solving systems of equations with multiple unknowns. The value of pi is taken to be equal to three in both texts. 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. Mathematics was developed to solve practical problems in the time such as division of land or problems related to division of payment. 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. The Book of Computations and The Nine Chapters on the Mathematical Art provide numerous practical examples that would be used in daily life.

## 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. It was discovered together with other writings in 1984 when
archaeologist Archaeology or archeology is the scientific study of human activity through the recovery and analysis of material culture. The archaeological record consists of artifacts, architecture, biofacts or ecofacts, sites, and cultural landscapes ...
s opened a tomb at Zhangjiashan in
Hubei Hubei (; ; alternately Hupeh) is a landlocked province of the People's Republic of China, and is part of the Central China region. The name of the province means "north of the lake", referring to its position north of Dongting Lake. The prov ...
province. From documentary evidence this tomb is known to have been closed in 186 BC, early in the Western
Han dynasty The Han dynasty (, ; ) was an Dynasties in Chinese history, imperial dynasty of China (202 BC – 9 AD, 25–220 AD), established by Emperor Gaozu of Han, Liu Bang (Emperor Gao) and ruled by the House of Liu. The dynasty was preceded by th ...
. 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. The Book of Computations contains many perquisites to problems that would be expanded upon in The Nine Chapters on the Mathematical Art. An example of the elementary mathematics in the ''Suàn shù shū'', the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
is approximated by using
false position method In mathematics, the ''regula falsi'', method of false position, or false position method is a very old method for solving an equation with one unknown; this method, in modified form, is still in use. In simple terms, the method is the trial and e ...
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." Furthermore, The Book of Computations solves systems of two equations and two unknowns using the same false position method.

## ''The Nine Chapters on the Mathematical Art''

'' The Nine Chapters on the Mathematical Art'' is a Chinese
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
book, its oldest archeological date being 179 AD (traditionally dated 1000 BC), but perhaps as early as 300–200 BC. 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. There are no formal mathematical proofs within the text, just a step-by-step procedure. The commentary of Liu Hui provided geometrical and algebraic proofs to the problems given within the text. ''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. It was later incorporated into ''The Ten Computational Canons'', which became the core of mathematical education in later centuries. This book includes 246 problems on surveying, agriculture, partnerships, engineering, taxation, calculation, the solution of equations, and the properties of right triangles. ''The Nine Chapters'' made significant additions to solving quadratic equations in a way similar to Horner's method. It also made advanced contributions to "fangcheng" or what is now known as linear algebra. Chapter seven solves
system of linear equations In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in th ...
with two unknowns using the false position method, similar to The Book of Computations. 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. The Nine Chapters solves systems of equations using methods similar to the modern Gaussian elimination and back substitution. 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. 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''. 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.

## 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. 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. Historians believe that this figure of pi was calculated using the 3:1 relationship between the circumference and diameter of a circle. Some Han mathematicians attempted to improve this number, such as Liu Xin, who is believed to have estimated pi to be 3.154. 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. 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. There is no explicit method or record of how he calculated this estimate.

## Division and root extraction

Basic arithmetic processes such as addition, subtraction, multiplication and division were present before the Han Dynasty. ''The Nine Chapters on the Mathematical Art'' take these basic operations for granted and simply instruct the reader to perform them. 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''. 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. This process of successive approximation was then extended to solving quadratics of the second and third order, such as $x^2+a=b$, using a method similar to Horner's method. 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.

## Linear algebra

''The Book of Computations'' is the first known text to solve systems of equations with two unknowns. 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. 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. 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''. To solve for the lesser of the two unknowns, simply add the minor terms together. Chapter Eight of ''The Nine Chapters on the Mathematical Art'' deals with solving infinite equations with infinite unknowns. This process is referred to as the "fangcheng procedure" throughout the chapter. 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 Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...
today. In this chapter, the process of Gaussian elimination and back-substitution are used to solve systems of equations with many unknowns. Problems were done on a counting board and included the use of negative numbers as well as fractions. The counting board was effectively a matrix, where the top line is the first variable of one equation and the bottom was the last.

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

Liu Hui Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu ( The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state ...
's commentary on ''The Nine Chapters on the Mathematical Art'' is the earliest edition of the original text available. 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. 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. In his commentary, Liu Hui finds a more accurate estimation of pi using the
method of exhaustion The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area be ...
. 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. From this method, Liu Hui asserted that the value of pi is about 3.14. 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.

# Mathematics in the period of disunity

In the third century
Liu Hui Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu ( The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state ...
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. 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 Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arith ...
during the 3rd century CE. 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 The ''Book of Sui'' (''Suí Shū'') is the official history of the Sui dynasty. It ranks among the official Twenty-Four Histories of imperial China. It was written by Yan Shigu, Kong Yingda, and Zhangsun Wuji, with Wei Zheng as the lead aut ...
, 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 $\tfrac$ 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" 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 Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
,
matrix algebra In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, '' ...
, 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 Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...
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. 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 The Tang dynasty (, ; zh, t= ), or Tang Empire, was an imperial dynasty of China that ruled from 618 to 907 AD, with an interregnum between 690 and 705. It was preceded by the Sui dynasty and followed by the Five Dynasties and Ten Kingd ...
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 The Sui dynasty (, ) was a short-lived imperial dynasty of China that lasted from 581 to 618. The Sui unified the Northern and Southern dynasties, thus ending the long period of division following the fall of the Western Jin dynasty, and lay ...
and Tang dynasty ran the "School of Computations". Wang Xiaotong was a great mathematician in the beginning of the
Tang Dynasty The Tang dynasty (, ; zh, t= ), or Tang Empire, was an imperial dynasty of China that ruled from 618 to 907 AD, with an interregnum between 690 and 705. It was preceded by the Sui dynasty and followed by the Five Dynasties and Ten Kingd ...
, and he wrote a book: Jigu Suanjing (''Continuation of Ancient Mathematics''), where numerical solutions which general cubic equations appear for the first time The Tibetans obtained their first knowledge of mathematics (arithmetic) from China during the reign of Nam-ri srong btsan, who died in 630. The
table Table may refer to: * Table (furniture), a piece of furniture with a flat surface and one or more legs * Table (landform), a flat area of land * Table (information), a data arrangement with rows and columns * Table (database), how the table da ...
of sines by the Indian mathematician,
Aryabhata Aryabhata ( ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the ''Aryabhatiya'' (which ...
, were translated into the Chinese mathematical book of the '' Kaiyuan Zhanjing'', compiled in 718 AD during the Tang Dynasty.Needham, Volume 3, 109. Although the Chinese excelled in other fields of mathematics such as solid
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
,
binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
, and complex
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
ic formulas, early forms of
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
were not as widely appreciated as in the contemporary Indian and Islamic mathematics.Needham, Volume 3, 108-109. 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. 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. Four outstanding mathematicians arose during the
Song Dynasty The Song dynasty (; ; 960–1279) was an imperial dynasty of China that began in 960 and lasted until 1279. The dynasty was founded by Emperor Taizu of Song following his usurpation of the throne of the Later Zhou. The Song conquered the rest ...
and
Yuan Dynasty The Yuan dynasty (), officially the Great Yuan (; xng, , , literally "Great Yuan State"), was a Mongol-led imperial dynasty of China and a successor state to the Mongol Empire after its division. It was established by Kublai, the fifth ...
, particularly in the twelfth and thirteenth centuries:
Yang Hui Yang Hui (, ca. 1238–1298), courtesy name Qianguang (), was a Chinese mathematician and writer during the Song dynasty. Originally, from Qiantang (modern Hangzhou, Zhejiang), Yang worked on magic squares, magic circles and the binomial theo ...
, 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 Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refe ...
's pivotal condensation. Qin Jiushao (c. 1202–1261) was the first to introduce the zero symbol into Chinese mathematics. Before this innovation, blank spaces were used instead of zeros in the system of
counting rods Counting rods () are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient East Asia. They are placed either horizontally or vertically to represent any integer or rational number. The written ...
. 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?" Qin also solved a 10th order equation. 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. 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 Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
, it had a great impact on the development of Japanese mathematics.

## Algebra

### ''Ceyuan haijing''

'' Ceyuan haijing'' (), 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. "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 ..ome 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''

''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. There are many summation series equations given without proof in the ''Mirror''. A few of the summation series are: : $1^2 + 2^2 + 3^2 + \cdots + n^2 =$ : $1 + 8 + 30 + 80 + \cdots + =$

### ''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.

### Magic squares and magic circles

The earliest known
magic square In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number ...
s of order greater than three are attributed to
Yang Hui Yang Hui (, ca. 1238–1298), courtesy name Qianguang (), was a Chinese mathematician and writer during the Song dynasty. Originally, from Qiantang (modern Hangzhou, Zhejiang), Yang worked on magic squares, magic circles and the binomial theo ...
(fl. ca. 1261–1275), who worked with magic squares of order as high as ten. He also worked with magic circle.

## Trigonometry

The embryonic state of
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
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. The
polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a substantial number of subjects, known to draw on complex bodies of knowledge to solve specific pro ...
Chinese scientist, mathematician and official
Shen Kuo Shen Kuo (; 1031–1095) or Shen Gua, courtesy name Cunzhong (存中) and pseudonym Mengqi (now usually given as Mengxi) Weng (夢溪翁),Yao (2003), 544. was a Chinese polymathic scientist and statesman of the Song dynasty (960–1279). Shen wa ...
(1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs. 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'' + 2''v''2/''d'', where ''d'' is the
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...
, ''v'' is the versine, ''c'' is the length of the chord ''c'' subtending the arc.Katz, 308. Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for
spherical trigonometry Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...
developed in the 13th century by the mathematician and astronomer
Guo Shoujing Guo Shoujing (, 1231–1316), courtesy name Ruosi (), was a Chinese astronomer, hydraulic engineer, mathematician, and politician of the Yuan dynasty. The later Johann Adam Schall von Bell (1591–1666) was so impressed with the preserved astro ...
(1231–1316).. As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used
spherical trigonometry Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...
in his calculations to improve the calendar system and Chinese astronomy.Gauchet, 151. 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).Needham, Volume 3, 109–110. 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).Needham, Volume 3, 110.

# Ming mathematics

After the overthrow of the
Yuan Dynasty The Yuan dynasty (), officially the Great Yuan (; xng, , , literally "Great Yuan State"), was a Mongol-led imperial dynasty of China and a successor state to the Mongol Empire after its division. It was established by Kublai, the fifth ...
, China became suspicious of Mongol-favored knowledge. The court turned away from math and physics in favor of
botany Botany, also called , plant biology or phytology, is the science of plant life and a branch of biology. A botanist, plant scientist or phytologist is a scientist who specialises in this field. The term "botany" comes from the Ancient Greek wo ...
and
pharmacology Pharmacology is a branch of medicine, biology and pharmaceutical sciences concerned with drug or medication action, where a drug may be defined as any artificial, natural, or endogenous (from within the body) molecule which exerts a biochemic ...
.
Imperial examination The imperial examination (; lit. "subject recommendation") refers to a civil-service examination system in Imperial China, administered for the purpose of selecting candidates for the state bureaucracy. The concept of choosing bureaucrats by ...
s 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.
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. 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. 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. 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.) 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. Other missionaries followed in his example, translating Western works on special functions (trigonometry and logarithms) that were neglected in the Chinese tradition. However, contemporary scholars found the emphasis on proofs — as opposed to solved problems — baffling, and most continued to work from classical texts alone.

# 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. At Kangxi's direction, Mei Goucheng and three other outstanding mathematicians compiled a 53-volume ''Shuli Jingyun'' he Essence of Mathematical Study(printed 1723) which gave a systematic introduction to western mathematical knowledge. At the same time, Mei Goucheng also developed to ''Meishi Congshu Jiyang'' he 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. The enterprise sought to alleviate the difficulties for Chinese mathematicians working on Western mathematics in tracking down citations. However, no sooner were the encyclopedias published than the
Yongzheng Emperor , regnal name = , posthumous name = Emperor Jingtian Changyun Jianzhong Biaozhen Wenwu Yingming Kuanren Xinyi Ruisheng Daxiao Zhicheng Xian()Manchu: Temgetulehe hūwangdi () , temple name = Shizong()Manchu: Šidzung () , house = Aisin Gioro ...
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 The ''Yongle Encyclopedia'' () or ''Yongle Dadian'' () is a largely-lost Chinese ''leishu'' encyclopedia commissioned by the Yongle Emperor of the Ming dynasty in 1403 and completed by 1408. It comprised 22,937 manuscript rolls or chapters, in ...
'' and several other mathematical works from Han and Tang dynasties. 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. 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.

# Western influences

In 1840, the
First Opium War The First Opium War (), also known as the Opium War or the Anglo-Sino War was a series of military engagements fought between Britain and the Qing dynasty of China between 1839 and 1842. The immediate issue was the Chinese enforcement of the ...
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''. 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. As Martzloff notes, "from 1911 onwards, solely Western mathematics has been practised in China."

## 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 Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
and one of the greatest mathematicians of the twentieth century and was awarded the
Wolf prize The Wolf Prize is an international award granted in Israel, that has been presented most years since 1978 to living scientists and artists for ''"achievements in the interest of mankind and friendly relations among people ... irrespective of natio ...
for his immense number of mathematical contributions. * 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 Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathe ...
, his contributions have influenced both
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rela ...
and mathematics, and he has been active at the interface between geometry and
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experime ...
and subsequently awarded the Fields medal for his contributions. *
Terence Tao Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes ...
, an ethnic Chinese child prodigy who received his master's degree at age 16, was the youngest participant in the
International Mathematical Olympiad The International Mathematical Olympiad (IMO) is a mathematical olympiad for pre-university students, and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, excep ...
'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 In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime ...
(the product of two primes) which is now called Chen's theorem . 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. During the chaos of the
Cultural Revolution The Cultural Revolution, formally known as the Great Proletarian Cultural Revolution, was a sociopolitical movement in the People's Republic of China (PRC) launched by Mao Zedong in 1966, and lasting until his death in 1976. Its stated goal ...
, 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 Guo Moruo (; November 16, 1892 – June 12, 1978), courtesy name Dingtang (), was a Chinese author, poet, historian, archaeologist, and government official. Biography Family history Guo Moruo, originally named Guo Kaizhen, was born on November ...
'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. 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. 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.

## Performance at the IMO

In comparison to other participating countries at the
International Mathematical Olympiad The International Mathematical Olympiad (IMO) is a mathematical olympiad for pre-university students, and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, excep ...
, China has highest team scores and has won the all-members-gold IMO with a full team the most number of times.

# Mathematical texts

Zhou Dynasty ''
Zhoubi Suanjing The ''Zhoubi Suanjing'' () is one of the oldest Chinese mathematical texts. "Zhou" refers to the ancient Zhou dynasty (1046–256 BCE); "Bì" literally means " thigh", but in the book refers to the gnomon of a sundial. The book is dedicated to ...
'' 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 ''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest sur ...
'' 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 In mathematics, the ''regula falsi'', method of false position, or false position method is a very old method for solving an equation with one unknown; this method, in modified form, is still in use. In simple terms, the method is the trial and e ...
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.Christopher Cullen, "Numbers, numeracy and the cosmos" in Loewe-Nylan, ''China's Early Empires'', 2010:337-8.

* Chinese astronomy *
History of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments ...
**
Indian mathematics Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, ...
** Islamic mathematics ** Japanese mathematics * List of Chinese discoveries * List of Chinese mathematicians *
Numbers in Chinese culture Some numbers are believed by some to be auspicious or lucky (吉利, ) or inauspicious or unlucky (不吉, ) based on the Chinese word that the number sounds similar to. The numbers 3, 6, and 8 are generally considered to be lucky, while 4 is c ...

# References

## Sources

* * * 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. * * ; Public domain * *