Early Chinese mathematics
Simple mathematics on Oracle bone scriptdate back to the Shang Dynasty (1600 BC-1050 BC). One of the oldest surviving mathematical works is the Yi Jing, which greatly influenced written literature during the Zhou Dynasty (1050 BC-256 BC). For mathematics, the book included a sophisticated use of hexagrams. Leibnitz pointed out, the I Ching 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.[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.
Mathematics was one of the Liù Yì (六艺) or Six Arts, students were required to master during the Zhou Dynasty (1122 BC – 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 BC-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 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.” 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 along with principles of space and bounded space It also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch The book provided definitions for circumference, diameter, and radius, along with the definition of volume
The history of mathematical development lacks some evidence. There are still debates about certain mathematical classics. For example, the Zhou Bi Suan Jing dates around 1200-1000BCE, yet many scholars believed it was written between 300-250BCE. The Zhou Bi Suan Jing contains an in-depth proof of the Gougu Theorem (Pythagorean Theorem) but focuses more on astronomical calculations.
Qin mathematics
Not much is known about Qin dynasty mathematics, or before, due to the burning of books and burying of scholars.
Knowledge of this period must be carefully determined by their civil projects and historical evidence. The Qin dynasty created a standard system of weights. Civil projects of the Qin dynasty were incredible feats of human engineering. Emperor Qin Shihuang（秦始皇）ordered many men to build large, lifesize statues for the palace, tomb along with various other temples and shrines. The shape of the tomb is designed with geometric skills of architecture. It is certain that one of the greatest feats of human history; the great wall required many mathematical “techniques.” All Qin dynasty buildings and grand projects used advanced computation formulas for volume, area and proportion.
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 called chousuan,consisted of only nine symbols, a blank space on the counting board stood for zero. The mathematicians Liu Xin (d. 23) and Zhang Heng (78–139) gave more accurate approximations for pi than Chinese of previous centuries had used. Zhang also applied mathematics in his work in astronomy.
Suan shu shu
The Suàn shù shū (writings on reckoning) 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 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. 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. Some linguistic hints point back to the Qin dynasty.
In an example of an elementary mathematics in the Suàn shù shū, the square root is approximated by using an “excess and deficiency” 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.”
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 1000BC), but perhaps as early as 300-200 BC. Although the author(s) are unknown, they made a huge contribution in the eastern world. The methods were made for everyday life and gradually taught advanced methods. It also contains evidence of the Gaussian elimination and Cramer’s Rule.
It was one of the most influential of all Chinese mathmatical books and it is composed of some 246 problems. 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.Boyer Chinese Math China and India estimates concerning the Chou Pei Suan Ching,generally considered to be the oldest of the mathematical classics,differ by almost a thousand years.A date of about 300 B.C. would appear reasonable,thus placing it in close competition with another treatise,the Chiu-chang suan-shu,composed about 250 B.C., that is, shortly before the Han dynasty (202 B.C.).Almost as old at the Chou Pei,and perhaps the most influential of all Chinese mathematical books,was the Chui-chang suan-shu,or Nine Chapters on the Mathematical Art.This book includes 246 problems on surveying, agriculture, partnerships, engineering, taxation, calculation, the solution of equations, and the properties of right triangles.Chapter eight of the Nine chapters is significant for its solution of problems of simultaneous linear equations,using both positive and negative numbers.The last problem int the chapter involves four equations in five unknowns,and the topic of indeterminate equations was to remain a favorite among Oriental peoples. The earliest known magic squares appeared in China.Boyer Magic Square China and India.The Chinese were especially fond of patters,as a natural outcome of arranging counting rods in rows on counting board to carry out computation;hence,it is not surprising that the first record(of ancient but unknown origin) of a magic square appeared there.The concern for such patterns left the author of the Nine Chapters to solve the system of simultaneous linear equations by performing column operations on the matrix to reduce it to The second form represented the equations 36z = 99, 5y + z = 24, and 3x + 2y + z = 39 from which the values of z, y, and x are successively found with ease.In Nine Chapters the author solves a system of simultaneous linear equations by placing the coefficients and constant terms of the linear equations into a magic square (i.e. a matrix) and performing column reducing operations on the magic square.
Mathematics in the period of disunity
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 acomplishment in the mathematical surveying exceeded those accomplished in the west by a millenium. 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 integral and the differential calculus 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, 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 till 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 used He Chengtian’s interpolation method for approximating irrational number with fraction in his astronomy and mathematical works, he obtained 355/113 as a good fraction approximate for pi; Yoshio Mikami commented that neither the Greek, nor the Hindus nor Arabs knew about this fraction approximation to pi, not until the Dutch mathematician adrian Anthoniszoom rediscoved it in 1585,”the Chinese had therefore been possessed of this the most extraodinary of all fractional values over a whole millenium earlier than Europe Along with his son, Zu Geng, Zu Chongzhi used the Cavalieri Method to find an accurate solution for calculating the volume of the sphere. 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 around 400 CE contained the most detailed step by step description of multiplication and division algorithm with counting rods. The earliest record of multiplication and division algorithm using Hindu Arabic numerals was in writing by Al Khwarizmi in early 9th century. Khwarizmi’s step by step division algorithm was completely indentical to Sunzi division algorithm described in Sunzi mathematical classic four centuries earlier Khwarizmi’s work was translated in to Latin in the 13th century and spread to the west, the division algorithm later evolved into Galley division. The route of transmission of Chinese place value decimal arithematic know how to the west is unclear, how Sunzi’s division and multiplication algorithm with rod calculus ended up in Hindu Arabic numeral form in Khwarizmi’s work is unclear, as al Khwarizmi never given any Sankrit source nor quoted any Sanskrit stanza.
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.Wang Xiaotong was a great mathematician in the beginning of the Tang Dynasty, and he wrote a book: Jigu suanjing (Continuation of Ancient Mathematics),in which cubic equation appeared first time
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. 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 I-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
Song and Yuan mathematics
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 C.E). Li Zhi on the other hand, investigated on a form of algebraic geometry. His book; Ce Hai Yuan Jing revolutionized the idea of inscribing a circle into triangles, which could be calculated using equations with the 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 is 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.[16] Before this innovation, blank spaces were used instead of zeros in the system of counting rods.[17]. One of the most important contribution of Qin Jiushao was his method of solving high order numerical equations. Refering 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 ?” [18].Qin also solved a 10th order equation[19]
Pascal’s triangle was first illustrated in China by Yang Hui in his book Xiangjie Jiuzhang Suanfa (详解九章算法), although it was described earlier around 1100 by Jia Xian.[20] Although the Introduction to Computational Studies (算学启蒙) 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.
Precious Mirror of the Four Elements
Si-yüan yü-jian《四元玉鑒》, or Precious Mirror of the Four Elements, was written by Chu Shi-jie in 1303 A.D. and it 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. The Ssy-yüan yü-chien 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.[22]
The Precious Mirror opens with a diagram of the arithmetic triangle (Pascal’s triangle) using a round zero symbol, but Chu Shih-chieh denies credit for it. A similar triangle appears in Yang Hui’s work, but without the zero symbol.[23]
There are many summation series equations given without proof in the Precious mirror. A few of the summation series are
Algebra:
Ts’e-yuan hai-ching (pinyin: Cèyuán Hǎijìng) (Chinese characters:測圓海鏡), or Sea-Mirror of the Circle Measurements, is a collection of some 170 problems written by Li Zhi (or Li Ye) (1192 – 1272 A.D.). He 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.[24]
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 A.D.) and with the invention of a method of solving simultaneous congruences, it marks the high point in Chinese indeterminate analysis.[24]
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.[25]. He also worked with magic circle.
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