The ''Book on Numbers and Computation'' (), or the ''Writings on Reckoning'', is one of the earliest known Chinese mathematical treatises. It was written during the early Western

Han dynasty The Han dynasty (, ; ) was an imperial dynasty of China (202 BC – 9 AD, 25–220 AD), established by Liu Bang (Emperor Gao) and ruled by the House of Liu. The dynasty was preceded by the short-lived Qin dynasty (221–207 BC) and a warr ...

, sometime between 202 BC and 186 BC.Liu et al. (2003), 9. It was preserved among the

Zhangjiashan Han bamboo texts The Zhangjiashan Han bamboo texts are ancient Han Dynasty Chinese written works dated 196–186 BC. They were discovered in 1983 by archaeologists excavating tomb no. 247 at Mount Zhangjia () of Jiangling County, Hubei Province (near modern Jing ...

and contains similar mathematical problems and principles found in the later

Eastern Han The Han dynasty (, ; ) was an imperial dynasty of China (202 BC – 9 AD, 25–220 AD), established by Liu Bang (Emperor Gao) and ruled by the House of Liu. The dynasty was preceded by the short-lived Qin dynasty (221–207 BC) and a warr ...

period text of ''

The 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 ...

''.

Discovery

The text was found in tomb M247 of the burial grounds near Zhangjiashan,

Jiangling County Jiangling () is a county in southern Hubei province, People's Republic of China. Administratively, it is under the jurisdiction of Jingzhou City. History The county name derived from the old name of Jingzhou. Liang dynasty Prince Xiao Yi 蕭繹 ( ...

, 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, excavated in December–January 1983–1984. This tomb belonged to an anonymous civil servant in early West Han dynasty. In the tomb were 1200 bamboo strips written in ink. Originally the strips were bound together with string, but the string had rotted away and it took Chinese scholars 17 years to piece together the strips. As well as the mathematical work the strips covered government statutes, law reports and therapeutic gymnastics. On the back of the sixth strip, the top has a black square mark, followed by the three characters 筭數書, which serve as the title of the rolled up book.

Content

The Suàn shù shū consists of 200 strips of

bamboo Bamboos are a diverse group of evergreen perennial flowering plants making up the subfamily Bambusoideae of the grass family Poaceae. Giant bamboos are the largest members of the grass family. The origin of the word "bamboo" is uncertain, bu ...

written in ink, 180 strips are intact, the others have rotted. They consist of 69 mathematical problems from a variety of sources, two names Mr Wáng and Mr Yáng were found, probably two of the writers. Each problem has a question, an answer, followed by a method. The problems cover

elementary arithmetic The operators in elementary arithmetic are addition, subtraction, multiplication, and division. The operators can be applied on both real numbers and imaginary numbers. Each kind of number is represented on a number line designated to the type. ...

;

fractions A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...

; inverse proportion; factorization of numbers;

geometric progression In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For e ...

s, in particular

interest rate An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...

calculations and handling of errors; conversion between different units; 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 er ...

for finding roots and the extraction of approximate

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 ...

s; calculation of the

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...

of various 3-dimensional shapes; relative dimensions of a

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

and its inscribed

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

; calculation of unknown side of rectangle, given area and one side. All the calculations about circumference and area of circle are approximate, equivalent to taking π = 3. Calculations of pi were made more accurate with the work of Liu Xin (c. 46 BC – 23 AD),

Zhang Heng Zhang Heng (; AD 78–139), formerly romanized as Chang Heng, was a Chinese polymathic scientist and statesman who lived during the Han dynasty. Educated in the capital cities of Luoyang and Chang'an, he achieved success as an astronomer, ma ...

(78–139 AD),

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 o ...

(fl. 3rd century AD), and

Zu Chongzhi Zu Chongzhi (; 429–500 AD), courtesy name Wenyuan (), was a Chinese astronomer, mathematician, politician, inventor, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3. ...

(429–500). Prior to discovery the oldest Chinese mathematical text were 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 ...

and ''

'' which dates from around 100 CE. Many topics are covered in both texts, however, error correction problems only appear in the Suàn shù shū, and the last two chapter of the nine chapters have no corresponding material in the Suàn shù shū. The text has been translated to English by

Christopher Cullen Christopher Cullen is an English sinologist. He has an MA from University of Oxford in engineering and a PhD from the School of Oriental and African Studies in classical Chinese. He is Director Emeritus of the Needham Research Institute and Gene ...

, director of the

Needham Research Institute The Needham Research Institute (NRI; zh , t = 李約瑟研究所 ), located on the grounds of Robinson College, in Cambridge, England, is a centre for research into the history of science, technology and medicine in East Asia. The institute is n ...

.The Suan shu shu 筭數書 'Writings on Reckoning'
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Notes

References

*Christopher Cullen: The Suan shu shu Writings on reckoning, Needham Research Institute, pdf free downloa

* * Dauben, Joseph W. (2004). "The Suan Shu Shu (A Book on Numbers and Computation), A Preliminary Investigation" in ''Form, Zahl, Ordnung'', 151–168. München: Franz Steiner Verlag. . *Dauben, Joseph W. (2007). "Chinese Mathematics" in ''The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook'', 187–384. Edited by Victor J. Katz. Princeton: Princeton University Press. . * Guilin Liu, Lisheng Feng, Airong Jiang, and Xiaohui Zheng. (2003). The Development of E-mathematics Resources at Tsinghua University Library (THUL)," in ''Electronic Information and Communication in Mathematics'', 1–13. Edited by Fengshen Bai and Bernd Wegner. Berlin: Springer. . * Stephanie Pain, ''Histories: China's oldest mathematical puzzles'',

New Scientist ''New Scientist'' is a magazine covering all aspects of science and technology. Based in London, it publishes weekly English-language editions in the United Kingdom, the United States and Australia. An editorially separate organisation publishe ...

, 30 July 2006

* Péng Hào, ''Zhāngjiāshān Hànjiǎn "Suàn shù shū" zhùshì (The Hàn dynasty book on wooden strips "Suàn shù shū" found at Zhāngjiāshān with a commentary and explanation)'' Beijing, Science Press, (2001). *

Wu Wenjun Wu Wenjun ( zh, s=吴文俊; 12 May 1919 – 7 May 2017), also commonly known as Wu Wen-tsün, was a Chinese mathematician, historian, and writer. He was an academician at the Chinese Academy of Sciences (CAS), best known for the Wu's method of ...

ed, Zhong Guo Shu Xue Shi Da Xi(The Grand Series of History of Chinese Mathematics) vol 1, chapter 2, "Suan Shu Shu". *Rémi Anicotte (2019). ''Le Livre sur les calculs effectués avec des bâtonnets – Un manuscrit du -IIe siècle excavé à Zhangjiashan'', Paris: Presses de l'Inalco. http://www.inalco.fr/publication/livre-calculs-effectues-batonnets-manuscrit-iie-siecle-excave-zhangjiashan

External links

Full text of book
Christopher Cullen: Suan shu shu ''Writings on reckoning''

{{DEFAULTSORT:Book On Numbers And Computation Archaeological artifacts of China Mathematics manuscripts Chinese mathematics Han dynasty texts Han dynasty literature 2nd-century BC books Mathematics books