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Chapter 4

Chinese Number System

4.1 HISTORY AND MATHEMATICS

China represents the mysterious Orient to many, and myths and legends of China certainly abound. Chinahas a long and storied history. To fix some dates, we note that

• Sun Tzu wrote The Art of War in the sixth century bc.

• Confucius, whose philosophy has come to be woven into Chinese thought, lived between 554 and479 bc.

• Laozi (Lao-Tzu), who founded Daoism (Taoism), lived around the fourth century bc.

• The Great Wall of China was built, rebuilt, and maintained between the fifth century bc and thesixteenth century ad.

• Ts’ai Lun is credited with inventing paper around ad 105, although recent archaeological findingsplace the invention some 200 years earlier.

• Gunpowder was discovered in China in the ninth century ad.

• The Mongol invasion of China occurred in the thirteenth century.

• Marco Polo (1254–1324) left Venice with his father and an uncle in 1271 to travel to China alongthe Silk Road; they arrived in 1275 and were welcomed by Kublai Khan (1214–1294). KublaiKhan, the grandson of Ghenghis Khan (originally Temujin; 1167–1227), founded the Yuan Dynasty(1279–1368).

• Jesuit missionaries began to arrive during the late Ming period (1368–1644).

• The Tiananmen Square protests took place in 1989.

Much of China’s history is marked by dynastic rule, beginning with the Xia Dynasty (2205–1766 bc)until the Qing Dynasty (ad 1644–1911). During this long period, the borders of China were not fixed.(See figure 4.1.) Finally, the fall of the Qing, the last dynasty of China, in the Xinhai Revolution sawthe founding of the Republic of China with Sun Yat-sen as the head of the newly established ChineseNationalist Party or Kuomintang (or Guomindang; 1912–1927); however, China was fragmented duringthis period. Eventually, the Republic of China was consolidated under the leadership of Chiang Kai-shek,who had succeeded Sun Yat-sen as the head of the Chinese Nationalist Party (1927–present). When underthe leadership of Chiang Kai-shek, the Chinese Nationalist Party was forced to retreat to the island ofTaiwan (Formosa) by the Chinese Communist Party in the Communist Revolution led by Mao Zedong.

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36 CHAPTER 4. CHINESE NUMBER SYSTEM

Figure 4.1: Antique map of China. (Source: c© Thinkstock.)

This has come to be called the “Kuomintang Debacle of 1949.” With this, the People’s Republic of Chinawas founded (1949–present).

The Republic of China is today commonly known as Taiwan, while the People’s Republic of China istoday commonly referred to as Communist China or Mainland China, and is what many today would thinkof at the mention of “China.” The Chinese Nationalist Party is still the ruling political party in Taiwan.Both the Republic of China and the People’s Republic of China claim there is one China, and each claimsto be the “real China”; in fact, during the early Cold War years (ca. 1945–1991), many Western nations,as well as the United Nations, recognized the Republic of China (Taiwan) as the legitimate China. Thisresulted in nervous relations between the two governments over the years.

Mathematics in written form in China came about during the late Shang period. Because of naturalbarriers—the Pacific Ocean to the east, jungles and mountains to the south and west, and the Gobi Desert tothe north—China remained isolated for many years. As a result, for most of Chinese history, mathematicsdeveloped without any outside influence. Nevertheless, the Chinese made great strides in mathematics,developing techniques in arithmetic and algebra, including root finding methods for solving higher-degreeequations. And, according to Calinger [35], Chinese algebra “reached its pinnacle in the thirteenth-centurywork of Qin Jiushao, Li Zhi, Yang Hui, and Shu Shijie.” With a focus on computational techniques,theoretical geometry in China was to wait until the Jesuit Matteo Ricci (1552–1610) and Xu Guanqi(1562–1633) had translated the first six books of Euclid’s Elements into Chinese in 1607. This was part ofa broader introduction of Western mathematics, including trigonometry and logarithms, into China [35].The remaining seven books of the Elements were not translated until much later in 1856 by Li Shanlan(1811–1882) and Alexander Wylie (1815–1887) [108].

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4.2. ROD NUMERALS 37

The oldest known Chinese text that is exclusively on mathematics is Jiu zhang suan shu by an unknownauthor. Interestingly, there are several translations of the title of this work: “Nine Chapters onMathematicalProcedures,” “Arithmetic in Nine Sections,” “Nine Chapters on the Mathematical Art,” “ComputationalPrescriptions in Nine Chapters,” and “Nine Categories of Mathematical Methods” are some examples[44]. However, the work is commonly referred to in short as the Nine Chapters, and we do so also. (Seepage 292 for a description of the contents.)

(a) Nine Chapters stamp. (b) Liu Hui.

Figure 4.2: Jiu zhang suan shu (Nine Chapters) and Liu Hui. (Source: Courtesyof Jeff Miller.)

The Nine Chapters is believed to have been written during the late Qin (221–206 bc) or early Han(206 bc–ad 220) Dynasty [39, 72], and perhaps even completed before the “burning of the books” in 208bc [44]. There is no surviving original copy of the Nine Chapters. What we have is a compilation byZhang Cang and Geng Shouchang (first century bc) to which was later provided an extensive commentaryby Liu Hui (third century ad) [44]. Liu Hui, who is also known for his other classic work, Hai dao suanjing (Sea Island Mathematical Classic), is often compared to Euclid, and the Nine Chapters to Euclid’sElements [44, 108]. Liu’s version of the Nine Chapters was included among the Shi bu suan jing (TenBooks of Mathematics Classics) with a commentary by Li Chunfeng (ad 604–672) and others. Ten Bookswas the mathematics text at the Imperial College during the Tang Dynasty (ad 618–907). It is to thisversion of the Nine Chapters included in Ten Books that current editions may be traced [44].

4.2 ROD NUMERALS

Ancient writings and archeological finds dating back to the second century ad in China have unearthedbone strips used as counting rods. Ancient carvings on tortoiseshells of Chinese rod numerals have alsobeen found. Dating back to at least the 4th century bc, early rod numerals were actual rods made primarilyof bone or bamboo. Lam and Ang [77, p. xx] tell us that,

As far back as the Warring States period (475–221 bc), the Chinese used straight rods or sticksto do their calculation. They formed numerals from the rods, and they did their addition, subtraction,multiplication and division with these rod numerals. The performance of a multiplication problemsuch as the above [3508 × 436] with these rods would be commonly known at a very early time notonly among mathematicians, but also among officials, astronomers, traders and others. The rods were

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38 CHAPTER 4. CHINESE NUMBER SYSTEM

carried in bundles and, whenever calculation was required, they were brought out and computation wasperformed on a flat surface such as a table top or a mat. After the results were obtained, they wouldprobably be recorded and the rods would be put away.

The earliest known written use of rod numerals has been found on wooden artifacts excavated from theHan Dynasty (206 bc–ad 220). For example, on an artifact found in Hubei from that period is the script當利二月定算▔▕ , where the T at the end is the rod numeral for the number 6.

Rod numerals were later transmitted from China to Japan, Korea, and Vietnam. With the adventand efficiency of the abacus, the use of physical rods for calculating gradually died out during the MingDynasty (ad 1368–1644). A number of texts were written that used the abacus as the main method forcalculations, with the most influential text being Cheng Dawei’s Suanfa tongzong (Systematic Treatise onArithmetic; 1592). The written form of rod numerals, however, continued to be used along with traditionalwritten numerals in China for many centuries and can sometimes be found in old markets even today.

According to Lam and Ang [77, p. 10], “The reason for the decline of Chinese mathematics after the14th century was because it underwent a change of foundations, from mathematics based on rod numeralswith its step by step reasoning, to mathematics based on the abacus with its emphasis on learning by rotemethod.” Lam and Ang also advance the thesis that the Indo-Arabic number system (section 6.3) has itsorigins in the Chinese rod numeral system, and provide their evidence for this.

The ancient Chinese rod numeral system is additive and positional, yet does not have a zero or aplace holder. They were placed on a counting board, like a checkerboard, which allowed for a very cleardelineation of the place values. The orientation of the rods also alternated between being placed verticallyand horizontally from one place value to the next, beginning with being placed vertically in the one place.Over time, the rods eventually became written numerals and were written using very clear spacing. Table4.1 displays the rod numerals as shown in Burton [22, pp. 29, 258]. Furthermore, at first, a vacant spaceon the counting board represented a zero in that place value, but a circular symbol© later appeared inprint in the 1200s. Lam and Ang [77, p. 152], and others, tell us that red rods were used to representpositive numbers (zheng 正) and black rods to represent negative numbers (fu 負).1 Moreover, if rods ofonly one color were available, then an extra rod would be laid across the last nonzero digit to indicate

that the number is negative. Lam and Ang give the example \ for −642. As another example,is 4379.

Used in place value 1 2 3 4 5 6 7 8 9

Unit, hundred,. . .

Ten, thousand,. . .

Table 4.1: Chinese rod numerals.

Not only were rods used for recording numbers and for addition and subtraction, but they were also usedto express fractions, as well as to carry out fairly advanced operations such as extracting cube roots, solvingproportions, and solving systems of linear equations. To express a fraction, which typically appeared as theresult of a division problem on a counting board, the numerator would be placed above the denominatorjust as we do today, but without a fraction line or fraction bar. So, for example [77, pp. 79–80],

represents46.

1Burton tells us that black rods were used to represent positive numbers and red rods were used to represent negative numbers.

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4.2. ROD NUMERALS 39

Like all things, however, the Chinese rod numerals evolved over time. The ones described here representan early version. As the rod numerals evolved, they became more elaborate and started to resemble moremodern Asiatic numerals.

Now You Try 4.1

1. Rewrite the following rod numbers using our numerals.

a)

b)

c) @@

d) © ©

2. Write the following numbers in Chinese rod numerals.

a) 2345 b) 7089 c) −506 d) −888

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Think About It 4.1 Explain why a place holder is not needed in Chinese rod numerals. Can you think ofsome numbers for which Chinese rod numerals would not show place values as clearly as the examplesabove? �