349-60045_Shell_ch01_4P July 24, 2015 12:07 Page 35 ¨ © 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. China has 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 and 479 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 the © © sixteenth century ad. • Ts’ai Lun is credited with inventing paper around ad 105, although recent archaeological findings place 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 along the Silk Road; they arrived in 1275 and were welcomed by Kublai Khan (1214–1294). Kublai Khan, 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 saw the founding of the Republic of China with Sun Yat-sen as the head of the newly established Chinese Nationalist Party or Kuomintang (or Guomindang; 1912–1927); however, China was fragmented during this 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 under the leadership of Chiang Kai-shek, the Chinese Nationalist Party was forced to retreat to the island of Taiwan (Formosa) by the Chinese Communist Party in the Communist Revolution led by Mao Zedong. 35 ¨ © 349-60045_Shell_ch01_4P July 24, 2015 12:07 Page 36 ¨ © 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 China was founded (1949–present). The Republic of China is today commonly known as Taiwan, while the People’s Republic of China is today commonly referred to as Communist China or Mainland China, and is what many today would think of 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 claims to 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. This resulted 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 natural barriers—the Pacific Ocean to the east, jungles and mountains to the south and west, and the Gobi Desert to the north—China remained isolated for many years. As a result, for most of Chinese history, mathematics developed 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-degree equations. And, according to Calinger [35], Chinese algebra “reached its pinnacle in the thirteenth-century work 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 of a 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]. ¨ © 349-60045_Shell_ch01_4P July 24, 2015 12:07 Page 37 ¨ © 4.2. ROD NUMERALS 37 The oldest known Chinese text that is exclusively on mathematics is Jiu zhang suan shu by an unknown author. Interestingly, there are several translations of the title of this work: “Nine Chapters on Mathematical Procedures,” “Arithmetic in Nine Sections,” “Nine Chapters on the Mathematical Art,” “Computational Prescriptions 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. (See page 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: Courtesy © © of 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 208 bc [44]. There is no surviving original copy of the Nine Chapters. What we have is a compilation by Zhang Cang and Geng Shouchang (first century bc) to which was later provided an extensive commentary by Liu Hui (third century ad)[44]. Liu Hui, who is also known for his other classic work, Hai dao suan jing (Sea Island Mathematical Classic), is often compared to Euclid, and the Nine Chapters to Euclid’s Elements [44, 108]. Liu’s version of the Nine Chapters was included among the Shi bu suan jing (Ten Books of Mathematics Classics) with a commentary by Li Chunfeng (ad 604–672) and others. Ten Books was the mathematics text at the Imperial College during the Tang Dynasty (ad 618–907). It is to this version 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 unearthed bone strips used as counting rods. Ancient carvings on tortoiseshells of Chinese rod numerals have also been found. Dating back to at least the 4th century bc, early rod numerals were actual rods made primarily of 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 sticks to 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 problem such as the above [3508 × 436] with these rods would be commonly known at a very early time not only among mathematicians, but also among officials, astronomers, traders and others. The rods were ¨ © 349-60045_Shell_ch01_4P July 24, 2015 12:07 Page 38 ¨ © 38 CHAPTER 4. CHINESE NUMBER SYSTEM carried in bundles and, whenever calculation was required, they were brought out and computation was performed on a flat surface such as a table top or a mat. After the results were obtained, they would probably 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 the Han Dynasty (206 bc–ad 220). For example, on an artifact found in Hubei from that period is the script v)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 advent and efficiency of the abacus, the use of physical rods for calculating gradually died out during the Ming Dynasty (ad 1368–1644). A number of texts were written that used the abacus as the main method for calculations, with the most influential text being Cheng Dawei’s Suanfa tongzong (Systematic Treatise on Arithmetic; 1592). The written form of rod numerals, however, continued to be used along with traditional written 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 the 14th century was because it underwent a change of foundations, from mathematics based on rod numerals with its step by step reasoning, to mathematics based on the abacus with its emphasis on learning by rote method.” Lam and Ang also advance the thesis that the Indo-Arabic number system (section 6.3) has its origins 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 a place holder. They were placed on a counting board, like a checkerboard, which allowed for a very clear delineation of the place values. The orientation of the rods also alternated between being placed vertically and 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. Table 4.1 displays the rod numerals as shown in Burton [22, pp. 29, 258]. Furthermore, at first, a vacant space on the counting board represented a zero in that place value, but a circular symbol later appeared in ¨ print in the 1200s.