Ancient Chinese Mathematics

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Ancient Chinese Mathematics

Overview

China has one of the world's oldest traditions in mathematical discovery, comparable to those of Egypt and the Middle East. The first Chinese mathematics text is of uncertain age, some dating it as early as 1200 b.c. and others over a thousand years later, but there is little doubt that relatively advanced mathematical concepts were discovered and practiced in China well before the birth of Christ. For much of its history, China has been in contact with the West, albeit intermittently, and Chinese and Western mathematicians influenced each other for centuries. Although it is sometimes difficult to determine who influenced whom, some Chinese contributions clearly predate those of the West, or are so obviously different that it is apparent they arose independently. In any event, the history of Chinese mathematics is both long and distinguished.

Background

There is no way to know when mathematics first appeared as a separate discipline in China. In fact, even dating the first Chinese civilization is difficult; various accounts place the first Chinese empire between about 2700 b.c. and 1000 b.c., and the earlier date is not considered unreasonable. These dates make it clear that Chinese civilization is probably neither much older nor much younger than those of the Middle East, although many aspects of the latter are better documented and, thus, easier to verify.

It is unlikely that a mathematical tradition could develop in the absence of some sort of civilization and, indeed, nothing more complex than counting seems to have developed anywhere in the world except where civilization already existed. This may be because there is little need, much less time, to develop sophisticated mathematical structures in the absence of cities and some sort of government. For example, many early mathematical documents deal with issues such as the proper measuring and allocation of properties, constructing buildings, calculating taxes, business computations, and the like. None of these sorts of transactions are likely to occur in the simpler (and smaller) tribal structures that predated the first cities and civil governments. In addition, in the absence of larger structured civil organizations, virtually everyone was responsible for sustaining themselves and their family. It was only with the rise of more centralized and larger governments that agriculture could progress to the point where a relatively large number of people could devote themselves to administration, business, crafts, or research. Thus, mathematics as a discipline would not have developed in the absence of civilization for another reason: there would be no need for it.

Given the above, it is also likely that mathematics began to advance rapidly once civilizations were established. One of the apparent hallmarks of civilization is the appearance of cities, a central government, and coordination of the activities of much larger populations. These, in turn, prompt the need for businesses, taxes to support the government, formal rules for land use, and so forth. It was these that seem to have driven mathematics to appear in short order in Egypt, China, Mesopotamia, India, and Latin America. And, once established as a formal discipline, in each of these places mathematics seems to have migrated to issues such as keeping calendars, making sense of astronomical phenomena, and so forth, becoming more abstract over time.

This certainly seems to have been the pattern in ancient China. Many of the first works in Chinese mathematics, including the first book,Nine Chapters on the Mathematical Art, and followed by The Ten Classics of Mathematics, concentrate on methods for solving practical problems in mathematics. However, these books also make it clear that many of these problems had existed (and been solved) for many years, or even centuries. What is also clear in these and other contemporaneous books is that Chinese mathematicians had already reached a high level of abstraction and sophistication by a few centuries b.c. In fact, by this time the Chinese had far surpassed Western mathematics in many areas. Tragically, much of this knowledge was lost on at least a few occasions when Chinese emperors ordered books burned and libraries destroyed. These actions not only caused Chinese mathematics to regress, but even kept many later Chinese scientists and mathematicians from knowing what their predecessors had done. Only recently have these early Chinese advances come to light; unfortunately, too late for them to have had the impact that would otherwise have been their due.

Impact

As noted above, for several centuries Chinese and Western mathematicians exchanged information, in spite of the slow rate of communications at the time. This makes assigning proper credit for various mathematical discoveries difficult, but not necessarily impossible.

The list of accomplishments that can be credited to Chinese mathematicians is impressive, and most of these appear to have arisen in China first, or at least independently. These include an impressive estimate of the value of π, use of zero, use of decimals and decimal fractions for calculations, use of negative numbers, the algebraic treatment of geometric problems, methods for solving problems with many factors, and methods for extracting square and cube roots. There are more, but even this short list is impressive. In the third century a.d. Chinese mathematicians calculated the value of π to 10 decimal places, a feat that would not be matched elsewhere for another 1,400 years.

In its simplest form, simply writing a number as 123 is decimal notation. Decimal refers to the fact that each place represents a larger factor of ten, so the 3 represents 3 × 1, the 2 represents 2 × 10, and the 1 means 1 × 100. This is similar to what is meant by positional notation, although having one does not necessarily dictate the other. Other civilizations, such as the Maya and the Hindu, developed these concepts as well, although the Maya used a base other than 10. It's interesting to note that some advanced civilizations did not develop these concepts. For example, the Greeks used letters to represent their numbers and the Romans never advanced beyond Roman numerals. To appreciate the difference between the decimal system and Roman numerals, consider the difficulty of multiplying XXVIII times XIII as compared to 28 times 13. This, of course is a simple problem to which the answer is 364, but calculating even this relatively simple problem in Roman numerals is not a trivial task.

Decimal fractions are simply extending this use to numbers less than 1. For example, the number 0.5 is a decimal fraction (for ½) as is the number 2.25 or 10.4. Developed in the first century a.d. in China, they were not widely used in the West for over 1,500 years.

Chinese mathematicians were very much in advance of their Western counterparts in the use of negative numbers. As late as the fifteenth century, Western mathematicians felt that negative numbers simply could not exist and many refused to even consider them. Today, of course, we recognize that negative numbers do exist, as anyone who has overdrawn their bank account knows. This has not always been the case. To their credit, the first mention of negative numbers in Chinese mathematics dates to at least the second century b.c., while they do not appear in Western mathematics for another 1,700 years. However, some texts suggest that Chinese mathematicians, while not worried about using negative numbers in calculations, did not view them as having an actual physical meaning. Therefore, they should be given due credit for the mathematical advance, while continuing to recognize their seeming unwillingness to fully accept the implications of this advance.

The other Chinese advances can, to some extent, be grouped together into developments in problem solving through algebra. There is a great deal of evidence that many algebraic techniques were developed in China, spreading to India and, from there to the Islamic scholars of the seventh and eighth centuries. This is not to take credit from Islamic mathematicians, who did a superb job in seeing the utility of these techniques, gathering them together, and adding to them their own unique contributions. The result is that, what we call algebra today is effectively a collaboration of Chinese, Hindu, and Islamic mathematical insights and advances, compiled and given utility by Islamic scholars.

Among the techniques developed by Chinese mathematicians are methods for extracting the roots of equations that were to be rediscovered in Europe up to a millennium later. They also learned to solve systems of linear equations, developed some very basic matrix algebra, and described geometric problems using equations instead of pictures. This last advance stands in interesting counterpoint to some of the Western mathematical traditions. In Egypt and Mesopotamia, for example, mathematics seems to have sprung from geometry, as early mathematicians attempted to find new ways of solving some problems. In China, by comparison, it appears as though geometry may have arisen from algebra as Chinese mathematicians tried to find ways to illustrate their mathematical discoveries. What makes this contrast particularly interesting is that both mathematical traditions arrived at fundamentally the same conclusions, giving a great deal of credence to the final results.

It is likely that many Chinese discoveries made their way to the West in during ancient and early medieval times. In particular, China is known to have had extensive contact with the West during the Han and T'ang dynasties (about 200 b.c. to a.d. 200 and a.d. 618 to 906, respectively). During these times, it is likely that China exported some mathematics to the West, although it is not certain what. Some claim that the ideas of decimal notation and zero originated in China, spreading to the West during these periods, but this may never be known for certain. What is certain is that China seemingly forgot her mathematical achievements towards the end of the first millennium a.d. and, by the Renaissance, had been eclipsed by Europe. In one of history's ironies, subsequent Chinese mathematicians learned mathematics from Europe that had been originally discovered in China and either exported to Europe or lost. There is no way to know what world mathematics would be like today if this episode of mathematical amnesia had not occurred, but it is certain to have been very different.

P. ANDREW KARAM