The science of mathematics is looked at with such importance in China that it is considered one of the six basic arts, along with ritual, music, archery, horsemanship and calligraphy (Yan, Du Shiran, 1987, p.22). It is the importance that the Chinese place on mathematics that caused it to be one of the most influential cultures in the history of the world in terms of mathematical breakthroughs. Unfortunately for the Chinese, it is only recently that they are beginning to receive the credit they deserve for their achievements. Much of the work that they did had been overlooked due to the fact that many scholars thought that the records that had been kept were not trustworthy, and there was a lack of satisfactory translations(Scott, 1969, p.80). It was Joseph Needham’s Science and Civilisation in Chinese Society Volume 3 that began to open the world’s eyes of the accomplishments the Chinese had made in the field of mathematics.

    The history of Chinese mathematics is very difficult to pin point. The reason being is that there is no definitive date as to when they began to theorize about mathematics. The two dates that are most widely excepted differ due to people’s different definition of mathematics. Many scholars believe that the Chinese began their study of mathematics as early as 3000 B.C.. This is when the Chinese began to make astronomical calculations. The other widely excepted date is around 400 B.C.(http://aleph0.clarku.edu). This is when they developed a standard numerical notation system. This system was based on counting rods, which will be looked at more closely later. It was after the invention of a universal numerical system that China began to make its most significant contributions to mathematics.

    If one takes the belief that mathematics started with astronomy then the most influential figure would be the mythical Yellow Emperor (2698-2598 B.C.). Whether he existed or not it was during this time period that exploration of the skies really began. The reason behind the studying of the skies was to create a calendar and to get a grasp of the seasons. The person who was regarded as the mathematical scholar of the time was Li Shou. He was placed in charge of creating arithmetic. The validity of one man creating arithmetic is very slim and this is why Li Shou is considered a legend among the Chinese people(Yan, Shiran, 1987, pp.1-2).

    Li Shou was not the only legend of Chinese mathematics. Quipu knots are looked at as the earliest way of counting things. Quipu knots were used in as way to remind people to do things (Yan, Shiran, 1987, pp.2-3). Much like the western tradition of tying a string around one’s finger. In terms of the knots, the bigger the knot the more important the task was, and the number of knots kept track of how many tasks one needed to accomplish. Since there is no real text from this time period it is impossible to determine what is truth and what is fiction. The only known text from this time the Book on Ancestories has been lost, and is only known to exist because of references to it in later works (Yan, Shiran, 1987, p.1).

    The first known mathematical text is another subject that has been debated by scholars for years, but the most accepted one is The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven (Zhoubi Suanjing)(Yan, Shiran, 1987, p.25). This text was written during the Zhou and Qin period. It is essentially a collection of discoveries and calculations that were made during the aforementioned time period. The other text that is looked at as the earliest is titled Chou-pei Suan-king. This text contains information on calendering, as well as some simple ideas on shadow reckoning. Shadow reckoning was a technique that was used to determine the distance of the sun using an eight foot high stake (see example A). It was based on a hypothesis that if the stake is moved from north to south, for every thousand miles the shadow changes by one inch(Yan, Shiran, 1987, p.30). It was later proved to be false due to the fact that the calculations were being made with the idea that the earth was flat instead of spherical and the original hypothesis was proved incorrect during the Tang Dynasty. The math itself, however, was correct.

    These texts play a very minor role in the history of mathematical texts in China when compared to what may be the most important math text ever written. The Nine Chapters on the Mathematical Art (Jiuzhang suanshu) will be discussed in great detail later in the paper. There may have been a great deal many more mathematical texts written, but they will never be known due to the massive book burning and burying of scholars under the leadership of Emperor Ch’in Shih Huang Ti (245-210 B.C.). It is after his reign comes to an end that starts significant growth for China in the field of technology. This is lead by the advancements that were made in mathematics.

    The history of mathematics cannot be fully appreciated without knowing the history of numbers. The first sign of numbers was used to keep track of time. This was done by what is known as oracle script. The Chinese would use bone as their writing tablet. Numbers began to take further shape when the Chinese began relating them to geometric figures. Oracle numeration was based on a compound number system. Instead of having one set term for a number the number was broken down. For example the number 1999 would be written with the symbols for 1,1000,9,100,9,10,9, meaning 1 times 1000 plus 9 times 100 plus 9 times 10 plus 9(http://alepha0.clarku.edu).

    Compound numbers were phased out in the Han dynasty. This created the first modern numeral system. The system was a base ten system like that of the Romans and the West. Unlike the Romans, the system was also a place value system. At this point in the numerical system there was still not representation of the number zero. It was shown in calculations by just leaving a blank spot. It was not until the thirteenth century A.D. when a man named Qin Jiushao began to represent this blank space with ‘0’(Reserve reading). Even without a number to represent the absence of a place value calculations were beginning to be worked out thanks largely in part to the invention of counting rods.

    Counting rods (chou) were the first tool the Chinese used to calculate. They were a distinctive feature of ancient Chinese mathematics. Calculating with the rods was a system known as chou suan. They were made mainly of bamboo, though some were made of bone. They were the precursor to the abacus. Counting rods were regarded as being approximately six Chinese inches long and one tenth of an inch in diameter (A Chinese inch ranged from 22.5mm to 33.3mm)(Yan, Shiran, 1987, p.7). As time went on the rods began to get shorter. This was done to make them easier to work with.

    There is no clearly defined time of when counting rods were brought into Chinese mathematics, but evidence seems to point to the first half of the Warring States (481-221 B.C.) Counting rods can be represented either horizontally or vertically. This is done in order to make it easier determining which place value a certain number is in. Units are represented vertically, the tens place is represented horizontally, the hundreds place vertically. This pattern of alternation continues to whatever place is needed (see example B). Unlike written text which in China is read from top to bottom, all numbers are read from left to right.

    The counting rod system very much resembles the tally system we are accustomed to using in the West. This system of calculation is based on a visual thought process. While western calculation is memorization based. If one was to look at the calculation of 4+3 the only way to tell the answer is 7 is if you know the number 3 equals three units and the number 4 equals four units. In a visual based calculation, as with counting rods you are able to do the calculation by looking at the physical nature of the number. Four is represented by //// and three is represented by ///. In order to come up with the correct answer of seven all one needs to do is count the rods.

    Numbers in China are read left to right as was mentioned above. The exception to this rule is fractions. For example, the number twelve and one-fourth would be written in a column (see example C). The top number would be the quotient, the middle number would be the remainder, and the bottom number would be the divider. Numbers now had a strong representation. This in turn led to the Chinese making significant discoveries in the field of calculations.

    Though the Chinese were the first to develop a modern numerical system, it was how they manipulated those numbers that made their advancements in mathematics extraordinary. As was mentioned above, the legend of Li Shou was the first documented case of someone doing complex calculations. It is, however, legends like this that have caused some experts to question how much truth is behind all of these mathematical breakthroughs. Such breakthroughs come in calculations as simple as addition and subtraction. It is assumed that people always understood the concept of addition and subtraction. This is probably why it was never explained in the Chinese’s math texts. However, what stemmed from the concept of addition and subtraction were the ideas of multiplication and division. The Chinese developed a way of multiplying by working from left to right. In other words the Chinese start with the biggest number and work towards the smallest number.

    Their multiplication table is taught at an early age by using the “Nine-nines rhyme”(Yan, Shiran, 1987, p.13). It was titled this because, when it was taught one would start with nine times nine and work down to one times one. In today’s school in China they start with one times one and work their way up.

    Calculations were began to be made in regards to fractions. At first fractions were just in tenths in order to keep things simple. It was not until the Nine Chapters was published was the concept of lowest common multiple put to use. It is thought by many that this concept was developed by the Italian mathematician Leonardo Pisano during the thirteenth century, but this is not true(Yan, Shiran, 1987, p.38). It was in fact thought of well over a thousand years prior in China. It is discoveries such as these that put the Chinese way ahead of their time.

    The Nine Chapters on the Mathematical Art is without argument the most important text ever written in Chinese mathematics. Like much of the ancient Chinese science texts, the origin of it is still a mystery (Smith, 1958, p.96). The rough estimate for its time of completion is approximately the first century. It is not known who assembled the book. A widely excepted theory is that the text was a long work in progress and compiles the mathematical knowledge of many from the Zhou, Qin, and Han periods. Broken up into nine chapters containing 246 question and answer problems, it became the basis for modern mathematics.

    Chinese math scholars continue to start their research with this text and write commentaries on it. The most famous of these commentaries was written in 263 A.D. by Liu Hui. Entitled Sea Island Mathematical Manual (Haidao suanjing) it focused its commentary on the ninth chapter of the text which covers the Gougu theorem and introduces quadratic equations. The other chapters cover as follows: Chapter one focuses on field measurement by discussing areas of shapes such as rectangles, triangles and circles. Chapters two, three and six are essentially very similar in that they deal with proportions and fair taxes. Chapter four deals mainly with square and cube routes. Chapter five is the least accurate of the chapters due to a poor knowledge of the value of pi. This leads to some approximations. Chapter seven known as ‘Excess and Deficiency’ deals with a concept that became known in the Middle Ages in Europe as double false position (see example D). The eighth chapter of the text is the most remarkable. It deals with solving three or more simultaneous linear equations (see example E). This concept was 1500 years ahead of anybody else. It is the advanced nature of the algebra that makes this book truly remarkable. Other topics included the manipulation of fractions and the concept of positive versus negative numbers (Yan, Shiran, 1987, pp.39-40). They also figured out the formula for extracting both square and cube roots (see example F). All of these concepts were being practiced in China well over 500 years before anyone else even thought of the concepts.

    Due to the Nine Chapters advanced nature it was used as a text book in many countries. In fact the only major topic of mathematics that it did not cover was the study of trigonometry. The concept of plane and spherical trigonometry was not acquired until Persian astronomers brought it to the country in the third century A.D. (Scott, 1969, p.82). This was one of the few outside influences on ancient Chinese mathematics. Despite all the similarities there is no evidence showing that the Chinese had any knowledge of what the Greeks were doing and vise versa. The only other outside educational influence that could have affected mathematics was the introduction of Buddhism in the mid first century (Scott, 1969, p.81). Buddhist missionaries may have brought with them some Hindu mathematical knowledge. It is more likely however that the Chinese were ahead of the Buddhists in mathematical studies.

    The mathematical knowledge of the Chinese was so ahead of their time it is not surprising that scholars had a hard time believing it. This does not damper the accomplishments of the Chinese as the majority of these criticisms came about before Needham published his works in 1959. The rest of the world in many areas found themselves a millennium behind. To put in today’s standards, just think if we ever realize that another civilization had already thought of the internet.