Ancient History of Chinese Math - Term Paper Example

Topic: ANCIENT HISTORY OF CHINESE MATH (Name) (Institution’s Name) 10/03/08 Introduction There are ancient drawings that indicate the knowledge of measurement and mathematics of ancient time which is based on the stars. Early attempts to quantify time have been found in various places in the world. For example, ochre rocks found in a cave in South Africa which date back to 70,000 B.C show some form of geometric patterns. Early counting has also been thought to have started with the women who kept the record of their biological functions. The knowledge of the Babylonian mathematics comes from clay tablets unearthed in 1850's. They were written in Cuneiform and the tablets were inscribed while the clay was still moist. Ancient Sumerian’s give us the evidence of written mathematics. The Sumerian's contributed greatly in building the ancient civilization in Mesopotamia. The purpose of the essay will be to explore the rise and development of ancient Chinese mathematics, its relations with the Egyptian, Greeks and the Islamic mathematics. In addition the essay will also focus on the benefit of ancient Chinese mathematics and why it was needed. The essay will have several subheadings or sections and it will also have a summary or conclusions of the whole argument at the end. Ancient Chinese Mathematics In China mathematics emerged independently by 11th century B.C. Simple mathematics concepts which were inscribed in tortoise shells date back to the Shang Dynasty. The oldest surviving mathematical concepts and works is the I Ching. This influenced written literature to a larger extent during the reign of the Zhou Dynasty. The ancient Chinese mathematicians developed large negative numbers, a binary system, a decimal system geometry, calculus and decimals. Most scholars have held believe that ancient Chinese mathematics developed independently until the time when the nine chapters were completed. Various discoveries suggest that ancient Chinese mathematics predate the western mathematics. Pythagorean Theorem which is also called the Pythagoras theorem is a good example of Chinese mathematics that predates the western mathematics. Controversy has ensued about the presence of such knowledge in China although evidence of Pythagorean science have been discovered in the oldest Classical Chinese texts called the King Wen sequence. This was a series of about sixty four binary figures which made a hexagram. Each comprised of 6 lines broken (yin) or unbroken (yang). This evidence show that the knowledge of Pythagoras theorem existed in ancient China. The ancient Chinese people were one of the most advanced mathematicians who created enormous numbers and mathematical computations. The evidence of the knowledge of Pascal triangle also existed in China long before the Pascal himself came up with the idea on the same. The focus was mostly on astronomy and making the calendar perfect and they were not so much concerned on establishing the proof. The oldest geometrical work in China came from the Mohist canon philosophy of 330 B.C. This was compiled by the followers of Mozi in 470-390 B.C. This philosophy provided a wealth of information on mathematics and gave the atomic definition of a geometric point. It stated that a line is divided into several parts and that the line with no remaining parts can not be divided into other smaller parts. It also stated that the extreme end of the line was made up of a point. The Mo jing further explained that a point is the smallest unit and it can not be cut into halves since it is impossible to halve nothing. He also offered definitions of and comparison of parallels and lengths and explained that two lines of equal length always finish at the same place. The ancient Chinese geometrical mathematics also gave the fact that planes without the quality of thickness could not be piled up since they can not touch mutually. The Mo jing also gave several definitions of diameter, circumference, radius and volume. The nine chapters on mathematical Art is an ancient Chinese mathematics book that is composed of generations of scholars in the 2nd and 1st centuries. The book laid down an approach to mathematics that centered on finding general methods of solving problems. The contents of the nine chapters include the following, Fang tian or the rectangular fields. In this chapter the work of finding the areas of various shapes and fields and manipulation of the vulgar fractions are found. The su mi chapter explains the pricing mode of different commodities and rice and millet were taken as the exchange commodities. Cui fen chapter explains the proportionality concepts. This includes the distribution of money and commodities at proportional rates. The Shao guang chapter describes extraction of cube roots and squares. The determination of volume of circles and sphere as well as the division by mixed numbers are also found in this chapter. The Shang gong chapter gave light into the determination of volumes of solids in various shapes. The Jun shu chapter gives the light into solving problems on equitable taxation. The Ying bu zu chapter helped in solving linear problems. This chapter was later developed in the west and gave rise to the principle known as the rule of false position. The eighth chapter was the Fang cheng which provided an explanation into solving problems with several unknowns. This was later solved using similar principle in the west called the Gaussian elimination (Burton, 1997). The Gou gu chapter gave the principle of solving problem regarding base and altitude. The ancient mathematics in China was very important especially in construction. It was also used in astronomy field. The right angled triangles and the Pythagoras theorem were very important and prominent in Chinese writing. These were both in practical science and mathematical treatises. They grasped a lot of principles regarding the right-angled triangle and applied these principles to practical problems. In the later development of mathematics in China the Chinese performed calculations using very small bamboo counting rods. This led to the emergence and use of the rod numerals as well as a positioning system for writing numbers. The three main mathematicians were Zhen Luan in the 6th century, Li Chunfeng in the 7th century and Zhao Shang in the 3rd century. The original texts written by the three mathematicians were basic and had complex computations which were without any indications on how to solve problems. The Zhao bi used the knowledge of right angled triangles in order to explain the astronomy. His knowledge was also taken to offer the most ancient proof of Pythagoras theorem although this was refuted by many mathematicians. The Zhou bi astronomy followed the gai tian cosmology that stated that the heavens rotated above the earth since the earth is a flat plane. With this idea the sun’s height could be calculated using the gnomonic or the bi and its shadow. The idea of the shadow principle stated that for every 1000 li located away from the shadow spot the eight chi gnomon shadow increased by one cun. Early illustration of similarity of two triangles was also done by the Chinese (Cooke, 1997). Triangle AEB was taken to have gnomon or the bi as the altitude and the shadow as the base. Triangle ACD was taken as having the height of the sun as its altitude and the base was taken to be the distance from the end of the shadow to the no shadow spot. The Pythagoras theorem used showed that squaring the base and the altitude gave the hypotenuse. In the ancient Chinese mathematics the right angle was known as the gou or the kou which meant the leg. The height or the altitude was known as the gu or ku while the hypotenuse was known as the xian or the hsian which meant the bowstring. The equivalent to the ancient 3-4-5 triangle solving formula is the current formula, (a2 - (c-b)2) /(2(c-b)) = b. a2 - (c-b)2 -c2 +2bc -b2 +a2 2bc -2b2 when we know a2 + b2 = c2 = b so substitute a2 - c2 = -b2 2(c-b) 2c -2b 2c -2b which is used in finding one side of a triangle when the other is known or when finding the position of a triangle. Chinese Numerals These are characters that were and are still used in writing numbers in Chinese. The ancient Chinese systems included the Suzhou system, the traditional systems and the simplified system. The Ancient character system is still in use today and it is roughly analogous to writing numbers in a text. Characters that represented numbers zero through nine and others that represented tens, thousands and hundreds were developed. The only surviving system is the huama system and it was the main system that was used in the Chinese markets before 1990.The current Chinese mathematics system is written in Chinese language. Numbers zero through nine are represented by characters. There are also other characters that represent larger numbers such as thousands, hundreds and tens. Two types of Chinese numerals exist today in China. One is used for day to day writing and the other is used for financial and commercial contexts. T is taken to denote traditional characters and S is used to denote the simplified characters. For example the table below shows how the ancient Chinese used various characters to denote various values, Character Value Notes 漠 10-12 (Ancient Chinese) corresponds the SI prefix pico. 渺 10-11 (Ancient Chinese) 埃 10-10 (Ancient Chinese) 塵 10-9 (Ancient Chinese) 皮 corresponds the SI prefix pico. Others that were used in the ancient Chinese math and still in use today include, 厘 1/100 also 釐.still in use, corresponds the SI prefix centi. 分 1/10 still in use, corresponds the SI prefix decil in use, corresponds the SI prefix micro. 忽 10-5 (Ancient Chinese) 絲 10-4 (Ancient Chinese) 毫 1/1,000 also 毛.still in use, corresponds the SI prefix milli. 厘 1/100 also 釐.still in use, corresponds the SI prefix centi. 分 1/10 still in use, corresponds the SI prefix deciAncient Chinese) 奈 (T) or 纳 (S) corresponds the SI prefix nano. In old texts that have been found it has been discovered that a number like 114 was written as (100) (10) (4).A number like 12,246,579,902,235 was translated into 12,2465,7990,2235 which was written as (12) (1,0000,0000,0000) (2465)(1,0000,0000) (7990) (10000) (2235) More examples include number 206 which was structured as follows (2) (100) (0) (6) with four Chinese characters. Today the huama system is used explicitly in displaying prices in the markets in China. Special digits are used instead of the characters. For example, 〤 〇 〢 二 are positional and when written in the horizontal order the numerical value is in two rows. The above characters represent number 4022.The bottom row which has more than one Chinese character represent the unit of the first row digit. The top row consists of the numeric symbols. Relation with the Egyptian Mathematics According to Stillwell (2004) berlin papyrus and the Rhind mathematical papyrus texts show that ancient Egyptians had the basic concepts of geometry and algebra. They were able to use fractions to perform operations such as subtraction, addition, multiplication and division. Just like the Chinese mathematicians they were able to calculate the surface areas and volumes of different shapes such as the circles, spheres, triangles and rectangles. Unlike their ancient Chinese counterparts the Egyptians mathematics were able to calculate the volumes of complex shapes such as the frustum .They used the formula, (diameter)*(8/9) to calculate the area of a circle a formula that the Chinese mathematicians did not have. According to Derbyshire 2006, just like the Ancient Chinese mathematics the Egyptian mathematics was not based on proof and whatever was done was taken to be absolute truth with no need for proof. But unlike the Egyptians who used unit fractions the Chinese used very common fractions and they managed to find lowest common denominator for adding different fractions. In the same way both the Egyptians and the ancient Chinese mathematics had a collection of practical problems which gave the problem first, then the answer and sometimes the solution methods used. However they did not offer the proof for the problems solved. (Bowler, 1991) Relation with the Greek Mathematics Greek mathematics refers to the studied mathematics before the Hellenistic period. It was only limited to ancient Greece, Sicily, Asia Minor and Libya. It was written at a period between 500-1500 years after the Greeks finished composing their work. The most crucial center of learning Greek mathematics was Alexandria. It attracted many scholars composed of the Jewish, Phoenician, Persian and the Indian scholars. Most of the Greek mathematical articles have been found in Egypt, Mesopotamia, Greece and Sicily. Fundamental developments occurred in mathematics in Greece. Geometry and formal proof of solutions to problems was carried out. Integral calculus, mathematical analysis, applied mathematics and the number theory also originated with the Greece. Although the Chinese mathematicians had also the idea of Geometry, they never gave proof to their work. The Chinese did not come up with the integral calculus something the Greek mathematicians came up with and were able to give proof to their work. It was in Greece that Pythagoras and Thales brought the proper knowledge of Pythagoras theorem together with proofs for the problems in that field. Using this knowledge, the Greek mathematicians were able to solve the problems such as calculating the distance of ships from the shore and also the height of pyramids and triangles. Pythagoras properly stated the Pythagoras theorem and according to Proclus' comments on Euclid he was able to construct the triples algebraically. As compared to the Ancient Chinese mathematics the Greek mathematics was more advanced and well understood by many people from several countries. Much of the work done in the geometry field was an advancement of what the Chinese mathematicians had done although the Greek mathematicians were able to offer various proofs for the concepts used in the geometry field (Heath, 1981) Relation with the Islam Mathematics Islamic mathematics is also called the Arab mathematics. This refers to the mathematics that was developed in the period between 622-1600 B.C in the Islamic world. The dominant language during this period was Arabic although most scientists were Muslims. The center of Islamic mathematics was in present-day Iran and Iraq. Islamic mathematics borrowed heavily from Indian, Greek and Mesopotamia mathematics. The work of mathematicians such as Apollonius, Archimedes, Euclid, Aryabhata, Diophantus and Brahmagupta made the basis for development of the Islamic mathematics. Just like the Chinese mathematicians, the Islamic mathematics touched on geometry field and did not offer proof. But more than the Chinese mathematics, the Islamic mathematics touched on areas such as algebra, Euclidean theory of numbers, polynomial algebra, numerical analysis, combinational analysis and the numerical solution of equations. The Islamic scientists were able to carry out the geometric construction of equations something that had not been done by the Chinese scientists. By around 1000 AD the first known Islamic proof was carried out by Al-Karaji. He started by proving the arithmetic sequence followed by the binomial theorem, the Pascal triangle and later the integral cubes. The Chinese came up with the knowledge about the Pascal triangle but were not able to go further to proof their argument. It was the Islamic mathematics that came up with the truth of the statement for n=1(1=13) Conclusion The Ancient mathematics has come along way in its development. More work has been done by many Chinese mathematicians to ensure that they are up to date. They have been able to offer proof of the solution of various problems unlike in the ancient days where only the problem and the answer were given without giving the proof. There was much relationship with other mathematics such as the Greek mathematics, Egyptian mathematics and the Islamic mathematics. The Egyptian mathematics borrowed most of its principles from the Chinese mathematic as well as the Greek mathematics. There are a number of related topics that were common to all the three schools of thoughts such as the Pythagoras theorem which was very useful and still useful in day to day mathematics. Although the ancient Chinese mathematics did not offer proof to the solutions of the problems solved, the mathematicians were confident that whatever they did was right to the best of their knowledge. Today the current Chinese mathematicians are able to offer proof to problems in mathematics. Mathematics is a dynamic subject and it is still being improved on by the current mathematicians. It is still an important subject in many areas of life and that is why ancient mathematicians saw the need to come up with formulas that addressed various needs of life mathematically. References Bowler, C. (1991). A History of Mathematics in China (New York: John Wiley & Sons, In) Burton, D. (1997). The History of Mathematics in China: An Introduction (The McGraw-Hill Companies, Inc) Cooke, R. (1997). The History of Mathematics: A Brief Mathematical induction Course, (New York: Wiley-Interscience) Derbyshire, J. (2006). Unknown Quantity: A Real and Imaginary History of Algebra, (Joseph Henry Press) Heath, T. (1981). A History of Greek Mathematics, (Volume I, Dover publications) Heath, T. (1981). A History of Greek Mathematics, (Volume II, Dover publication) Stillwell, J. (2004). Mathematics and its History in Egypt (Springer Science Business Media Inc.) Read More