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Home , Chinese mathematics, Jia Xian, Li Ye (mathematician), Qin Jiushao, Sunzi Suanjing, Zhoubi Suanjing

V Chinese

The ancient record of civilization in China began in the more northern Yellow River Valley where the Yangshao culture existed from about 5,000 to 2700 B.C.E. These people lived in the middle and lower parts of the river valley. They cultivated rice and millet, kept pigs, wove baskets and made pottery without a wheel. Their pottery was dyed red with black geometric designs painted on it, and also pictures of fish and human faces. Archeologists have found pottery from these people with the symbols for 1, 2, 5, and 7 painted on the dishes. The next culture, one which evolved from the Yangshao, was the Longshan culture, which thrived from 3,000 to 2,000 B.C.E., approximately. This culture domesticated the water buffalo, sheep and cattle. They made black pottery on a wheel. The pottery was not painted but decorated with grooved or raised rings. From about 2100 to 1600 B.C.E. the Xia (or Hsia) dynasty controlled the Yellow River Valley. Recent discoveries have unearthed a large city with a wall and a moat for defensive purposes surrounding the city. This city has been dated to the early Xia dynasty. It is in Province in today’s China.

From about 1500 to 1000 B.C.E the Shang dynasty controlled the area with a capital city at Anyang, also in current day Henan province, just north of the southern bend in the Yellow River. They also used the walled and moated city of the Xia, mentioned above. By 1300 B.C.E. the Shang scribes had invented a brush for writing on bamboo strips, and had a system of writing numerals. We have records of their kings through oracle bones, bones with inscribed writing that were used to foretell the future. Their writing was pictographic and had pictures of the objects referred to in the writing as the symbols used to write. After the Shang dynasty we come to the period of classical , that of the Zhou (or Chou) dynasty. The early Zhou is usually dated from 1027 to 771 B.C.E. with the later Zhou extending from 770 to 256 B.C.E. During the later the territory controlled extended to include the more southern Yangtze River Valley. Also during this period the two social strata which formed the basis of Chinese society until 1911 were formed. These are the peasant farmer class and the scholarly scribes in more administrative positions. The Zhou nobility studied a curriculum of ritual, music, archery, horsemanship, calligraphy and mathematics. They had a sexagesimal calendar but a decimal for daily life. During the 700’s B.C.E. the first construction on the began, and iron was introduced into China. The period from 551 to 233 B.C.E. is known as the Hundred Schools Period of Chinese history. The great philosopher Lao Tsu seems to have predated Confucius in this period. His dates are a matter of speculation. A school of philosophy emerged from his teachings. He tried to reform the government. His philosophy was one of non-striving and non-interference, and came to be known as Taoism. He believed that his central principle could not be expressed in words, so his writings are contradictory and difficult to absorb. His ideas lead most people to a more passive role towards reform. He taught that the Tao is The Way, how the universe actually works, and trying to oppose it is fruitless activity. There were elements of mysticism in his world view. From about 550 to 470 B.C.E. the philosopher Kung Tsu (Confucius) expounded a moral code that included respect for the past, for elders, loyalty, universal education and responsible government that would bring a decent life to all of its citizens. He stayed away from speculations on the nature of existence and the shape of the universe. His writings espouse a more activist way of dealing with life. A very old book, The Book of Diverse Crafts (Kao Gong Ji in Chinese), was written by 476 B.C.E. with later additions in the Warring States Period. The book includes the state of Chinese thinking on engineering, administration, , and mathematics at that time. In the area of mathematics it contains material on , with tenths singled out as the commonest fractions in use, the of measurement for use in surveying and building, standard metrological units and angle measurements of o o o. 90 , 45 and 22.5 The third great philosopher of China is Master Mo, who lived about 470 to 390 B.C.E. He was a master engineer and an expert on fortifications. So he was sought out by rulers to help them become dominant rulers. He had a pacifist philosophy and did not think one should blindly follow the principles of the past, as they were innovative in their own day. He urged people to lead lives of self-restraint, avoiding both material and spiritual excesses. He evaluated actions based on their utility to promote the good for people. He argued for a universal benevolence and love for all people, whereas Confucius argued for deeper love for ones’ own parents and superiors. During the Warring States Period, from 403 to 221 B.C.E. the Chinese used a system of to calculate. This used bamboo, wood or ivory rods shaped like the numerals and arranged in columns by place value, with the larger powers of 10 going to the left. This has a natural place value arrangement, and has no need for a zero, as one merely left that column empty. It carried on the earlier base 10 arithmetic. Another old book is “The Records and Rites of the Zhou Dynasty (Liji in Chinese). In this book we find the curriculum for the Zhou nobility mentioned above as well as mention of the two main classes of administrative officials, the Sihuai who were statistical arithmeticians and the Chouren who were astronomers in charge of the calendar. In the second section we find mathematical problems included to support the development of mathematical innovation. The Book of Master Mo (The in Chinese) contains the beginnings of theoretical geometry in China. At the start are definitions of point, circle, line, surface and solid figures. In this book is the principle that “A stick, though half of it be broken off each day, will never be exhausted.” This suggests some interest in the idea of a limit 1 1 1 1 to us, and in the infinite series + + + + + = 1. 2 4 8 2n After the Zhou Dynasty the Qin (221 to 206 B.C.E.) rose to power in China. This Dynasty was a severe and ruthless government, which destroyed the old nobility and instituted large scale public works. With their military they were the first to unify all of China, but they did not last long. They standardized the characters used in writing as well as Chinese metrology. They considered scholars dangerous to their power, and burned all books that were not for engineering or agriculture. The Imperial Library was destroyed in 206 B.C.E. The famous terra cotta army, a collection of 8,099 larger than life ceramic soldiers was buried with the first Qin emperor in an ornate mausoleum. This is near the Qin capitol city of Xi’an, on the Yellow river near the current city of Xianyang in Shaanxi province. The avoided the excesses of the Qin, and lasted from 206 B.C.E. to 220 A.D. The Han reunited China, and extended it to cover most of central Asia. They reassembled old books from fragments and reopened schools. By the first century A.D. they were trading with Persia, Alexandrian Egypt and the Roman Empire. The famous was thriving. Papermaking was invented in China by 100 A.D. It was in Egypt by 900 A.D. and in Spain by 1150 A.D. There are two venerable texts on which date from this Han Dynasty. They are the (The Zhou Shadow Guage manual) and the Jiuzhang Suanshu ( The Nine Chapters on the Mathematical Art.) The Zhoubi Suanjing was primarily a book about astronomy which included plenty of mathematics. It was probably written between 100 B.C.E. and 100 A.D. There were two competing theories of astronomy at that time in China. Gai theory held that the earth is an inverted basin bounded by four seas with a heaven that is a concentric hemisphere overhead. HunTian theory held that a rotating sphere contains the heavenly lights and is centered around the earth. The text records the measurements of the moon’s movements, those of the , and uses the Gougu Theorem ( ) to calculate distances. Here Gu refers to the length of a vertical stick in the ground, while Gou refers to the shadow cast by the stick. The theory of similar triangles is part of the mathematics used in this book. A picture like the following for 3, 4, 5 triangles appears to justify the Gougu Theorem. It is claimed that this is just a numerical check, but 9 + 16 = 25 is an easier check. It seems that this picture, which generalizes, can be thought of as a proof.

4

3 5 5

5

5 3

4

The Zhoubi Suanjing, probably written around 100 A.D. but based on much earlier material, gives rules for calculating with decimal fractions and approximates square roots, without explicitly recording how to do this. In the second half of this treatise, a character named Master Chen lectures on the importance of learning to argue both inductively from particular cases to general principles and deductively, from hypotheses to conclusions that follow logically from these hypotheses. It is thus clear that mathematics had reached a sophisticated level by the time this work was written. The calendar in the Zhoubi Suanjing has a 365.25 day year, with a lunar month of 29 & 499/940 days. This is 29.53085106 which compares well with our current figure of 29.5308796 for the length of the lunar period. The Jiuzhang Suanshu is the analog of the Elements of for Chinese culture. It consists of 246 problems arranged in nine chapters. It is a practical treatise, designed to be used to train engineers, architects and planners for their duties in the workplace. Here is an outline of the contents of the nine chapters:

1. Areas of squares, rectangles, triangles trapezoids, circles and annuli. Calculations with fractions. 2. Proportions, focusing on the exchange of millet and rice. 3. Proportions again. This time focusing on taxation and distribution of properties. 4. Given a figure with a certain area or volume, to find the dimensions of its sides. Includes a trial and error method for finding square and roots. 5. Exact and approximate volumes of prisms, cylinders, pyramids, circular cones and tetrahedra. (assumes = 3) 6. Fair taxes. Proportions again. 7. solving linear . 8. Solving simultaneous linear equations and how to calculate with both positive and negat ive numbers. (red counting rods for +, black for -) 9. Problems involving the Gougu Theorem and an introduction to solving quadratic equations.

The Jiuzhang Suanshu has the old Chinese algorithm for approximating square roots to any desired degree of accuracy in chapter four. This used to be taught in 8th grade schools in the U.S. with no explanations, just do the rote divisions. Here it is. Suppose you want to find the of 1145.

First break 1145 into two-digit pieces working from the decimal point:

11 45 (The answer goes in the open rectangle.)

Now we start the algorithm: Write the nearest smaller square root of the 11 in the answer box and subtract its square 11 45 3 from 11. 9 6 (3) 2 45 Then double the 3 in the answer box, 6, and write it 1 89 out in front of the 2. Bring down the 45, and guess 56 the best number of the form 6x, which “divides” our 245. It is 63. 3 x 63 = 189 subtract it from 245, and write its multiplier, 3, in the answer box, 33. 33 x 33 = 1089, and this is the best smaller answer.

We can go on:

11 45. 00 00 33.83 9 To get more accuracy, keep 63 2 45 on the same way, doubling the 33 1 89 and seeking a number 66x which 668 56 00 divides 5600. 53 44 This number is 668, 668 x 8 = 5344 which gets subtracted from 5600. 6763 2 56 00 20289 6763 x 3 = 20289 6766(?) 5411 We stop. 33.83 x 33.83 = 1144.4689

Here is what is going on. 2 The whole area of the big square is (10A + B) = 100A2 + 2(10A + B) + B2 Which one can see in the pieces of the picture.

10A

2(10A + B) = 106 + 6 100A2 10A 10A + B 1089 = 332= (103 + 3)2 which is where the 23 comes from. so A = 3 and B = 3

10A + B B

B

One can draw a cubic picture of a similar sort and develop a algorithm, which the Chinese did, and which is in chapter four of the Jiuzhang Suanshu.

Here is the place to stop and do worksheet #9, the Chinese Square Root Algorithm.

Chapter eight has an algorithm for solving simultaneous linear equations. It is called Gauss-Jordan elimination in the west and was discovered by around 1800, long after the Chinese were using it. It is the basis of the best computer software for solving systems of linear equations. It can work for any number of unknowns and any number of equations. It is in most Linear courses.

Here is an example. Solve the system:

x + 2y + z = 4 3x + 8y + 7z = 20 2x + 7y + 9z = 23

One can add or subtract any multiple of any from any other equation without changing what the solution is. We first use the first equation to obtain zero in the first terms of the last two equations:

x + 2y + z = 4 2y + 4z = 8 second Eqn -3( first Eqn ) 3y + 7z = 15 Third Eqn -2( first Eqn)

x + 2y + z = 4 y + 2z = 4 ½ of second Eqn. 3y + 7z = 15

x + 2y + z = 4 y + 2z = 4 third Eqn – 3(second Eqn) z = 3

We do this to get zeros in the first two places of the third equation. Now since z = 3, substitute in the second equation to get y = -2, and substitute both z and y values in the first equation to get x = 5. We have the solution.

Here we work on worksheet #10, Guassian Elimination.

After the Han Dynasty there is a period of disorder until the from 598 to 618 A.D. But there are three very brilliant mathematicians who are active despite the unrest. They are who was active in about 260 A.D., Sun Zi from the late 200’s to the 300’s A.D. and Chongzhi. (429 to 500 A.D.)

Liu Hui wrote The Sea Island Mathematical Manual and the Commentary on the Jiuzhang Suanshu. In the latter he systematizes the presentation, provides brief ideas of the proofs of why the calculations are valid and gives underlying principles. He uses inscribed 96 and 192-gons to approximate the value of " . He gets " = 3.141024 (We have 3.14159265. ) He gets close to the idea of a limit in his work on the number . He correctly calculated the volume of a frustrum of a square pyramid, and of a tetrahedron. He uses the ideas in the old Chinese square root algorithm to solve equations of higher degree. He fails to get a proof for the formula! f or the volum! e of a sphere. But his version of the Jiuzhang Suanshu is still one that scholars use to this day.

Sun Zi wrote Master Sun’s Mathematics Manual, the Sunzi Suanjing, in which we find the Chinese Remainder Theorem. His treatise became a standard text for civil servants later on. The theorem is used in and is often in first courses in the subject. The Chinese used it in astronomical problem solving. For instance, at one point in time the winter solstice, the new moon and the start of the 60-day cycle used in ancient Chinese dating all coincided. When will this occur again? One has to reconcile counting by 365.25’s, 29.5’s and 60’s. This would be solved by reconciling 36525, 2950 and 6000. An example shows this process better than an abstract explanation. Here is one involving smaller numbers.

A farmer is taking as load of eggs to the local town one morning when an oxcart accidentally bumps his basket and breaks a lot of the eggs. Luckily the farmer had his children count the eggs before leaving home. One child remembers that she counted by 3’s as 3 is her favorite number. There were 2 eggs left over in this count. A son used the fingers on one hand and counted by 5’s and had 3 left over. His wife used a broken egg carton and counted by 7’s and had 2 left over. How many eggs did the farmer originally have, so he can negotiate with the owner of the oxcart?

We write N for the number of eggs. N = 3X + 2 N = 5Y + 3 N = 7Z + 2

First find a number that is a multiple of 5, a multiple of 7 and leaves 2 when divided by 3. 5 x 7 = 35 which will work fine.

Secondly, find a number that is a multiple of both 3 and 5 and leaves 2 when divided by 7. 15 is no good, but 30 works fine.

Third find a number that iis a multiple of both 3 and 7 but which leaves 3 when divided by 5. 21 is no good, 42 is no good but 63 works fine.

So, adding the numbers we found, 35 + 30 + 63 = 128 solves the problem.

Notice however, that there are infinitely many different correct solutions to this problem, making it what is called an indeterminate system of equations. If you add 3 x 5 x 7 = 105 to 128, or any multiple of 105, you get a perfectly satisfactory solution. So 233, 338, 443, and so on will all work. Presumably the farmer is honest, and only had room for 128 eggs in his basket. So the number 23 = 128 – 105 also works.

Work on worksheet # 11 now, which is a Chinese Remainder Theorem Problem.

Homework VIII

1. Given the following system of four linear equations in four unknowns, use the method of the Jiuzhang Suanshu to find the solution.

2u " v = 0 "u + 2v " w = 0

" v + 2w " z = 0 " w + 2z = 5

2. A certain number leaves a remainder of 5 when divided by 11, a remainder of 3 when divided by 17 and !a remainder of 2 when divided by 19. Find the smallest positive solution, and give two other solutions.

3. Use the algorithm from the Jiuzhang Suanshu to find the square root of 2 to 3 decimal places.

The next mathematical star we study is Zhu Chongzhi, a very talented astronomer and engineer. In addition to his mathematical achievements, he wrote 10 books in a genre that would later be called novels. He improved Liu Hui’s calculation of to 3.1415926 which is the same as the current 7-place approximation. He also proved 4"r3 that the volume of a sphere of radius r is V = , using the idea of infinitesimally 3 thin slices which was a method attributed to Cavalieri in Europe in the 1600’s A.D. This is a start of the integral .

The Sui Dynasty from 598! to 618 A.D. restored unity to China, engineered the Grand Canal from the Yangtze to the Yellow River, and created a civil service system based on merit examinations. Block printing was invented in China around 600, using the paper that had then been in use for 500 years in China. The more long lasting T’ang Dynasty ruled China from 619 to 907 A.D. The civil service examination system was continued, an Imperial Academy was established in 754. BY the late T’ang Dynasty the mathematics curriculum at the Imperial Academy had evolved into a completed system based on the Ten Books of Mathematical Classics. These included the Jiuzhang Suanshu, the Sea Island Mathematical Manual and Master Sun’s Mathematical Manual. Trade with foreign nations flourished along the Silk Road and a capital city was established at Chi’ang-an, near Xi’an. The first true porcelain china was produced during this era. The great Chinese poet Po (701-762) wrote beautiful verse in the tradition that extends back at least to the later Zhou Dynasty. The extended from 960 to 1270 A.D. This was an autocratic form of government. During this dynasty tea, cotton and early ripening rice were harvested and gunpowder, the magnetic compass and stern-post rudders were invented. The mathematician practiced about 1050. He recognized the importance of what we call binomial coefficients in extending the material in the fourth chapter of the Jiuzhang Sunashu to finding roots of equations of the fourth and fifth degree. He wrote down the array of binomial coefficients that we know in the west as Pascal’s Triangle. Pascal studied this in 1655 A.D. We should call it Jia’s triangle, although there is evidence of this array in Indian and Persian mathematics even before Jia. Jia created iterative methods for solving third and fourth roots and worked with equations having only positive coefficients. Jia noticed the following: If one wants to solve x 3 " 26, one guesses the largest 3 digit in the answer, say 2. Then (x + 2) " 26 will have the same root as the original 3 equation, except reduced by 2. (x 2 + 2) 26 = x 3 26. But he expanded 3 3 2 !2 3 (x + 2) " 26 as x + 3(x 2) + 3(x (2) ) + 2 26 and got familiar with the ! 3 binomial coefficients for (x + h) and also for higher powers of x + h. Liu Yi practiced from about 1080 to 1120 A.D. and extended Jia Xi’an’s ! methods to include both positive and negative coefficients. In 1167 A.D. the nomadic Mongols living to the north of China invaded and conquered the S!ong in the north. The were led by Temujin, or Ghengis Khan. Temujin’s grandson, conquered the remaining Song empire in the south, moved his capital to Beijing in 1264 A.D., and hosted Marco Polo from Venice from 1225 to 1292. His dynasty was called the . Mathematics reached its highest level of development during this time. Four master mathematicians were active during this thirteenth century. The first of these was (1202-1261) a brilliant, talented, athletic, womanizing character. In 1247 he wrote a work called The Mathematical Treatise in Nine Sections. In it he extends the Chinese remainder Theorem to work for moduli (the numbers used like 3, 5, and 7 in the problem on the previous page) that are not necessarily relatively prime. He introduces the use of a round zero symbol. He solves higher degree equations, including one of degree 10. He allows positive, negative and fractional coefficients in the equations he solves, and allows both positive and negative numbers as solutions. The equivalent work in our algebraic language, of the Chinese work to evaluate higher degree polynomials and to factor them, is called synthetic evaluation and division. The two are handled the same way. ax 3 + bx 2 + cx + d can be factored and written as ((ax + b)x + c)x + d. Then you calculate ax + b, multiply by x, add c etc. Suppose we have 5x 3 + 7x 2 3x 11 as our higher order polynomial. To evaluate this polynomial for x = 2 one can organize the calculation as follows: !

2 5 7 -3 -11 (Here, bring down the 5, 2x5 =10, add, 10 34 62 2x17=34, add, etc.) 5 17 31 51 The answer is 51

The same tabular calculation can be used to divide 5x 3 + 7x 2 " 3x "11 by 51 (x – 2). You have to read the answer as 5x 2 + 17x + 31 + (x " 2) ! In 1248, one year after Qin Jiushao wrote the Mathematical Treatise in Nine Sections, another gifted mathematician, Li Zhi (1192-1279) wrote The Sea Mirror of Circle Measurements and a w!or k called The Old Mathematics in Expanded Sections. The first of these works consists of 170 problems having to do with the incircles and excircles of a right triangle. These are the two circles you met in Quiz number 2. Like Qin Jiushao he solved equations of higher degrees, using the iterative methods generalized from the square root and cube root algorithms. He was given the same name as the third T’ang emperor, whom he did not respect. He changed his name to . He was a government official in Henan Province until the Mongols overran this part of China in 1232. He then became a scholar-hermit and lived an extremely simple life with two friends near Fenglong Mountain. They became known as “The Friends of Fenglong Mountain.” Kublai Khan invited, and then forced, Li Ye to accept a position in a recognized academy, but Li Ye then got away because he was an old man. He died near Fenglong Mountain in the city of . The third outstanding mathematical talent during the thirteenth century in China was . He lived in the south of China during the Song Dynasty. Not much is known with certainty about his life, but he wrote “A Detailed Analysis of the Mathematical methods in the Nine Chapters” in 1261, and a work called “The Method of Computation of Yang Hui” in seven volumes during 1274 and 1275. His writings consolidate what has gone before and are an important source for understanding ancient Chinese mathematics, even today. His careful reworking of problems, and his organizing them according to increasing degrees of difficulty brings his expositions close to being proofs. He also gives examples of magic squares up to 10 by 10. A is a square array of numbers with the property that the sum of all of the numbers in any column, or any row, or any diagonal, is always the same number. The oldest magic square is the Lo Shu, dating back to at least 300 B.C.E. during the warring states period. Legend has it that the Emperor, Yu Huang had a flood to deal with in the Yellow River area, along a tributary called the Lo River. A turtle crawled out of the Lo River, and the magic square called the Lo Shu was arranged on the scutes forming the shell on the back of this turtle. The Emperor used the Lo Shu to have the flood subside. The Lo Shu is below: 4 9 2 3 5 7 The magic Number here is 15. 8 1 6

Yang numbers are odd and yin numbers are even. Chinese philosophy advises us to strive for a balance of Yin and Yang energies in life. The 5 in the center is the average of any two numbe!r s on opposite sides of the 5.

We use worksheet # 12 Magic Squares at this point.

Here is a way to generate magic squares of odd order. Start with your 1 in the top center position. Keep putting the next integer in the place to the upper right of where you put the last number, but scrolling around, like a word processor does when you go beyond the end of the line. So the 2 is at the bottom, and the 4 is scrolled across from the 3. x x 1 x x 17 x 1 8 15 17 24 1 8 15 x 5 x x x x 5 7 14 16 23 5 7 14 16 4 x x x x 4 6 13 x x 4 6 13 20 22 x x x x 3 10 12 x x 3 10 12 19 21 3 x x x 2 x 11 x x 2 9 11 18 25 2 9

One more rule. When you get stuck, like following the 5 by trying to place it where the 1 already is, put the 6 directly under the 5, and go on. The 11 is to the upper ! right of the 15 b y sc!rol ling, so the 16 goes below !the 15. Our magic number here is 65. A famous 4x4 magic square appears in the engraving “Melancholia” by the German artist Albrecht Durer. The year in which he created this piece of art was 1514, the number in the bottom center of the magic square which appears on the wall behind the perplexed angel in the engraving.

16 3 2 13 5 10 11 8

9 6 7 12 4 15 14 1

!

Homework IX

1. Use the wrap-around method to construct a magic square of order seven. Show three intermediate steps. What is the magic number of this magic square?

2. a. Use the method of synthetic division to divide 3x 3 " 25x 2 + 38x " 70 by (x – 7). What is the answer?

b. In the above polynomial, what do you get for an answer when you substitute x = 7? !

The fourth mathematician from this century is who was most active from about 1280 to 1303 A.D. He was respected as a mathematician and a teacher in China. he wrote “Introduction to Mathematical Studies” for students. This text covers all of the fields of Chinese mathematics and has the problems ordered by the degree of difficulty. He also wrote a more advanced book with his own original contributions to mathematics in it, called “The Jade Mirror of the Four Elements.” In this work he goes beyond the mathematics of Qin Jiushao and Li Ye. He not only solves equations of higher degree, but equations with up to four unknowns, say x, y, z, w. Zhu Shijie represents the climax of mathematics and algebra in classical Chinese culture. In the ensuing centuries, western contacts brought in more outside influences.