China - THE GROWTH OF MATHEMATICS TO 1600 - MATHEMATICS IN HISTORY - Mathematics for the liberal arts

Mathematics for the liberal arts (2013)

Part I. MATHEMATICS IN HISTORY

Chapter 2. THE GROWTH OF MATHEMATICS TO 1600

[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.

GALILEO GALILEI (1564–1642)

So far, we have looked at the ancient roots of mathematics, primarily around the Mediterranean Sea. In this chapter we will look further east, to the two great Asian cultures of China and India, before returning to Western Asia and Europe. The mathematics developed in these cultures was in each case influenced by others, although less so in the case of China. We will trace some of the connections among the different traditions, but must acknowledge up front that much remains unknown about this fascinating topic.

2.1 China

The Master said: Learning without thinking is useless. Thinking without learning is dangerous.

CONFUCIUS (551–479 BCE)

China.1

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Major Chinese dynasties, from the Qin to the Ming.

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History to 1600: An Overview

China is dominated by two great river systems. The Yellow River and the Yangtze River both originate in the Tibetan Plateau, and flow eastward to the Pacific Ocean. The Yellow River is in northern China, the Yangtze further south.

The earliest known Chinese civilizations arose in the Yellow River valley. By 4500 BCE villagers there were growing millet and raising pigs. The earliest known writing dates from the Shang dynasty (c. 1550–1046 BCE), on “oracle bones”—animal bones and turtle shells sometimes used for divination—and on bronze. The scribes also wrote on bamboo strips, but these didn’t last very long. The Shang dynasty only encompassed a small area in northern China.

The successor to the Shang was the Zhou dynasty (c. 1046–256 BCE), a semifeudal society with a somewhat larger geographical extent. Work on an early version of the Great Wall of China, intended to protect the northern border against invasion by nomads, commenced in the 8 th century BCE. Culture flourished in the 6 th century BCE, with the founding of several academies of scholars. The most famous philosopher of this time was Confucius (Kung Fu-zi, “Master Kung”), who valued education, loyalty, moderation, and his version of the Golden Rule: “What you do not wish for yourself, do not do to others.” Also dating from this period is the Taoism of Lao-Tzu, a more mystical philosophy and an important influence on early Chinese science.

After the Zhou dynasty was the short-lived but important Qin dynasty (221–206 BCE). This dynasty was initiated by one of the most remarkable rulers of any time and place, Qin Shi Huangdi (“First August and Divine Emperor of Qin”). Qin united all of China for the first time. He reorganized the government, destroying the power of the feudal lords, and he established a central bureaucracy. He engaged in massive infrastructure projects, building roads and irrigation systems, constructing a canal linking the Yangtze and Pearl River systems in the south, and completing the Great Wall in Inner Mongolia. The current Great Wall was built later on the foundations of this one. The construction of the wall cost the lives of thousands of workers and helped lead to the end of the Qin dynasty a few years after its founder’s death.

Qin Shi Huangdi standardized the Chinese system of weights and measures, and the Chinese written language too. After this time, regardless of the local spoken language, educated Chinese used the same written characters. The emperor distrusted scholars and banned private ownership of books. For his tomb, he employed about 700,000 workers over a period of more than 30 years to build a massive underground palace in wood, clay, and bronze. In 1974 local farmers made a remarkable discovery at his tomb. He had “protected” his underground palace with an army of 8000 life-size soldiers and horses modeled in clay. The army was complete with generals and chariots, and each soldier was unique. Qin Shi Huangdi died at age 49 in 210 BCE. His dynasty was overthrown four years later.

The Han dynasty (206 BCE–220 CE) succeeded the Qin. The most notable period in this dynasty was the rule of Emperor Wu, which lasted over 50 years, from 140-87 BCE. During this time, Wu extended his empire from Vietnam in the south to northern Korea, and westward into central Asia. The Silk Road became a major trading route, reaching all the way to Rome. Chinese culture flourished. In the year 2 CE, a census put the population of China at almost 60 million people. Sometime in the first century paper was invented. Education, based on Confucianism, became the route to advancement in the bureaucracy, via a new civil service examination system. This system was maintained, with brief disruptions, into the 20th century.

The period after the Han dynasty (220–598 CE) was a time of political disunity, characterized by a succession of short-lived dynasties and invasions from the north. In fact, much of Chinese history was characterized by invasions from the north; the Great Wall was built and rebuilt in an effort to protect against northern invaders. Although the wall was not always successful, the invaders did not replace the native Chinese culture, instead typically adopting it themselves.

There were several notable developments in this period, which comprised the Three Kingdoms Period (220–265) and the Northern and Southern Dynasties (265–589). One was the invention of paired stirrups in western China, which made the cavalry a more effective fighting force. This was also the time that the first porcelains were made. Although Buddhism was first imported from India in the Han dynasty, it became more important in these centuries. Also in this time, there was a great migration of people from northern China to the south, many as refugees. Thus the southern, rice-growing region of China became more important.

The empire was restored in the short-lived Sui dynasty (589–618). Like the Qin dynasty, the Sui is noted for its huge projects and its unpopularity. In addition to work on the Great Wall, the Sui reportedly conscripted five million peasants to work on the Grand Canal linking the Yangtze valley to northern China. This canal, over 1 0 0 0 miles long and still the longest in the world, was used mainly to transport grain from the south to the north.

Block printing was invented around 600. In this method, a wooden block was carved in such a way that the desired characters stood out. The block was then coated with ink and pressed upon paper or other substance. With this method the same page could be printed repeatedly.

Next up was the T’ang dynasty (618–907), one of the high points in Chinese civilization. Poetry and the fine arts flourished. The capital at Chang’an was the largest city in the world, with almost two million people. The Imperial Academy was founded in 754 to prepare scholars for public service. The curriculum was based on classical Confucian literature. Gunpowder was invented in the 8 th century.

After the T’ang ensued a half-century of disunity, ending with the establishment of the Song dynasty (960–1279), another brilliant period. The Song rule of northern China was somewhat tenuous; a northern invasion forced them to move their capital south to Hangzhou. Nonetheless, the arts, trade, and urban culture flourished. Agricultural production doubled. Tea and cotton cultivation expanded. Neo-Confucianism expanded its influence at the expense of Buddhism.

A couple of important advances in shipping date from the Song dynasty. One is the stempost rudder, which provided increased maneuverability for sailing ships. Another is the magnetic compass, which may actually have been invented earlier but was first widely used in shipping in this time.

Movable type printing also dates from the Song dynasty. In block printing an entire page is carved at one time. The new method involved carving individual characters, which then could be assembled into a page for printing. The characters could be reused.

Starting around 1200 the Mongols under Genghis Khan built one of the greatest empires ever, covering most of northern and western Asia, stretching into Eastern Europe and the Middle East. His empire was larger in area and population than the Roman empire. Kublai Khan, Genghis’s grandson, moved the capital of his empire to what is now Beijing, in 1264. From there, he completed the conquest of southern China, and formed the Yuan dynasty (1279–1368).

Chinese contacts with the West increased during Mongol rule. Moslems, Tibetan Buddhists, Nestorian Christians, and Roman Catholics were all invited to China. The most famous European visitor was the Venetian Marco Polo, who visited Kublai Khan in Beijing 1275–1292 and wrote about this amazing place upon his return to Europe, which was at the time backward compared to China. Chinese mathematics attained its greatest achievements at this time.

The foreign rule of the Yuan was never popular, and it was overthrown by an ex-Buddhist monk named Zhu Yuanzhang, who established the Ming dynasty (1368–1644). The Ming rulers rebuilt the Great Wall and built a new southern capital at Nanjing.

In the early 14th century, there was a remarkable series of naval expeditions launched from China, seven in all, led by the eunuch Zheng He (c. 1371–1433). These preceded by several decades the more famous European voyages of exploration. Zheng He traveled throughout the Indian Ocean, as far as the Persian Gulf, the Red Sea, and eastern Africa. These were not entirely unknown lands to the Chinese; they had traded with them before. So Zheng He’s were not entirely voyages of discovery. Instead, he was interested in advertising the might of China, and collecting treasure and tribute. His fleet contained as many as 300 ships with 28,000 crewmen. Some of the ships were huge, up to 400 feet in length. By comparison, Columbus’ flagship was 85 feet long. In addition to treasure, Zheng He collected exotic animals, such as giraffes, lions, and zebras.

In the end, these voyages were terminated by politics. The conservative Confucian faction won out over the eunuch faction at court. If that had not happened, one can only imagine the different course of history if the first Europeans voyagers had encountered Chinese imperial power in the spice islands.

Later in the Ming dynasty, Chinese farmers began growing new crops from America, such as maize (com), potatoes, and peanuts. There was increasing contact with western culture. A notable figure at this time was the Jesuit missionary Matteo Ricci, who traveled to Macao in southern China in 1582 and to Beijing in 1601. He became a Confucian scholar himself, and he introduced China to many western ideas and inventions. He was valued at court for his knowledge of astronomy. After this time, Chinese mathematics was no longer isolated from Western mathematics.

Early Chinese Mathematics

Chinese mathematicians.

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The Chinese number system was decimal, based on 10. The earliest writings, on oracle bones, had separate symbols for 10, 100, and so on, but by the Han period a place-value system like ours was used, with separate symbols only for 1 ,…, 9. Zero was represented by a space, later by a dot, then finally by a circle in the 1 2 th century.

Calculation was usually carried out on a counting board, using small rods to represent the numerals. There were two sets of counting rod numerals, shown in Figures 2 .1 and 2.2.

Figure 2.1 Vertical counting rods.

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Figure 2.2 Horizontal counting rods.

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These rods were arranged in horizontal rows on the counting board, one row to a number. These numbers could be manipulated to carry out whatever computation was required. To avoid confusion, the horizontal and vertical numbers were alternated within a row, and the rightmost digit also alternated between horizontal and vertical. Some examples are shown in Figure 2.3. Negative numbers could be handled easily: a different color rod was used, black for negative, red for positive.

The earliest mathematical texts we have are from the Han period, at which time Chinese mathematics was already relatively sophisticated. The first work, the Suan shu shu (Book of Numbers and Computation) was only discovered in 1984, in a tomb. Two others of importance are the Zhoubi suanjing (Arithmetical Classic of the Gnomon and the Circular Paths of Heaven), an astronomical text, and the Jiuzhang suanshu (Nine Chapters on the Mathematical Art). The latter, often referred to as simply the Nine Chapters, remained important in China for many centuries.

Figure 2.3 Numbers on a counting board.

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These early books, and many later ones, were primarily collections of problems, along with their solutions and commentary. The problems concerned practical matters such as surveying, engineering, and taxation. They were probably used in the education of bureaucrats.

Liu Hui (lyoo hwimages) (c. 220–280)

Liu Hui worked in the Three Kingdoms period immediately following the end of the Han dynasty. He lived in the northern Kingdom of Wei. Nothing else is known of his life.

Liu edited the standard version of the Nine Chapters, adding his own commentaries on the material and also a tenth chapter, the Sea Island Mathematical Manual. His reputation as one of the greatest mathematicians of ancient China rests on this work.

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Geometry

The Chinese had formulas for the area and volume of a number of geometrical figures. The Nine Chapters gives several methods of computing the area of a circle, all of which use the value of 3 for π. In his commentary, however, Liu Hui noted that this was incorrect. He used a succession of polygons inscribed in a circle to obtain a better approximation: 3.141024. (Compare to what we now know: 3.141592653 …) This method is essentially the same as that used by Archimedes (see Section 1.4 and Figure 1.15 in Section 1.3), although Liu Hui did not provide the rigorous proof that Archimedes did.

The Nine Chapters makes clear that Chinese mathematicians were quite familiar with the Pythagorean Theorem and knew how to apply similar triangles to surveying problems. It also contains a rule for determining the diameter of a sphere from its volume. Here again, Liu notes that this rule was incorrect but he was not able to determine the correct formula.

Zu Chongzhi (429–500) and Zu Geng (c. 480–525)

Zu Chongzhi came from a family of distinguished astronomers and mathematicians. He himself was a prominent astronomer, engineer, and mathematician. He spent most of his life as a court official, working mainly in Jiankang (modem Nanjing). He also wrote ten novels.

Zu Geng was the son and collaborator of Zu Chongzhi. He followed in the family tradition as astronomer, mathematician, and court official (as did Zu Geng’s son).

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Zu Chongzhi improved on Liu Hui’s estimate for 7r, obtaining the inequalities 3.14159127 < π < 3.1415926. This estimate was not bested for more than a thousand years.

Zu Chongzhi is best known for revising the calendar. His revision was based on a new estimate of the length of the year, an estimate that was accurate to 50 seconds. He had trouble implementing the new calendar, however. A court rival accused him of “distorting the truth about heaven and violating the teaching of the classics.” Zu Chongzhi answered that his calendar came

not from spirits or from ghosts, but from careful observations and accurate mathematical calculations. … People must be willing to hear and look at proofs in order to understand truth and facts.

Zu Geng continued his father’s campaign to revise the calendar, and finally succeeded in having it adopted by the emperor, in 510. His main claim to fame, however, was finding the correct formula for the volume of a sphere, thereby completing the task that Liu Hui was unable to accomplish.

Chinese astronomers prepared trigonometric tables in the 8 th century. Apparently, these tables were based upon the work of Indian mathematicians. There was considerable contact between China and India at this time, particularly in the form of Buddhist monks. As we shall see later in this section, Indian astronomers were more advanced, partly based on earlier Greek influence.

Although Chinese geometers gave proofs for some of their results, they never built up the careful logical structure that characterized Greek mathematics.

Algebra

One of the problems in the Nine Chapters is this.

There are three classes of grain, of which three bundles of the first class, two of the second, and one of the third make 39 measures. Two of the first, three of the second, and one of the third make 34 measures. And one of the first, two of the second, and three of the third make 26 measures. How many measures of grain are contained in one bundle of each class?

If we denote the measures of grain in the three bundles by x, y, and z, we get the following three equations.

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(Check!) The problem is to solve for x, y, and z. The text gives instructions for the solution, using a counting board. First, write the coefficients of the three equations in columns, from right to left. Using modem notation, this yields the following arrangement.

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Multiply the middle column by 3,

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and subtract twice the right-hand column from the middle column.

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Note that we now have a 0 at the top of the second column. We next obtain a 0 at the top of the first column: multiply the first column by 3,

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then subtract off the third column.

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Multiply the first column by 5, then subtract off 4 times the second column.

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(Again, check.) This gives us another 0. Now, we write the equations that correspond to these columns, reversing how we originally got the columns.

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Notice that the third equation is easy: z = 99/36 = images. Substituting this into the second equation gives images, which we can solve for images. Finally, substituting our values for y and z into the first equation and solving gives images. (As always, check.)

Do you see the pattern? By appropriately adding or subtracting multiples of the columns, we produced enough 0s so that the resulting set of equations was easy to solve. Chinese mathematicians noticed that this way of combining columns did not change the solutions. In other words, the new set of equations has the same solution as the old one. This method can be generalized, and was in the Nine Chapters, to solve more complicated problems with more unknowns and equations. This algorithm is essentially the same as what is now known as Gaussian elimination, independently invented by Carl Friedrich Gauss in 1800.

As early as the Nine Chapters, Chinese mathematicians also had methods for solving some quadratic and cubic equations. Recall that a general quadratic equation is of the form ax2 + bx + c = 0 for some numbers a, b, and c. Similarly, a cubic equation is of the form ax3 + bx2 + cx + d = 0. The Chinese had methods for approximating x using counting rods. Their method for solving quadratic equations was not the same as our quadratic equation, however. It used the binomial expansion (x + y)2 = x2 + 2xy + y2.

In the 11th century, Jia Xian generalized this method to approximate roots of higher order equations, i.e., when the highest power of x is more than 2 or 3.

Jia Xian (jyä shimages’an) (c. 1010–1070)

Almost nothing is known of Jia Xian’s life, other than that he was a government official and studied with the mathematician and astronomer Chu Yan. He reportedly wrote at least two books, but they are now lost. His work is known to us primarily through later texts.

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Jia Xian’s method of extracting roots was based on the expansion of the binomial (x + y)n. Here are the expansions for n = 0, 1, 2, 3.

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Consider the coefficients of the terms on the right of these equations. If we write only the coefficients, we get this triangle.

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These are the first few rows of what is now known as Pascal’s triangle. (Blaise Pascal was a 17th century mathematician.) Jia Xian discovered this triangle, and the following easy method to generate it. Except for the Is at the ends of each row, every number is the sum of the two numbers above it. Thus the next row is 1 4 6 4 1, and the corresponding binomial expansion is

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Jia’s method of approximating roots of polynomials was essentially the same as that developed in Europe by William Homer and Paolo Ruffini some 750 years later.

Li Ye (limages yimages) (1192–1279)

Li Ye was bom in Zhending, in Hebei Province in northern China. He passed the civil service examination and worked in the government until it fell to the Mongols. He then moved to the foot of Mt. Fenglong, where he lived in seclusion, and often in poverty. Later in his life, he was invited by Kublai Khan to serve in the Mongol government, which he did only briefly, after which he returned to Mt. Fenglong.

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Li Ye published several works, of which the most famous is Ceyuan haijing (Sea Mirror of Circle Measurements), which he wrote in 1248. The Sea Mirror of Circle Measurements contains 170 problems based on one geometric diagram of a circular city wall circumscribed by a right-angled triangle. Although the problems are of geometric origin, each of them led to a quadratic equation which Li would then solve, both algebraically and geometrically.

Yang Hui (yäng hwimages) (c. 1238–1298)

Yang Hui was from southern China. Almost nothing else is known about him, although it is likely that he was a minor civil servant.

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Yang Hui wrote two major works which still survive: Xiangjie jiuzhang suanfa (A Detailed Analysis of the Mathematical Methods in theNine Chapters’) from 1261, and Yang Hui suanfa (Yang Hui’s Methods of Computation) from 1274–75. The former work is our main source of the algebra of Jia Xian on solving equations, mentioned earlier.

Yang Huis Methods of Computation is a collection of seven volumes. It contains work on a variety of mathematical topics, including the solution of quadratic equations. His writing is notable for its careful exposition of methods. It also provides examples of magic squares (see the exercises).

Zhu Shijie images (c. 1260–1320)

Zhu Shijie was bom near modern-day Beijing. After extensive travels as an itinerant mathematics teacher, he settled down in modern-day Yangzhou, where he attracted students “like clouds from the four quarters.”

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Zhu Shijie wrote two major works: Suanxue qimeng (Introduction to Mathematical Science) and Siyuan yujian (Precious Mirror of Four Elements). The first was an elementary book.

The Precious Mirror is considered the summit of Chinese algebra. It is most notable for its treatment of equations with more than one variable. Zhu adapted earlier methods of solving polynomials to equations with up to four unknowns. He also dealt with series (sums), for example, giving the rule for adding the first n counting numbers,

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and the first n squares,

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Congruences

Some time between 300 and 500 CE, a book called Sunzi suanjing (Mathematical Classic of Master Sun) was written. In it is the following problem.

Suppose we have an unknown number of objects. When counted in threes, 2 are left over, when counted in fives, 3 are left over, and when counted in sevens, 2 are left over. How many objects are there?

A method of solving this problem is given.

Multiply the number of units left over when counting in threes by 70, add to the product of the number of units left over when counting in fives by 21, and then add the product of the number of units left over when counting in sevens by 15. If the answer is 106 or more then subtract multiples of 105.

Let’s use the method. First we multiply 2 by 70, to get 140. To this we add 3 times 21, getting 140 + 63 = 203, then add 2 o 15, giving us 233. This is more than 105, so subtract 105 to get 128. This is still too large, so subtract 105 again to get our answer: 23. It is easy to check that 23 works.

This problem uses what we now call congruences. Let a and b be integers, and m a positive integer. We say that a is congruent to b modulo m if a and b have the same remainder when divided by m. We write this as ab (mod m).

Using this notation, we can write the problem this way. Find a number x that satisfies the following three congruences.

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Congruence problems arise naturally when working with calendars. Say that your calendar has 365 days in the year, 30 days to a month. If we assume that day 1 is the beginning of a year and a month, then when will the 15th of the month fall on the 100th day of the year? In this case, we want a number x such that x ≡ 15 (mod 30) and x ≡ 100 (mod 365).

Qin Jiushao (chin jyoo shou) (c. 1202–1261)

Qin Jiushao is one of the more colorful characters in the history of mathematics. He was bom in southern China late in the Song dynasty, and lived in the time when the Mongols were in the process of conquering China. His father was a government official in the imperial capital, and Qin Jiushao was able to study at the Imperial Astronomical Bureau.

As well as being one of the best Chinese mathematicians ever, Qin was an accomplished poet and expert in music and architecture. He was in the military for a while, and was known for his skill in fencing, archery, and riding. He served in a number of government positions, which he used to make himself rich. He was known for his corruption, apparently including poisoning his opponents. He was also known for his love affairs. According a Chou Mi, a contemporary biographer, in his mansion Qin had a “series of rooms for lodging beautiful female musicians and singers.”

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Qin is best known for his Shushu jiuzhang (Mathematical Treatise in Nine Sections). In it is the first extant explanation of Jia Xian’s method of solving polynomials. It is most famous, however, for Qin’s work on congruences.

Qin addressed this problem: given integers m1, m2, …, mn and remainders r1, r2, …, rn, find a number x such that xri (mod mi) for each i = 1, 2, …, n. An example of this problem is the one at the beginning of this section, with r1 = 2, m1 = 3, r2 = 3, m2 = 5, r3 = 2, m3 = 7.

Such systems of congruences do not always have a solution. For example, consider the system x ≡ 1 (mod 2) and x ≡ 2 (mod 4). If x is a solution of the first congruence, it must be odd. But no odd number can solve the second congruence. Qin gave conditions for when a solution exists, in what became known later as the Chinese Remainder Theorem. He also invented a method that could solve any problem for which there was a solution.

More about congruences can be found in Chapter 5.

The 13th century, with the work of Li Ye, Qin Jiushao, Yang Hui, and Zhu Shijie, was the acme of medieval Chinese mathematics. The only notable event after this time was the invention of the modem form of the Chinese abacus, which appeared sometime before 1400. In an abacus, numbers were represented by beads on strings or rods. The beads in an upper section represent five, in a lower section one. Each column holds one digit. Beads count if they are moved to the divider (see Figure 2.4). The abacus allowed for much faster computations than the counting board, and was widely used in Asia until very recently.

Figure 2.4 An abacus showing 314,159.

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During the Ming dynasty, there were no major mathematical developments. Not only did the field not advance, but some of the earlier mathematics was lost. Then, in 1607, Matteo Ricci and Xu Gaunqi translated the first six books of Euclid’s Elements into Chinese, ushering in a new age of Chinese mathematics.

EXERCISES

2.1 What would the following numbers look like on a counting board? Arrange them as in Figure 2.3.

a) 23

b) 517

c) 890

d) 6004

2.2 Use the method from the Nine Chapters to solve this system of equations for x, y, and z.

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2.3 There are three classes of grain, of which three bundles of the first class, one of the second, and two of the third make 29 measures. Four of the first, one of the second, and one of the third make 28 measures. And two of the first, three of the second, and five of the third make 46 measures. How many measures of grain are contained in one bundle of each class?

2.4 Write out the first six rows of Pascal’s triangle. Use your results to expand (x + y)5 and (x + y)6.

2.5 Use the formulas of Zhu Shijie to compute.

a) 1 + 2 + 3 + … + 100

b) l2 + 22 + 32 + … + 102

2.6 Decide whether each of the following congruences is true or false.

a) 24 = 3 (mod 7)

b) 6 = 10 (mod 17)

c) 242 = 2 (mod 10)

d) 240001 = 3 (mod 2)

2.7 Find a value of x that satisfies the following congruences.

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2.8 Find a value of x that satisfies the following congruences.

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2.9 Suppose we have an unknown number of objects. When counted in threes, 2 are left over, when counted in fives, 2 are left over, and when counted in sevens, 6 are left over. How many objects are there?

a) Write this problem using the ab (mod m) notation.

b) Solve the problem.

2.10 A band director wants his band to march in rows of 7, but finds that, when he lines them up, he has only 5 people in the last row. So, after some careful figuring, he thinks maybe rows of 15 are OK. But when he lines them up in rows of 15, the last row has only one person in it.

a) Write this problem using the ab (mod m) notation.

b) Assuming that there are no more than 100 people in the band, how many people are there?

c) What if there might be more than 100 people?

As early as about 2800 BCE, a type of pattern called a magic square appears in Chinese literature. A magic square of order n is a square array of the numbers 1, 2, …, n2 with the property that all rows, all columns, and the two diagonals have the same sum. Below is a magic square of order 3, called the Lo Shu Square. All rows, columns, and diagonals add up to 15.

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2.11 Find another magic square of order 3.

2.12 Show that there is no magic square of order 2. (There are magic squares for every order greater than 2.)

2.13 Show that every magic square of order 3 has row sum 15. (Hint: the sum of all rows is 1 + 2 + 3 + · · · + 9.)

2.14 Find a magic square of order 4.

2.15 Find a formula for the row sum of a magic square of order n. (Hint: the sum of all rows is 1 + 2 + 3 + · · · + n2.)