Ancient Chinese Mathematics (a very short summary)

• Documented civilization in China from c.3000 BCE. Un-til c.200 CE, ‘China’ refers to roughly to the area shownin the map: north of the Yangtze and around the Yellowrivers.

• Earliest known method of enumeration dates from theShang dynasty (c.1600–1046 BCE) commensurate withthe earliest known oracle bone script for Chinese char-acters. Most information on the Shang dynsaty comesfrom commentaries by later scholars, though many orig-inal oracle bones have been excavated, particularly fromAnyang, its capital.

• During the Zhou dynasty (c.1046–256 BCE) and especially towards the end in the Warring Statesperiod (c.475–221 BCE) a number of mathematical texts were written. Most have been lost butmuch content can be ascertained from later commentaries.

• The warring states period is an excellent example of the idea that wars lead to progress. Rapidchange created pressure for new systems of thought and technology. Feudal lords employeditinerant philosophers1) The use of iron in China expanded enormously, leading to major changesin warfare2 and commerce, both of which created an increased need for mathematics. Indeedthe astronomy, the calendar and trade were the dominant drivers of Chinese mathematics formany centuries.

• The warring states period ended in 221 BCE with the victory of victory of the Qin Emperor ShiHuang Di: famous for commanding that books be burned, rebuilding the great walls and forbeing buried with the Terracotta Army in Xi’an. China was subsequently ruled by a successionof dynasties until the abolition of the monarchy in 1912. While this simple description mightsuggest a long calm in which Chinese culture and technology could develop in comfort, inreality the empire experienced its fair share of rebellions, schisms and reprisals. The empire ofthe Qin dynasty was tiny in comparison to the extent of modern China, and its expansion didnot happen smoothly or without reversals.

• East Asia (modern day China, Korea, Japan, etc.) is geographically isolated from the rest of theworld and, in particular, from other areas of developing civilization. To the north is Siberia, tothe west the Gobi desert, southwest are the Himalaya and to the south is jungle. During theHan dynasty (c.200 BCE–220 CE) a network of trading routes known as the silk road connectedChina, India and, through Persia, Europe. Indeed part of the purpose of the Great Wall wasthe protection of these trade routes. Knowledge moved more slowly than goods, and there isvery little evidence of mathematical and philosophical ideas making the journey until many

1Confucius was one such and served for a period as an advisor to Lu, a vassal state of the Zhou. He died at the beginningof the warring states period, around 479 BCE, but his followers turned his teachings into a major philosophical school ofthought Confucianism. Its primary emphasis was stability and unity as a counter to turmoil. The other major philosophicalsystem dating from this time is Taoism which is more comfortable with change and adaptation.

2Sun Tzu’s military classic The Art of War dates from this time.

centuries later. For instance, there is no evidence of the Chinese using sexagesimal notationto aid their calculations, which meant that essentially none of Babylonian and Greek work onastronomy made it to China. Similarly, there are many mathematical ideas which saw no ana-logue in the west until many centuries after Chinese mathematicians had invented them. Itseems reasonable to conclude that Chinese and Mediterranean mathematics developed essen-tially independently.

Important Mathematical Texts

The oldest suspected3 mathematical text is the Zhou Bi Suan Jing (The Mathematical Classic of theZhou Gnomon4 and the Circular Paths of Heaven). It was probably compiled some time in the period500–200 BCE. The text contains, arguably, the earliest statement of Pythagoras’ Theorem as well assimple rules for computing fractions and conducting arithmetic. The book was largely concernedwith astronomical calculations and presented in the form of a dialogue between the 11th century BCEDuke of Zhou5 and Shang Gao (one of his ministers, and a skilled mathematician).

As with other early texts, there is no rigorous notionof proof or axiomatics, merely strong assertions or ‘ob-vious’ appeals to pictures. For instance, the ‘proof’of Pythagoras’ Theorem is purely pictorial: taking thesides of the triangle to be a, b, c in increasing order oflength, the picture essentially claims that

c2 = 4 · 12

ab + (b− a)2

giving the familiar result when multiplied out.

Of course the Chinese do not attribute this result to Pythagoras! Instead it is known as the gou guwhere these words refer first to the shorter and then the longer of the two non-hypotenuse sides ofthe triangle.

There are several other early works; here are two of the most important.

Suanshu Shu (A Book on Arithmetic) Compiled c.300–150 BCE, covering, amongst other topics, frac-tions, the areas of rectangular fields, and the computation of fair taxes.

Jiu Zhang Suan Shu (Nine Chapters on Mathematical Arts) Written c.300 BCE–200 CE. Many top-ics are covered, including square roots, working with ratios (false position and the rule of three6),simultaneous equations, areas/volumes, right-angled triangles, etc. The Nine Chapters is a hugelyinfluential text, in no small part due to the creation of a detailed commentary and solution manual toits 246 problems written by the mathematician Liu Hui in 263 CE.

3Ancient Chinese texts are extremely difficult to date: we have no original copies and most seem to have been compiledover several hundred years. Most of what we know of these texts is in the form of commentaries by later authors.

4Gnomon: “One that knows or examines” — strictly the elevated piece of a sun/moondial which would have been usedfor measuring said circular paths of heaven.

5Credited with writing the I Ching, the ‘classic of changes.’6Given equal ratios a : b = c : d, where a, b, c known, then d = bc

a .

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The style of both text consists of laying out methods of solution which have wide application, ratherthan on proving that a particular method is guaranteed to work. Indeed there is no notion of ax-iomatics on which one could construct a proof in the modern sense of the word.

Liu made other contributions to mathematics, including accurateestimates of π made similarly to Archimedes. He made particu-lar use of the out-in principle which essentially describes how tocompare areas and volumes:

1. Areas and volumes are invariant under translation.

2. If a figure is subdivided, the sum of the areas/volumes ofthe parts equals that of the whole.

For instance, Liu gave the argument shown in the picture as analternative proof of the gao gu: the green square is subdividedand the in pieces Ai, Bi, Ci translated to new positions Ao, Bo, Coto assemble the required squares.

Ao

Ai

Bo

Bi

Co

Ci

Liu extended the out-in principle to analyze solids, comparing the volumes of four basic solids:

• Cube (lifang)

• Right triangluar prism (qiandu)

• Rectangular pyramid (yangma)

• Tetrahedron (bienuan)

These could be assembled to calculate the volume of, say, a truncated pyramid:

The Bamboo Problem This problem is taken from the Nine Chapters:indeed the picture shows the problem as depicted in Yang Hui’sfamous Analysis of the Nine Chapters from 1261.

A bamboo is 10 chi high. It breaks and the top touches the ground 3chi from the base of the stem. What is the height of the break?

In modern language, if a, b, c is the triangle, we know that b + c = 10and a = 3. We want b.

The solution is b =12

(b + c− a2

b + c

)=

91100

chi: think about why. . .

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Chinese Enumeration

The Chinese had two parallel systems of enumeration. Both are essentially decimal.

Oracle Bone Script and Modern Numerals The earliest Chinese writing is known as oracle bonescript dates from around 1600 BCE. The numbers 1–10 were recorded with distinct symbols, withextra symbols for 20, 100, 1000 and 10000. These were decorated to denotes various multiples. Someexamples are shown below.

The system is quite complex, given all the possibilities for decoration, and therefore more advancedthan other contemporary systems. Standard modern numerals are a direct descendant of this script:

Observe the similarity between the expressions for the first 10 digits. The second image denotesthe number 842, where second and 4th symbols represent 100’s and 10’s respectively: literally eighthundred four ten two. No zero symbol is required as a separator: one could not confuse 205 with 250.The system is still partly positional: the symbol for, say, 8 can mean 800 if placed correctly, but onlyif followed by the symbol for 100.

Rod Numerals The second dominant form of enumeration dates from around 300 BCE and wasin very wide use by 300 CE. Numbers were denoted by patterns known as Zongs and Hengs. Theserepresented alternate powers of 10: Zongs denoted units, 100’s, 10000’s, etc., while Hengs were for10’s, 1000’s, 100000’s, etc.

Rod numerals were immensely practical. In extremis they could easily be scratched in the dirt. Morecommonly they were created using short bamboo sticks or counting rods, of which any merchantworth their salt would carry a bundle. They were often used in conjunction with a counting board:a grid of squares on which sticks could be placed for ease of calculation. This technology was in-valuable for keeping calculations accurate and facilitating easy trade. They also made for severalcalculation methods which will seem familiar. There was no need for a zero in this system, as anempty space did the job.

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Addition and subtraction are straightforward by carrying and borrowing in the usual way. Thesmallest number was typically placed on the right. Multiplication is a little more fun: for instance tomultiply 387 by 147, one would set up the counting board as follows:

3 8 7

1 4 7Arrange rods: use modern numerals for clarity

3 8 74 4 11 4 7

441 = 147× 3: note the position of 147 under the 3

8 74 4 11 1 7 6

1 4 7

Delete 3, move 147 and multiply: 1176 = 147× 8

8 75 5 8 6

1 4 7Sum rows

75 5 8 6

1 0 2 91 4 7

Delete 8, move 147 and multiply: 1029 = 147× 7

5 6 8 8 91 4 7

Sum rows: 387× 147 = 56889

The algorithm is essentially long-multiplication, but starting with multiplication by the largest digitinstead of the units as we are used to.

Division essentially works like long-division: to divide 56889 by 147 one might have the followingsequence of boards

5 6 8 8 91 4 7

−→3

5 6 8 8 91 4 7

−→3 8

1 2 7 8 91 4 7

−→3 8 7

1 0 2 91 4 7

In the first two boards. 147 goes 3 times into 568.In board 3, we subtract 3× 147 from 568 to leave 127, shift the 147 over and observe that 147 goes 8times into 1278. In the final step we have subtracted 8× 147 from 1278 to leave 1029 before shiftingthe 147 to its final position. Since 147 divides exactly seven times into 1029, we are done.There is nothing stopping us from dividing numbers where the result is not an integer: simply con-tinue as in long-division.

Similtaneous equations The counting board could be set up to compute solutions to simultaneouslinear equations. Essentially the coefficients of a linear system were placed in adjacent columns and

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then column operations were performed. The method is thus identical to what you learn in a linearalgebra class, but with columns rather than rows. For instance, a linear system could be encodedthus: {

3x + 2y = 72x + y = 4

−→3 22 17 4

−→1 21 13 4

−→1 11 03 1

−→0 11 02 1

−→ x = 1, y = 2

This matrix method was essentially unique to China until the 1800’s.

Euclidean algorithm The counting board lent itself to the computation of greatest common divi-sors, which were used very practically for simplifying fractions. Here is the process applied to 35

91 :,

35 35 35 14 14 791 56 21 21 7 7

At each stage, one subtracts the smaller number from the larger. Once the same number is in eachrow you stop. You should recognize the division algorithm at work. . . Since gcd(35, 91) = 7, bothcould be divided by 7 to obtain 35

91 = 513 in lowest terms.

Negative numbers There is a strong case for arguing that the Chinese are also the oldest adoptersof negative numbers. These were not thought of as numbers per se, rather different colored rodscould be used to denote a deficiency in a quantity. Indeed the Nine Chapters describes using redrods for positives and black for negatives. This would commonly be used when adding up accounts.This practice was known by around 1 CE, roughly 500 years before negative numbers were used incalculations in India.

Music, Mysticism and Approximations Like the Pythagoreans, the Chinese were interested in mu-sic and pattern for mystical reasons. While the Pythagoreans delighted in the pentagram, the Chinesecreated magic squares7 as symbols of perfection. The notion of equal temprament in musical tuningwas first ‘solved’ in China by Zhu Zaiyu (1536–1611), some 30 years before Mersenne & Stevin pub-lished the same result in Europe. This required the computation of the twelfth-root of 2 which Zhucomputed using approximations for square and cube roots:

12√

2 =3

√√√2

Zhu’s approximation was correct to 24 decimal places!Indeed the Chinese emphasis on practical methods meant that they often had the most accurateapproximations for their time:

• Approximations to π including 227 ,√

10, 355113 , 377

120 . Most accurate in the world from 400–1400 CE.

• Methods for approximating square and cube roots were found earlier than in Europe. Approxi-mations for solutions to higher-order equations similar to the Horner–Ruffini method were alsodiscovered earlier.

• Pascal’s triangle first appears in China around 1100 CE. It later appeared in Islamic mathematicsbefore making its way to Europe.

7Grids where all rows and columns sum to the same total.

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Two famous problems

We finish with a discussion of two famous Chinese problems. The first is known as the Hundred FowlsProblem and dates from the 5th century CE. It was copied later in India and then by Leonardo da Pisa(Fibonacci) in Europe, thus showing how Chinese mathematics travelled westwards.

If cockerals cost 5 qians8 each, hens cost 3 qians each, and 3 chickens cost 1 qian, and if 100fowls are bought for 100 qians, how many cockerels, hens and chickens are there?In essence, the problem asks us to find non-negative integers satisfying{

5x + 3y + 13 z = 100

x + y + z = 100

The answers are simply stated as (4, 18, 78), (8, 11, 81), (12, 4, 84) while the possible solution (0, 25, 75)was not stated.9

Finally an example of the Chinese Remainder Theorem for solving simultaneous congruence equations.The Theorem dates from the 4th century CE: it travelled to India where it was described by Bhra-magupta and thence to Europe. This example comes from Qin Jiushao’s Shu Shu Jiu Zhang (NineSections of Mathematics) in 1247.

Three thieves entered a rice shop and stole three identical vessels filled to the brim withrice, but whose exact capacity was not known. The thieves were caught and their vesselsexamined: all that was left in vessels X, Y and Z were 1 ge, 14 ge and 1 ge respectively. Thethieves did not know the exact quantities they’d stolen. A used a “horse ladle” (capacity19 ge) to take rice from vessel X. B used a wooden shoe (capacity 17 ge) to take rice fromvessel Y. C used a bowl (capacity 12 ge) to take rice from Z. What was the total amount ofrice stolen?

In modern language the total amount of rice N satisfies

N ≡

1 (mod 19)14 (mod 17)1 (mod 12)

The answer is that N = 3193 ge. This is the smallest possible solution: all such are congruent modulo19 · 17 · 12 = 3876.

8A qian is a copper coin.9As a possible method, one substitutes z = 100− x− y in first equation to obtain 7x + 4y = 100, or

y = 25− 74

x

Since y ∈ Z, we must have x = 4m for some integer m. Thus all solutions have the form

x = 4m, y = 25− 7m, z = 75 + 3m, m ∈ Z

For m = 1, 2, 3 and 0 we obtain the above solutions. For m ≥ 4 we have y < 0, while m < 0 yields x < 0.

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