mathematics of china - brigham young...

Mathematics of China

Early Timeline

• Shang Dynasty: Excavations near Huang River, dating to 1600 BC, showed “oracle bones” – tortoise shells with inscriptions used for divination. This is the source of what we know about early Chinese number systems.

Early Timeline

Early Timeline

• Zhou Dynasty: The Shang Dynasty gave way to the Zhou Dynasty around 900‐1000 BC, which in turn dissolved into numerous Warring States. Around 500 BC, there was a great intellectual flowering, caused in part by the development of iron. The age of Confucius and other wise men who were advisors to various feudal lords.

Qin Dynasty (221 – 206 BC)

• Emperor Qin Shi Huangdi united China, imposed taxes, standardized weights, measures, and money, instituted a severe legal code, built the Great Wall, and (according to legend) demanded all the books be burned to avoid subversion. We don’t know if that was carried out. In any event, it’s likely some survived, or at least were copied from memory soon afterward.

Qin Dynasty

Han Dynasty ( 206 BC –220 AD)

• System of Education especially for civil servants, i.e. scribes.

• Two important books:• Zhou Bi Suan Jing (Arithmetical Classic of the Gnomon and the Circular Paths of Heaven)

• Jiu Zhang Suan Shu (Nine Chapters on the Mathematical Art)

Nine Chapters

• This second book, Nine Chapters, became central to mathematical work in China for centuries. It is by far the most important mathematical work of ancient China. Later scholars wrote commentaries on it in the same way that commentaries were written on The Elements. We’ll look at it in greater detail later.

Han Dynasty ( 206 BC –220 AD)

• Other works:– Ta Tai Li Chi (magic square)

– Commentary on Nine Chapters (– Shu Shu Chi Yi (Manual on the Traditions of the Mathematical Arts) (magic squares, first mention of abacus)

• Paper was invented.

Continuing On….

• Three Kingdoms (220 ‐ 280)• Jin Dynasty (280 ‐ 420)• Northern and Southern Dynasties (420 – 581)

• Liu Hui (Commentary on Jiu Zhang)• Sun Zu Suan Jing (Master Sun’s Mathematical Manual) ‐ Square roots, indeterminant analysis

• Zu Chongzhi (π = 355/113 ≈ 3.1415929)

Continuing on….

• Sui Dynasty (581 – 618): Reunification of China.

• Tang Dynasty (618 – 907) – Great period of openness to cultural influences. Cosmopolitan. Little mathematical work discovered from this period, however.

• Five Dynasties, Ten Independent States (907 –960) – Block Printing was developed.

Song Dynasty (900 – 1279)

• Long and productive.• Shu Shu Jiu Zhang (Nine Sections of Mathematics) – numerical solutions of equations of all degrees.

• Ce Yuan Hai Jing (The Sea Mirror of the Circle Measurements) – construction of equations from data.

Song Dynasty (900 – 1279)

• Two Books by Zhu Shijie had topics such as:– Pascal’s triangle (350 years before Pascal)– Solution of simultaneous equations using matrix methods

– “Celestial element method” of solving equations of higher degree.

• European algebra wouldn’t catch up to this level until the 1700’s.

Finally…..

• Yuan Dynasty (1280‐1367), founded by Kublai Khan, grandson of Genghis. European contacts were prevalent, but not much math (Marco Polo).

• Ming Dynasty (1368 – 1644). Restoration of indigenous culture; outside influences, like Euclid’s Elements, were not embraced.

• Qin Dynasty (1644 – 1911). Western mathematics begins to have its profound influence.

Timeline

• Like many ancient cultures, ebb and flow were typical, as was an emphasis on the utility of mathematics. But there was clearly some mystical and intellectual aspects to it as well.

Numeration

• Numerals on the Oracle Stones:

Numeration

Numeration

Hindu ‐Arabic

0 1 2 3 4 5 6 7 8 9 10 100 1000

Chinese 〇一二三四五六七八九十百千

Financial 零壹贰叁肆伍陆柒捌玖拾佰仟

Counting Rod System

Counting Rods

• Counting rods allowed for a number of very quick calculations, including the basic four arithmetic operations, and extraction of roots.

• The diagrams in the next few slides are taken from “Rod Calculus” in Wikipedia; you can check out the dynamic gif versions there.

Addition

Subtraction

OK, Subtraction with Borrowing

Subtraction

Multiplication

Division

Roots: Look it up

• But it is very similar to the algorithm used for years before the advent of easily available tables or calculators.

Fractions

• From Nine Chapters:• “If the denominator and numerator can be halved, halve them. If not, lay down the denominator and numerator, subtract the smaller number from the greater. Repeat the process to obtain the greatest common divisor (teng). Simplify the original fraction by dividing both numbers by the teng.

Fractions

• Addition and subtraction were done as we do them but without finding least common denominators – the common denominator is just the product of the two denominators. The fraction is simplified after adding or subtracting.

Fractions

• Multiplication was done as we do it.• Division was done by first getting common denominators, then inverting and multiplying so that the common denominators cancel. Then the fraction was simplified.

Linear Equations

• 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?

Linear Equations

• Solution: “Arrange the 3, 2, and 1 bundles of the 3 classes and the 39 measures of their grains at the right. Arrange other conditions at the middle and the left:”

1 2 32 3 23 1 126 34 39

Linear Equations

• “With the first class on the right multiply currently the middle column and directly leave out.” (That is, multiply the middle column by 3, and then subtract some multiple of the right column, to get 0).

1 0 32 5 23 1 126 24 39

Linear Equations

• Do the same with the left column:

0 0 34 5 28 1 139 24 39

Linear Equations

• “Then with what remains of the second class in the middle column, directly leave out.” In other words, repeat the procedured with the middle column and left column:

0 0 34 5 28 1 139 24 39

0 0 30 5 236 1 199 24 39

Linear Equations

• This was equivalent to a downward Gaussian reduction. The author then described how to “back substitute” to get the correct answer.

Negative numbers?

• Red and black rods, or rods laid diagonally over others.

• “For subtractions – with the same signs, take away one from the other; with different signs, add one to the other; positive taken from nothing makes negative, negative from nothing makes positive.”

• “For addition – with different signs subtract one from the other; with the same signs add one to the other; positive and nothing makes positive; negative and nothing makes negative.”

Method of Double False Position

• Or, “Excess and Deficit.”• A tub of capacity 10 dou contains a certain quantity of husked rice. Grains (unhuskedrice) are added to fill up the tub. When the grains are husked, it is found that the tub contains 7 dou of husked rice altogether. Find the original amount of husked rice. Assume 1 dou of unhusked rice yields 6 sheng of husked rice, with 1 dou = 10 sheng.

Our Method, Maybe

Let x be amount of husked rice, y be amount of unhusked rice. Then and

. So , and substituting

we have . Simplifying, we

get , and , or 2 dou, 5 sheng.

Method of Double False Position

• If the original amount is 2 dou, a shortage of 2 sheng occurs. If the original amount if 3 dou, there is an excess of 2 sheng. Cross multiply 2 dou by the surplus 2 sheng, and then 3 dou by the deficiency of 2 sheng, and add the two products to give 10 dou. Divide this sum by the sum of the surplus and deficiency to obtain the answer 2 dou and 5 sheng.

Double False Position

• ∙ ∙ Why does this work?

• We want to solve In general, we’ll examine a method for solving .


• So suppose we want to solve . We’ll do it by making two guesses and , with the respective errors

, and .

Then subtracting these equations gives . Next, multiplying

equation 1 by and equation 2 by we get:


, and .

Subtracting these equations gives:. Finally, dividing this

equation by gives us:

. Finally, if is a surplus and is

a deficit, we can say .

Magic Squares

Lo Shu

• The semi‐mythical Emperor Yu, (circa 2197 BC) walking along the banks of the Luo River, looked down to see the Divine Turtle. On the back of his shell was a strange design.

Lo Shu

• When the design on the back was translated into numbers, it gave the 3x3 magic square.

• Saying “the” 3x3 magic square is appropriate because it is unique up to rotations and reflections.

He Tu

• According to legend, the He Tu is said to have appeared to Emperor Yu on the back of (or from the hoof‐prints of) a Dragon‐Horse springing out of the Huang (Yellow) River.

He Tu

• When it was translated into numbers, it gave a cross‐shaped array.

• To understand its meaning is to understand the structure of the universe, apparently.

• Or, at least to understand that, disregarding the central 5, the odds and evens both add to 20.

72

8 3 5 4 916

Magic Squares

• Yang Hui, “Continuation of Ancient Mathematical Methods for Elucidating the Strange Properties of Numbers”, 1275.

Order 3

• Arrange 1‐9 in three rows slanting downward to the right.

14 2

7 5 38 6

9

Order 3


• Exchange the head (1) and the shoe (9).

94 2

7 5 38 6

1

Order 3



• Exchange the 7 and 3.

94 2

3 5 78 6

1

Order 3



• Exchange the 7 and 3.• Lower 9, and raise 1.

4 9 23 5 7

8 1 6

Order 3



• Exchange the 7 and 3.• Lower 9, and raise 1.• Skootch* in the 3 and 7*technical term

4 9 23 5 78 1 6

Order 3 – The Lo Shu

Order 4

• Write 1 – 16 in four rows.

1 2 3 45 6 7 89 10 11 1213 14 15 16

Order 4


• Exchange corners of outer square

16 2 3 135 6 7 89 10 11 124 14 15 1

Order 4



• Exchange the corners of inner square.

16 2 3 135 11 10 89 7 6 124 14 15 1

Order 4



• Exchange the corners of inner square.

• Voila! Sum is 34.

16 2 3 135 11 10 89 7 6 124 14 15 1

Order 4

• Other magic squares of order 4 are possible for different initial arrangements of the numbers 1 – 16.

13 9 5 114 10 6 215 11 7 316 12 8 4

4 9 5 1614 7 11 215 6 10 31 12 8 13

Order 5, 6, 7, ….

• Yang Hui constructed magic squares of orders up through 10, although some were incomplete.

A Little About Magic Squares

• Normalmagic squares of order n are n x n arrays containing each number from 1 through They exist for all .

• The sum of each row, column, and diagonal is the magic number M which for normal magic squares depends only on n.

• . For the first few n’s this is 15, 34, 65. 111, 175 . . .

• For n odd, the number in the central cell is

Approximations of π

• Liu Hui, 260 AD: 3.1416 (by inscribing hexagon in circle, using the Pythagorean Theorem to approximate successively polygons of sides 12, 24, ….,96).

• Zu Chongzhi, 480 AD: between 3.1415926 and 3.1415927 (by similar method, but moving past 96 to oh, say 24,576).

Chapters in … uh, the Nine Chapters

1. Field measurements, areas, fractions2. Percentages and proportions3. Distributions and proportions; arithmetic and

geometric progressions4. ;Land Measure; square and cube roots5. Volumes of shapes useful for builders.6. Fair distribution (taxes, grain, conscripts)7. Excess and deficit problems8. Matrix solutions9. Gou Gu – Gou ^2 + Gu ^2 = Xian ^2. Astronomy,

surveying

Gou Gu in Zhou Bi

Liu Hui’s Proof of Gou Gu

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