# Chinese Mathematics

Here we work on worksheet #10, Guassian Elimination.

After the Han Dynasty there is a period of disorder until the Sui Dynasty from

598 to 618 A.D. But there are three very brilliant mathematicians who are active
despite

the unrest. They are Liu Hui who was active in about 260 A.D., Sun Zi from the
late

200’s to the 300’s A.D. and Zhu 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 π. He gets π = 3.141024 (We

have 3.14159265. ) He gets close to the idea of a limit in his work on the
number π. 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 for the volume 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 number theory 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

π to 3.1415926 which is the same as the current 7-place approximation. He also
proved

that the volume of a sphere of radius r is
, using the idea of infinitesimally

thin slices which was a method attributed to Cavalieri in Europe in the 1600’s
A.D. This

is a start of the integral calculus.

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 Li Po (701-762) wrote

beautiful verse in the tradition that extends back at least to the later Zhou
Dynasty.

The Song Dynasty 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 Jia Xian 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

digit in the answer, say 2. Then
(x + 2)^{3} - 26 will have the same root as the original

equation, except reduced by 2.
(x - 2 + 2)^{3} - 26 = x^{3} - 26. But he expanded

(x + 2)^{3} - 26 as
x^{3} + 3 (x^{2} · 2) + 3 (x · (2)^{2} ) + 2^{3}
- 26 and got familiar with the

binomial coefficients for
(x + h)^{3} 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 Song in the north. The were led by Temujin, or Ghengis Khan.

Temujin’s grandson, Kublai Khan 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 Yuan Dynasty. Mathematics reached its

highest level of development during this time.

Four master mathematicians were active during this thirteenth century. The first

of these was Qin Jiushao (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:

(Here, bring down the 5, 2x5 =10, add, 2x17=34, add, etc.) The answer is 51 |

The same tabular calculation can be used to divide
5x^{3} + 7x^{2} - 3x -11 by

(x – 2). You have to read the answer as

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 work 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 Li
Ye. 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 Hebei.

The third outstanding mathematical talent during the thirteenth century in China

was Yang Hui. 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 magic

square 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 7The 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 numbers 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.

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 scrolling, 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.

**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 Zhu Shijie 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.