Contributions of Ancient Civilization: Chinese

CONTRIBUTIONS OF ANCIENT CIVILIZATION: CHINESE China representing one of the earliest civilizations in the world, has a recorded history of about 3600 years. It possesses rich historical documents and ancient relics. China passed through the stages of primitive society, slave society and feudal society. During middle decades of 19th century, capitalists forces of foreign countries invaded china and China was slowly transformed into a semi-colonial and semi- feudal society. Ancient Chinese Mathematics Relatively little is known about early Chinese mathematics because they wrote on bark and bamboo that were far more perishable than clay tablets or papyrus. Chinese produced some remarkable mathematical feats. Contained in this are a few but nowhere near all the details of development of ancient Chinese Mathematics. 1. The Chinese Remainder Theorem (Ta-Yen) The basic format of problems it was to solve was; N=a(mod b)=c(mod d)=…This meant that find N such that when divided by b gives a remainder of and when divided by d gives a remainder of c. Eg: N=10(mod 12)=0(mod 11)= 0(mod 10) =4(mod 09)= 6(mod 8)= 0(mod 7)= 4(mod 6) The method goes like this: a) Find the least common multiple of the moduli. In our eg, moduli are 12,11, 10, 9, 8, 7, 6 Reduce all moduli to a multiplication of prime numbers or their powers, unless they are prime or a power of a prime alone. 12=2*2*3, 11=11, 10=2*5,9=3*3,8=2*2*2, 7=7,6=2*3 Leave any modulus which a prime or only tho power of a prime as it is. In all the other, moduli cancel out any of these occurrences. If no numbers are left, assign it a 1. The new relatively prime moduli, are called the dingshu and are represented by mi for each i. m1=1, m2=11, m3=5, m4=9,m5=8, m6=7 m7=1 The least common multiple ( yenmu O) is the product of dingshu. In our case yenmu O =11*5*9*8*7=27720 b) Divide the yenmu by each of the dingshu inturn to get yenshu Represent these as Mi for each i. Mi={yenmu O)/ mi Mi=27720/mi, M1=27720/1=27720, M2=27720/ 11=2520, M3=27720/5=5544, M4=3080, M5=3465, M6=3960, M7=27720 c) Subtract from each of the yeshu as many copies of the corresponding dingshu as possible, and take the remainder Represent these as ni for each i N1=27720- 27720*1=0, N2=2520-229*11=1, N3= 5544-1108*5=4, N4=2, N5=1, N6=5, N7=0 d) Solve for xi using Nixi= 1(mod mi) These solutions are called the chenglu. When Ni=0, xi=0 x1=x7=0 x2= 1(mod 11), x2=1 x3= 1(mod 5), x3=4 x4= 1(mod 9), x4=5 x5= 1(mod 8), x5=1 x6= 1(mod 7), x6=3 e) Multiply each xi by the corresponding Mi and ri where ri is the original remainder. These solutions are called tsungshu rMx4= 4x 3080x 5= 61600 rMx5= 6x 3465x 1= 20790 f) Sum these values 61600+20790=82390 g) Reduce the sum by twice the yenmu to get N 82390-2x27720=26950 Therefore, the solution is N=26950 2. Fractions In adding fractions each numerator was multiplied by the denominators of the other fractions and then all the denominators multiplied by one another. The numerators were then added. In division by a fraction both numbers were multiplied by the denominators of the fraction and then rules similar to those of today were used. Subtraction was carried out by multiplying the numerators by the ultimate denominators, subtracting the smaller of the two products from the greater and dividing the remainder by the product of the denominators 3. Permutations and Combinations In relation to the discovery of permutations and combinations is the chess board problem. The invention of chess in its modern from involves work with series, permutations and combinations and probability. One of the best known European medieval problems related to the Chinese chess board is the problem involving the grains. Anothr famous problem is associated with a Thang monk by the name of I-Hsing. Shen Kua the author of Meng-Chhi Pi Than discusses I-hsing’s method. 4. Rule of Three Rule of Three is commonly considered as being of Indian origin. The proper founder is a Chinese man by the name of chiu chang. This rule is simply a proportion involving three knows from which an unknown quantity is found. Eg: ‘Two and one half picules of rice are purchased for three thirds of a tael of silver. How many can be purchased for 9 taels?’ If we alter this question by using the modern use of symbolism and the Rule of three, Solution is (5/2)/(3/7)=x/9 35/6=x/9 x=(35x9)/6 x=(105/2) picules 5. Systems of Linear equations Chiu cahngsuan ahu describes a method of solving systems of linear equations as simple, virtually identical to the method of Gaussian elimination and is presented in matrix form. Eg. ― There are three classes of grain, of which three bundles of first class, two of the second and one of the third makes 39 measures. Two of the first, three of the second and one of the third makes 34 measures. One of the first, two of the second and three of the third makes 26 measures. How many measures of grain are contained in the one bundle of each class? 3x+2y+z=39 2x+3y+z=34 x+2y+3z=26 Arranged as vertically 1 2 3 2 3 2 3 1 1 26 34 39 ‘With the first class on the right column multiply currently the middle column and directly leave out’ ie, multiply the middle column by 3 subtract from it 2 lots of the right column. This reduce middle column to 0. Same operation performed w r t to left column 003 452 811 39 24 39 Same operation performed using middle column and left 003 052 36 1 1 99 24 39 Equivalent to 3x+2y+z=39 5y+z=24 36z=99 Final results are found by back substitutions 6. Magic Squares Discovered by Chinese Emperor Yu the Great in 2000 B.C. Yu presented two charts or diagrams by miraculous animals, during his reign of governing the empire. They were deemed to possess magical properties. Yang Hui was the first to actually study magic squares or vertical and horizontal diagrams as they were called. Some of the magic squares he made are complicated. 7. Negative numbers They were familiar with the laws of signs as of about +1299. They realized the equations could result in negative anwers but considered this absurd and therefore ignored them. Two methods of writing negative numbers: in black to distinguish them from the red positive numbers or drew a diagonal stroke to through the right hand digit figure of the negative number. Chinese Medical Theory The primitive society of China divided into two time periods: Old New Stone Age. During Old age knives were made of stone and were medical properties. During the New stone Age stones were refined into served as instruments of healing. They were named bian stone- which sharp edged stone to treat disease. Stone Age and made for certain fine needles and means use of a a) Yellow Emperor’s Book of Internal Medicine The most ancient book on medicine Transformation from ancient magical healing to a system Basis of Chinese medical Literature b) Influences of Taoism Everything flows and is in a constant state of flux Essence of health is to follow the natural tao Ill health is caused by going against the flow. In the body Yin is active and acts as guardian of yang; yang is active outside but regulates Yin The flow of nature is split into five elements. Yin and Yang is absorbed from food. Moods related to five elements: anger effects the liver, worry affects spleen, fear and shock effects blood vessels. Acupuncture and Moxibustion Theory: Acupuncture is a very ancient form of healing which predates recorded history Thin needles, usually made of stainless steel. Relieve pain and improve health. Restores balance. Disease and pain all caused by imbalance. a) Human Body: Twelve physical systems (12 paired and 2 unpaired). Vital life force called chi flows through each meridian. 361 regular acupuncture points 36 extraordinary points. b) Moxibustion: Originated in pre historic China. Specific sites are simulated by heat instead of inserting needles. Small cones of dried leaves of wormwood are burned. Ashes were rubbed into the blisters that formed. Paper and Printing Technology Paper is made up by tearing up rags or the bark of trees or certain grasses and putting them into a large pot with water. The mixture is boiled and the pulp like mass that lies at the bottom is removed, strained and spread out to dry.The making of paper was improved with sizing, dyes and use of moulds made out of bamboo stripsThe invention of printing and improvements in papermaking led to the printing of a whole set of Buddhist sutras by 868. Gun Powder The first known recipe for saltpeter, the principal ingredient of gunpowder can be found in a Chinese military manual written by Wu Ching Tsung Yao. Chinese used gunpowder in fireworks for religious purpose, also used gunpowder in cannons, rockets, and guns The Compass The ability to magnetize iron by placing it near a loadstone was known to ancient civilizations. Chinese applied this theory of this principle of magnetism to create the compass. The early compasses were used in divination rather than in navigation.