The Earliest Applications Of Linear Algebra
Download as PPTX, PDF2 likes2,207 views
The document discusses the earliest applications of linear algebra in ancient civilizations like Greece, Egypt, China, and India, tracing its origins back to problems involving measurement, distribution of goods, and taxation that were solved using systems of linear equations. It presents Archimedes' famous "cattle problem" as an example tackled by Greek mathematicians and provides examples of early linear systems from Egyptian, Chinese, and Indian texts, demonstrating that the concepts underlying linear algebra have existed for millennia.
The Earliest Applications Of Linear Algebra
- 2. The Earliest Applications of Linear Algebra PREREQUISITES:- LINEAR SYSTEMS
- 3. Group Members:- Anas Ahmed Abdul Saboor Junaid Mustafa Tooba Anwar Umer 15B-211-EL Topic:- Greece 15B-199-EL Topic:- Egypt 15B-188-EL & 15B-196-EL Combined Topic:- China 15B-101-EL Topic:- India
- 4. HISTORY OF LINEAR ALGEBRA Emerged from the study of determinants, Leibniz Gabriel Cramer: Cramer’s Rule Gauss: Gaussian elimination They All were used to solve linear systems
- 5. Ancient Civilization & Linear Algebra “Linear systems can be found in the earliest writings of many ancient civilizations”. Practical problems of early civilizations included the measurement of land, distribution of goods, tracks of resources, and taxation. It Pre-dates to the Islamic mathematicians who created the field of algebra, which eventually led to the branch of Linear Algebra.
- 6. Greece Period (3rd century)
- 7. Greece & linear algebra 3rd century B.C Most famous problem: Archimedes’ Cattle problem It is a challenge given by Archimedes for Eratosthenes No solution from ancient times, not known how the Greeks solved it
- 8. Cattle Problem If thou art diligent and wise, O stranger, compute the number of cattle of the Sun, who once upon a time grazed on the fields of the Thrinacian isle of Sicily, divided into four herds of different colours, one milk white, another a glossy black, a third yellow and the last dappled. In each herd were bulls, mighty in number according to these proportions: Understand, stranger, that the white bulls were equal to a half and a third of the black together with the whole of the yellow, while the black were equal to the fourth part of the dappled and a fifth, together with, once more, the whole of the yellow. Observe further that the remaining bulls, the dappled, were equal to a sixth part of the white and a seventh, together with all of the yellow. These were the proportions of the cows: The white were precisely equal to the third part and a fourth of the whole herd of the black; while the black were equal to the fourth part once more of the dappled and with it a fifth part, when all, including the bulls, went to pasture together. Now the dappled in four parts were equal in number to a fifth part and a sixth of the yellow herd. Finally the yellow were in number equal to a sixth part and a seventh of the white herd. If thou canst accurately tell, O stranger, the number of cattle of the Sun, giving separately the number of well-fed bulls and again the number of females according to each colour, thou wouldst not be called unskilled or ignorant of numbers, but not yet shalt thou be numbered among the wise
- 9. Variables Used In Cattle Problem W=number of white bulls B=number of black bulls Y=number of yellow bulls D=number of dappled bulls W’= number of white cows B’=number of black cows Y’=number of yellow cows D’=number of dappled cows
- 10. Equations 1. W=(1/2+1/3)B+Y 2. B=(1/4+1/2)D+Y 3. D=(1/6+1/7)w+y 4. W’=(1/3+1/4)(b+b’) 5. B’=(1/4+1/5)(d+d’) 6. D’=(1/5+1/6)(y+y’) 7. Y’=(1/6+1/7)(w+w’)
- 11. After Some Calculations W=10,366,482k B=7,460,514k. This is the possible outcomes of Bulls and cows Y=4,149,387k D=7,358,060k w=7,206,360k b=4,893,246k y=5439213k d=3,515,820k
- 12. Egypt
- 13. EGYPT The earliest true mathematical documents of Egypt date to 1990–1800 BC. The Egyptian Mathematical Leather Roll, the Lahun Mathematical Papyri which are a part of the much larger collection of Kahun Papyri, all date to this period. The Rhind Mathematical Papyrus which dates to the Second Intermediate Period (ca 1650 BC) is said to be based on an older mathematical text from the 12th dynasty (1990-1800 B.C.)
- 14. Ancient Egyptian Mathematics Egyptian Numerals:-
- 16. The Ahmes (Rhind) Papyrus The Ahmes (or Rhind) Papyrus is the source of most of our information about ancient Egyptian mathematics.
- 17. The Ahmes (Rhind) Papyrus This 5-meter-long papyrus contains 84 short mathematical problems, together with their solutions, and dates from about 1650 B.C.
- 18. PROBLEM 40 IN THE AHMES PAPYRUS Divide 100 hekats of barley among five men in arithmetic progression so that the sum of the two smallest is one-seventh the sum of the three largest.
- 19. Simplification (a) is the least amount that any man obtains (d) is the common difference between them in the Arithmetic Progression a, a+d , a+2d , a+3d , a+4d Adding them up makes the equation:- Similarly
- 20. Method of False Assumption Reducing these two Equations:- Using the Method of False Assumption, i.e. assume any value of a (for example a=1) and substitute it in equation 2 It gives d = 11 2
- 21. Substitute the values a=1 and d= 11 into equation 1 which gives 60=100 2 Which is the initial guess for “a” i.e. a= 100/60 Substituting this value of “a” in equation 2 which gives d=55/6
- 22. Result of False Assumption Technique So, after calculating the amount of barley each of the 5 men receive is :- 10/6, 65/6 ,120/6 ,175/6 and 230/6 hekats. This technique of guessing a value of an unknown and later adjusting it has been used by many cultures throughout the ages.
- 23. China (A.D. 263)
- 24. Mathematics in China emerged independently by the 11th century BC. Chiu Chang Suan Shu in Chinese characters played big role in china matemtics history. The most important treatise in the history of Chinese mathematics is the Chiu Chang Suan Shu,or “The Nine Chapters of the Mathematical Art.” This treatise, which is a collection of 246 problems and their solutions, was assembled in its final form by Liu Hui in A.D. 263. Its contents, however, go back to at least the beginning of the Han dynasty in the second century B.C. The eighth of its nine chapters, entitled “The Way of Calculating by Arrays,” contains 18 word problems that lead to linear systems in three to six unknowns. The general solution procedure described is almost identical to the Gaussian elimination technique developed in Europe in the nineteenth century by Carl Friedrich Gauss.
- 25. First problem in the eighth chapter There are three classes of corn, 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? Let x, y, and z be the measures of the first, second, and third classes of corn. Then the conditions of the problem lead to the following linear system of three equations in three unknowns:
- 26. The solution described in the treatise represented the coefficients of each equation by an appropriate number of rods placed within squares on a counting table. Positive coefficients were represented by black rods, negative coefficients were represented by red rods, and the squares corresponding to zero coefficients were left empty. The counting table was laid out as follows so that the coefficients of each equation appear in columns with the first equation in the rightmost column:
- 27. Next, the numbers of rods within the squares were adjusted to accomplish the following two steps: (1) two times the numbers of the third column were subtracted from three times the numbers in the second column and (2) the numbers in the third column were subtracted from three times the numbers in the first column. The result was the following array: In this array, four times the numbers in the second column were subtracted from five times the numbers in the first column, yielding This last array is equivalent to the linear system
- 29. RELATED PROBLEMS A passenger jet took three hours to fly 1800 miles in the direction of the Jetstream. The return trip against the jetstream took four hours. What was the jet's speed in still air and the Jetstream's speed? You are told the area of a square of 100 square cubits is equal to that of two smaller squares, the side of one square is 1/2 + 1/4 of the other. What are the sides of the two unknown squares.
- 30. Facts and Figures “There is a conflict in between religions but not in between education.” “Religions are different but human being is a human being.” And “we are not concerned to follow/adopt so many religions but we are only concerned with education and being an educated person, we have to respect the humanity.”
- 31. Bakhshali Manuscript (India and Linear Algebra) The Bakhshali Manuscript is an ancient work of Indian mathematics Around the 4th century A.D Found only about 70 leaves or sheets of birch bark containing mathematical problems and their solutions in 1881 near the village of bakhshali. Now peshawar in pakistan. Many of its problems are so-called equalization problems that lead to systems of linear equations.
- 32. (India and Linear Algebra)
- 33. Earliest Example One merchant has seven asava horses, a second has nine haya horses, and a third has ten camels. They are equally well off in the value of their animals if each gives two animals, one to each of the others. Find the price of each animal and the total value of the animals possessed by each merchant.
- 34. Solution Given conditions: 5x+y+z=k ---(a) x+7y+z=k ---(b) x+y+8z=k ---(c) x=? y=? z=? K=? Subtract the sum of all variables from b/s: 4x=k-(x+y+z) - - - (1) 6y=k-(x+y+z) - - - (2) 7z=k-(x+y+z) - - -(3)
- 35. Continue 4x=6y=7z=k-(x+y+z) 4(6)(7)=k-(x+y+z) 168=k-(x+y+z) =>by substituting in eq(1),(2) and (3) we get, x=42,y=28,z=24 By putting these values either in eq(a),(b) or(c) we get, K=262 (total value of animals)
- 36. Exercise question 2. Solve the following problems from the Bakhshali Manuscript. B possesses two times as much as A; C has three times as much as A and B together; D has four times as much as A, B, and C together. Their total possessions are 300. What is the possession of A? Let a=x,b=2x,c=3(a+b),d=4(a+b+c), a=? Their total possessions= T= 300, a+b+c+d=T 60x=300 x=5 (value of a)
- 37. More of Bakhshali Manuscript Gives us formula for a square root: If a=6,b=5 then q=41; We will get the same value by either using under root 41 or by using ancient formula (i.e. 6.40)
- 38. Advantages of Bakhshali Manuscript Give us Equalization method to solve the problems Give us formula to calculate square root of (a^2+b).
- 39. Any Question