Published on July 15, 2014
Mathematics in the near and far east: Mathematics in the near and far east By: S. “John” Shia Math Near East (Arabs): Math Near East (Arabs) 7 th century (Islam ) – 13 th century (Crusade) contribution by Arab mathematicians. Initial impetus: emergence of Islam on Mohammed combined with religious need for accurate calendar. Caliphs(“successors”) initially in Damascus, then Baghdad. Standardization of language: Arabic. L egacies: preservation of Greek legacy, transmission of Hindu numbering system, and original contributions. The Algebra of al-KhowarizmI: The Algebra of al- KhowarizmI Mohammed ibn Musa al- Khowarizmı (circa 780–850 ) Original Arab manuscripts are lost Latin translations extant Brought Hindu numerals+algebraic approach to math to West “Book of Addition and Subtraction According to the Hindu Calculation” Latin translation Algoritmi de numero Indorum The Algebra of al-khowarizmi: The Algebra of al- khowarizmi “algebra” is the European corruption of al- jebr , part of the title of treatise ( from Arabic al- jebr meaning "reunion of broken parts). Hisab al- jebr w’al muqabalah “science of reunion and reduction” “reunion ” movement of negative terms “reduction” combination of like terms in the equation (in modern notation) 6 x^ 2 − 4 x + 1 = 5 x^ 2 + 3, “reunion” 6 x^ 2 + 1 = 5 x^ 2 + 4 x + 3, “reduction” x^ 2 = 4 x + 2 So the book loosely translated to “the science of reunion and reduction” Arab math : Arab math Arab algebra: no symbols like today. Both symbols and numbers were fully spelled out in words! Believed for argument (proof) to be sound, had to be geometric per Euclid. Quadratics, addressed only positive coefficients, knew about completing the square. Slowly moving away from purely geometric proof. However, they did it words!: However, they did it words! If you wish to subtract the root of 4 from the root of 9 until what remains of the root of 9 is a root of one number, then you add 9 to 4 to give 13. Retain it. Then multiply 9 by 4 to give 36. Take 2 of its roots to give 12. Subtract it from the 13 that was retained. One remains. The root is 1. It is the root of 9 less the root of 4 . ======== ================= Arab math : Arab math Arab math: fundamental contributions: Arab math: fundamental contributions Recognized irrationals roots of quadratic equations(listed only positive ones). Did not believe in negative solutions. Bhaskara (Indian) first affirmed negative solutions (11 th century). The reckoner of egypt: The reckoner of egypt Arab math : Arab math Thabit ibn Qurra (~836–901): Thabit ibn Qurra (~836–901) Translated bulk of Greek works “Book on the Determination of Amicable Numbers” 1 st completely original work in Arabic 10 propositions E.g., if p = 3·2^ n −1 , q = 3^(2 n − 1)−1 , and r = 9 · 2^(2 n − 1) − 1 are all prime numbers, then M = 2^ n * ( pq ) and N = 2^ n * r form a pair of amicable numbers. Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. A proper divisor of a number is a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3 . Smallest pair of amicable numbers is ( 220 , 284 ); for the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220. al-Karajı (1029) or Al-KarkhI: al-Karajı (1029 ) or Al-KarkhI “The Marvelous” Algebra of polynomials Binomial expansions Law of polynomials Predated Pascal’s Triangle (Arithmetic Triangle) al-Samaw’al (ca. 1180): al- Samaw’al (ca. 1180) Figured closed solution to Al- Karaji’s summation of square and cube For summation of square, summation of squares = (1/6) n ( n + 1)(2 n + 1 ) 1^2+…+10^2=? For summation of cube, summation of cubes = (1/4 )( n ( n + 1 ))^2 1^3+…+100^3=? Omar Khayyam (11th century): Omar Khayyam (11 th century) End of caliphate in Baghdad due to Moslem Seljuk Turks seizing Baghdad. Actual name: Umar al- Khayyami , famed for poetry Rubaiyat Positive solution to cubic equations! “Treatise on Demonstrations of Problems of al-Jabra and al- Muqabalah ” worked on cubics via conic sections “Commentaries on the Difficulties in the Premises of Euclid’s Book” attacked Euclid’s 5 th , try to define parallel postulate from different angle Omar Khayyam (11th century): Omar Khayyam (11 th century) first mathematician to solve every type of cubic equation having a positive root Nasır al-Dın al-Tusı (1201–1274): Nasır al-Dın al- Tusı ( 1201–1274) Worked with Mongul leader Hulagu Khan. Developed non-Ptolemaic(heliocentric) theory of planetary motions. Similar to work of Copernicus. Ghiyath al-Din al-Kashı (d. 1429): Ghiyath al-Din al-Kashı (d. 1429) Persian. Developed tables of precise tables of sines and cosines. Al- Kashı’s “ Treatise on the Circumference” expounded on the use of decimal fractions. π = 3.14159265358979324 Pi to 16 th place, record till 16 th century in Europe. End of Arab development : End of Arab development Crusade was the beginning of the end. Crusaders brought knowledge of Arab math/science/culture to Europe, beginning the start of European Renaissance. Arab attitude toward learning became reactionary, distrusting knowledge based on source/affiliation, and leaders who advocated open knowledge such as Ulugh Beg was assassinated. Chinese math : Chinese math Emphasized the practical most of the time. Practical problems connected with everyday life: calculation of areas of all kinds of shapes, volumes of various vessels, and dams. Profoundly algebraic, so geometric figures served only to transmute numerical information into algebraic form. Chinese math: Chinese math “ Arithmetic Classic of the Gnomon and the Circular Paths of Heaven” (300 BC but there might be even older copies, as far back as 1000 BC) “ gnomon” in the title referring to Chinese version of sundial. Discussion of right angle, including common 3:4:5 ratio. Derived Pythagorean theorem for specific case but apparently, also knew it could be extended. Chinese math: Chinese math “Nine Chapters on the Mathematical Art”: “Nine Chapters on the Mathematical Art” Oldest textbook on arithmetic in existence. Free of mystic cosmology unlike earlier book. Original destroyed in Burning of the Books in 213 B.C. Only commentary on the Nine Chapters , prepared by Liu Hui in A.D. 263 survives. Chinese mathematics was geared toward proficiency in algebraic manipulation and problem solving, so that there was little incentive to change a procedure that worked longevity of Nine Chapters. Nine Chapters: Nine Chapters O ne of the earliest printed textbooks when a printed version appeared in 1084. Government decreed its use throughout the universities for civil service examinations. 9 sections with 246 problems+solutions . R ules—areas of rectangles, triangles, trapezoids, segments of circles, and for the volumes of solids (spheres, cylinders, pyramids, cones, etc.) Special emphasis on fractions. Problem 32 of Chapter 1 (“Field Extensions”): Problem 32 of Chapter 1 (“Field Extensions”) Takes π to be 3 in the calculation Assumed area of circle to be 3/4d^2 Problem 3.20 Rule of three: Problem 3.20 Rule of three Rule of Three Finding the fourth term in a simple progression , that is, to solve the equation a / b = c / x for x Problem 3.20 There is a loan of 1000 qian with a monthly interest of 30 qian . Now there is a loan of 750 qian which is returned in 9 days. Find the interest. Chinese attempt at pi: Chinese attempt at pi Initially 3, by 3 rd century π = 142/45(3.1555) Using geometrical approximation (polygon with 3000+ sides to approximate circle), got 3.1415926 < π < 3.1415927 Chinese math: practical: Chinese math: practical 14 th Century, Chao Yu-chin used a polygon of 16382 sides to get π ≈ 3.1415926 Basically, accuracy of 16 th Century math in Europe. Favored algebra over geometry, used geometry to derive more difficult equations, worked only on positive roots. Nine Sections problem: Nine Sections problem “There is a circular walled city of unknown diameter with four gates. A tree lies 3 li north of the northern gate. If one walks 9 li eastward from the southern gate, the tree becomes just visible . Find the diameter of the city .” Ans. 9 li Negative numbers: Negative numbers Li Ye (1192–1279) Li Ye’s original contribution to Chinese mathematical notation was to indicate negative quantities by drawing a diagonal stroke through the last digit of the number in question . Previously, color was used. −8643 would appear as Old Mathematics in Expanded Sections: Old Mathematics in Expanded Sections Problem 8 There is a circular pond centered in the middle of a square field, and the area outside the pond is 3300 square pu . It is known only that the sum of the perimeters of the square and the circle is 300 pu . Find the perimeters of the square and circle. Solution: Solution π = 3 x= the diameter of the circular pond Per( sq ) + Per( cir ) = 300, Per=Perimeter A( sq ) – A( cir ) = 3300, A=Area pond’s circumference is 3 x perimeter of the square field 300 − 3 x (300 − 3 x )^2=16x A( sq ) 16(3 x^ 2/4 ) = 12 x^ 2 = 16x A( cir ) (300 − 3 x )^2 − 12 x^ 2 = 16x (A( sq ) – A( cir )) (300 − 3 x )^2 − 12 x^ 2 = 16 · 3300 = 52,800 Solution continued: Solution continued diameter of the pond x = 20 pu perimeter of the square field 240 pu perimeter of the circular pond 60 pu Chu Shih-chieh (1280–1303): Chu Shih- chieh (1280–1303 ) “Introduction to Mathematical Studies” in 1 299 “Precious Mirror of the Four Elements” in 1303 Covered Pascal’s triangle/binomial expansion. (x+1)^n up to 8 th power. Preceded Horner’s method(1819) for solving polynomials in West. Impacted development of math in Korea/Japan. Decline : Decline Started to decline in 14 th Century. Possibly due to lack of adequate symbolism, use of abacus, and/or lack of adequate records of earlier accomplishments. Foray into pure math primary on individual basis (itinerant mathematicians) since government funding was devoted to mostly practical application. Lacked development of formal logic which limited development of more formal, pure math albeit Catch-22 since more funding could have possibly resulted in parallel development of more formal logic as well as more symbolic notation. 16th century: 16 th century Introduction of European math along with Jesuit missionaries. Math+religion brought to China with math text infused with religious symbols. Both progressive and reactionary responses to Western math, judging merit by origin/affiliation. Traditionalist mathematician chose 3.16 as value of π, since “this exactness being in agreement with those of ancient authorities.” Opposition to barbarians/West. RECAP: RECAP Arab math - preserved Greek legacy - served as gateway for Hindu numbers - moved away from Greek emphasis on geometry toward algebraic system - original contribution, e.g. positive irrational roots of quadratic equations - development pretty much stopped at Crusade - Crusaders brought Arab/Hindu/Greek math/science/cultural/dietary legacy to Europe, starting Renaissance Chinese: Chinese H ighly practical, focused on practical applications. A lgebraic, used geometry to transform numbers into algebraic equations. P redated some European discoveries(1600-1800) by hundreds of years (e.g., Horner). Lacked formal logic and focus was mostly on practical application, not theory, due to lack of funding which in turn limited mathematical advances. CONCLUSION: CONCLUSION Religion+state played a major role in development of Arab math due to need for more accurate calendar, similar to Mayans. Funding played a major role in development of math, e.g., Chinese. Military conflict may or may not play a major role, e.g., Mongol invasion (positive) vs. Crusade(negative for Arabs, positive for Europe). Reactionary response in form of judging merit by ethnic/religious/historical basis almost always resulted in setback(negative). *** MORAL OF THE STORY *** Need communication+preservation of ideas/understanding for progress in math to be made. Funding needed for both practical and theoretical work.