CENTRAL ASIA ALGEBRA - CENTRAL ASIA: THE PLACE WHERE THE...

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Unformatted text preview: CENTRAL ASIA: THE PLACE WHERE THE ALGEBRA WAS BORN? ALGEBRA CENTRAL ASIA: CENTRAL THE PLACE WHERE ALGEBRA WAS BORN? Sponsored by the following: UNC Office for Diversity and Multicultural Affairs UNC Graduate School UNC Center for Slavic, Eurasian, and East European Studies Carolina Center for the Study of the Middle East and Muslim Civilizations UNC Cell and Developmental Biology Department Central Asian Focus Group CENTRAL ASIA: CENTRAL THE PLACE WHERE ALGEBRA WAS BORN? CENTRAL ASIA: CENTRAL THE PLACE WHERE ALGEBRA WAS BORN? CENTRAL ASIA: CENTRAL THE PLACE WHERE ALGEBRA WAS BORN? CENTRAL ASIA: CENTRAL THE PLACE WHERE THE WEST AND EAST MET CENTRAL ASIA CENTRAL THE PLACE WHERE THE WEST AND EAST MET The Battle of Issus (333 BC) CENTRAL ASIA CENTRAL THE PLACE WHERE THE WEST AND EAST MET Hellenistic States, Greek influence till 2nd century AD Hellenistic CENTRAL ASIA: CENTRAL THE PLACE WHERE THE WEST AND EAST MET ZIAN QUANG,138­126 BC THE SILK ROUTE THE KHORAZM KHORAZM CALIPHATE CALIPHATE Prophet Muhammad, Arabia Abu Bakr ­ “khalifa” (>caliph) – (prophet’s) successor (632­634), Syria and Iraq, campaign against Sassanid Iran and Byzantium Omar (634­644), Iran and Palestine Uthman (644­656) immediate contact with Central Asia Ali (656­661) Umayyad caliphs (661­750), further expansion into Central Asia Expansion into Central Asia Expansion Qutaiba ibn Muslim ­ 705­715 Romitan ­ 707 Varakhsha and Bukhara ­ 709 Shumon, Nasaf and Kesh ­ 710 Khorezm and Samarkand ­ 712 Shosh, Farghona and Koshghar ­ 715 CALIPHATE CALIPHATE Practical Necessity to Develop Science Science Finance, communication, calendar Trade relations, transportation, navigation Civil engineering, measurement of distances on the globe, direction to Mecca Levy tax, division of heritage justification of decisions Preservation and Development Preservation Library devoted to translation and preservation Persian works, first from Pahlavi (Middle Persian), then from Syriac, Sanscrit, and Greek, under Caliph al­Mansour (754­775). Bait al­Hikma – House of Wisdom ­ was established under caliph al­Ma’mun (786­833)­ shift to research in mathematics, astronomy, philosophy... Algebra – Why? Algebra Equations of various degrees and necessity to find solutions Babylon about 4000 years ago, by means of tables Egypt – Ahmes papyrus, about 2000­1700 BC Greece – geometric methods Diophantus – 3rd century CE, by means of tables India Maya, Aztec, Inca Breakthrough - Al-Khwārizmī Breakthrough Born around 780, died around 850 Abū’Abd Allāh (Abū Ja’far) Muhammad ibn Musa al­ Khwārizmī al­Majousi al­Katarbali. Al-Khwārizmī - Breakthrough Al-Khwārizmī Kitab al­mukhtasar fi hisab al­jabr wa'l­ muqabala (820) (The Compendious Book on Calculation by Completion and Balancing) ­ algebra Kitāb al­jam wa­l­tafrīq bi­ḥisāb al­hind (825)("The Book of Addition and Subtraction According to the Hindu Calculation") ­ arithmetic Al-Khwārizmī - Books Al-Khwārizmī Kitab surat al­ard ("Book on the appearance of the Earth" or "The image of the Earth"; (833), translated as Geography) Zīj al­sindhind ("astronomical tables") (820) Risāla fi istikhrāj ta’rīkh al­yahūd ("Extraction of the Jewish Era") Ma’rifat sa’at al­mashriq fī kull balad (on the morning width) Ma’rifat al­samt min qibal al­irtifā’ (the determination of the azimuth from a height) Al-Khwārizmī - Books Al-Khwārizmī Kitāb ar­Ruḵāma(t) (the book on sundials) Kitab al­Tarikh (the book of history) (the two have been lost) Al-Khwārizmī - Books Al-Khwārizmī Several Arabic manuscripts in Berlin, Istanbul, Tashkent, Cairo and Paris contain further material that surely or with some probability comes from al­Khwārizmī Al-Khwārizmī - Algebra Al-Khwārizmī al­Kitab al­mukhtasar fi hisab al­jabr wa'l­ muqabala (The Compendious Book on Calculation by Completion and Balancing) (820) al­jabr ­ “algebra” Geometric Method Geometric x2 + bx = c A2 S2/4 A1 A x2 = S1 bx = S2 B2 S2/4 D1 S1 D B1 B x S2/4 x C S2/4 D2 C2 C1 Geometric Method Geometric A2 S1 = x = surfaceABCD 2 S2/4 A1 b x 4 B2 A B S2/4 b S2 = 4 ⋅ x 4 x x D D1 2 b b 2 x + bx = c = ( AA2 B2 BB1C1CC2 D2 DD1 A1 ) = x + − 4 2 4 2 or S2/4 C S2/4 D2 2 b b x+ = c+ 2 4 B1 C2 2 C1 2 2 b b x+ = c+ 2 4 2 2 b x+ b = c+ 4 2 2 b b x+ = c+ 2 4 b2 b x = c+ − 42 b b2 x =− + c+ 2 4 Al-Khwārizmī - Algebra Al-Khwārizmī 1) 2) 3) 4) 5) 6) 5x2 = 40x 25/5x2=100 5x = 10 x2 + 10x = 39 x2 + 21 = 10x 12x + 288 = x2 Al-Khwārizmī - Algebra Al-Khwārizmī 1) 2) 3) 4) 5) 6) squares equal roots squares equal number roots equal number squares and roots equal number squares and number equal roots roots and number equal squares Al-Khwārizmī - Algebra Al-Khwārizmī 1) 2) 3) 4) 5) 6) ax2 = bx ax2=c bx = c ax2 + bx = c ax2 + c = bx bx + c = ax2 Al-Khwārizmī - Algebra Al-Khwārizmī 1) 2) 3) 4) 5) 6) 5x2 = 40x 25/5x2=100 5x = 10 x2 + 10x = 39 x2 + 21 = 10x 12x + 288 = x2 1. 2. 3. 4. 5. 6. squares equal roots ax 2 = bx squares equal number ax2=c roots equal number bx = c squares and roots equal number ax2 + bx = c squares and number equal roots ax2 + c = bx squares and number equal roots bx + c = ax2 Al-Khwārizmī - Algebra Al-Khwārizmī 2x2 + 50 – 20x = 29 + x2 – 10x x2 ­ 10x + 21 = 0 ax2 + bx + c = 0 Al-Khwārizmī - Algebra Al-Khwārizmī 2x2 + 50 – 20x = 29 + x2 – 10x Aljabr ­ completion 2x2 + 50 + 10x = 29 + x2 + 20x Almuqabala ­ balancing x2 + 21 = 10x Al­Khwārizmī We Divide the number of roots by two Multiply it by itself Subtract number of it Extract square root of it Subtract it from half number of roots b 22 b 2 2 b −c 2 2 b −c 2 2 x1= b − b −c 22 or add it to half number of roots 2 b ± b −c x1,2 = 22 2 b + b −c x1,2 = 22 Al-Khwārizmī - Ariphmetic Al-Khwārizmī Al-Khwārizmī - Ariphmetic Al-Khwārizmī Al-Khwārizmī - Ariphmetic Al-Khwārizmī I, II, III, IV, V, VI, VII, VIII, IX, X, L, C, D, M CDLXXXVII ­ 487 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ABACI ABACI Al-Khwārizmī - Ariphmetic Al-Khwārizmī 487 x 15 8 4 7 3 0 7 4 2 0 5 3 0 5 1 5 487 x 15 2435 487 7305 Al-Khwārizmī Al-Khwārizmī Impact on European Mathematics Al Kitab al muhtasar fi hisab aljabr va al muqabala • • • Robert of Chester, 1145 Gerard of Cremona, 12th century Leonardo of Pisa commented on 6 types of quadratic equations Al-Khwārizmī Al-Khwārizmī Impact on European Mathematics Kitab al jam va tafriq bi hisab al hind ­ “Algorizm’s book on arithmetical practice” ­ Johannes of Toledo, 12th century. 1135 to 1153, The book spread in Europe. 13th c., Socrobosco, Algorismus Vulgaris (The Simple Algorism). Danish scholar Ingversen, comments in 1290. 13th c., Demonstratto de algorismo (The explanation of algorism) by Iordan Nomoraria. Besides, Algorithmus demonstatus (The explained algorism) is known. Nurnberg, 1534, and Paris, 1570. In Italian Tractatus algorismi (Treatise on Algorism), by Giacobo of Florence, 1307. Italian Prosdocimo de Beldomodo’s Algorismi tractatus peritulis et necessarus (The very useful and necessary treatise on algorism) Padua, 1483, Venice, 1540. 13th., poetic book Carmende algorismo (The song on algorism), French Alexander, in French, English, and Iceland. 14th century the French mathematician Nicol Orem’s Algorismus proportionus (The algorism of proportions). 15th century Peierbach’s book Algorithmus ­ the main aid book in the Austrian universities. Al-Khwārizmī Al-Khwārizmī Impact on European Mathematics Kitab al jam va tafriq bi hisab al hind ­ “Algorizm’s book on arithmetical practice” ­ Johannes of Toledo, 12th century. 1135 to 1153, The book spread in Europe. 13th c., Socrobosco, Algorismus Vulgaris (The Simple Algorism). Danish scholar Ingversen, comments in 1290. 13th c., Demonstratto de algorismo (The explanation of algorism) by Iordan Nemoraria. Besides, Algorithmus demonstatus (The explained algorism) is known. Nurnberg, 1534, and Paris, 1570. In Italian Tractatus algorismi (Treatise on Algorism), by Giacobo of Florence, 1307. Italian Prosdocimo de Beldomodo’s Algorismi tractatus peritulis et necessarus (The very useful and necessary treatise on algorism) Padua, 1483, Venice, 1540. 13th., poetic book Carmende algorismo (The song on algorism), French Alexander, in French, English, and Iceland. 14th century the French mathematician Nicolas Orem’s Algorismus proportionus (The algorism of proportions). 15th century Peierbach’s book Algorithmus ­ the main aid book in the Austrian universities. Algebra after Al-Khwārizmī Algebra In the East • Ghiyās od­Dīn Abul­Fatah Omār ibn Ibrāhīm Khayyām Nishābūrī or Omar Khayyam (1048 – 1131) • Ulugbek (1394­1449) • Ghiyāth al­Dīn Jamshīd ibn Mas’ūd al­Kāshī (c. 1380 – 1429) Algebra after Al-Khwārizmī Algebra In the West • Niccolò Fontana Tartaglia (1499/1500 – 1557) cubic equations • Gerolamo Cardano (1501 ­ 1576) • Scipione del Ferro cubic equations • Lodovico Ferrari (1522 – 1565) • François Viète (or Vieta), (1540 ­ 1603), CENTRAL ASIA: CENTRAL THE PLACE WHERE ALGEBRA WAS BORN Questions? Thank you Thank you ...
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This note was uploaded on 01/12/2011 for the course MAT 1117 MAT 117 taught by Professor White during the Spring '09 term at University of Phoenix.

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