Some elements in the history of Arab mathematics Mahdi ABDELJAOUAD From arithmetic to algebra Part 1 Summary Introduction Domains studied by Arabs Arithmetic and number theory Algebra Conclusion Palerme, 25-26 novembre 2003 2 Quick Chronology of Islam • 622 : First year of Arabic Calendar • 632 : Death of Mohamed, the Prophet • 633 – 640 : conquest of Syria and Mesopotamia • 639 – 646 : conquest of Egypt. • 687 – 702 : conquest of North Africa. • 701 – 716 : conquest of Spain (Andalusia). • 640 – 750 : Reign of the Ommayads (Damascus) • 762 – 1258 : Reign the Abbassids. (Baghdad) • 1055 : Turks take over Baghdad • 1258 : Mongols take over Baghdad. • 1492 : Christians take over Grenada and arrive in America. Palerme, 25-26 novembre 2003 3 Palerme, 25-26 novembre 2003 4 From 750 up to 900 Palerme, 25-26 novembre 2003 5 From 900 up to 1000 Palerme, 25-26 novembre 2003 6 From 1100 up to 1300 Palerme, 25-26 novembre 2003 7 From 1300 up to 1500 Palerme, 25-26 novembre 2003 8 Subjects studied by Arab mathematicians • Geometry : Thabit ibn Qurra – Omar al-Khayyam – Ibn al-Haytham. • Sciences of numbers 1. Indian numeration : al-Uqludisi. 2. Business arithmetic : Abu l-Wafa. 3. Algebra : al-Khawarizmi 4. Decimal numbers : al-Kashi 5. Combinatorics : Ibn al-Muncim • Trigonometry : Nasir ad-Dine at-Tusi • Astronomy : al-Biruni • Science of music : al-Farabi. Palerme, 25-26 novembre 2003 9 Science of numbers « al – arithmétika » and «cIlm al-hisāb » • The first one is speculative or theoretical and is interested to abstract numbers and to pytagorical and euclidian arithmetic. • The second one is active or practical and is interested to concrete numbers and to the needs of merchants. Palerme, 25-26 novembre 2003 10 Speculative arithmetic Inspired from Aristotle philosophy Two approaches : 1. Euclid’s Elements Books VII – VIII and IX 2. Pythagoras through Nicomachus of Gerase’s Introduction to Arithmetic. Thabit ibn Qurra (d.901) Bagdad (a star worshipper) Palerme, 25-26 novembre 2003 11 Types of active arithmetic 1 . Al-hisāb al-hawā’i (air calculus) • Based solely on memory • Rethorical calculus uses only words in the text and no symbols. • It is digital : calculus uses fingers to compute and to do operations. • It uses unitary and sexagesimal fractions • How to solve problems: rule of three – algebra • A chapter on geometric mensurations • A great number of practical problems Palerme, 25-26 novembre 2003 12 Abu'l-Wafa (d.998) Bagdad Book on what Is necessary from the science of arithmetic for scribes and businessmen This book : ... comprises all that an experienced or novice, subordinate or chief in arithmetic needs to know, the art of civil servants, the employment of land taxes and all kinds of business needed in administrations, proportions, multiplication, division, measurements, land taxes, distribution, exchange and all other practices used by various categories of men for doing business and which are useful to them in their daily life. Palerme, 25-26 novembre 2003 13 Abu'l-Wafa (d.998) Bagdad Part I: On ratio. Part II: Arithmetical operations (integers and fractions). Part III: Mensuration (area of figures, volume of solids and finding distances). Part IV: On taxes (different kinds of taxes and problems of tax calculations). Part V: On exchange and shares (types of crops, and problems relating to their value and exchange). Part VI: Miscellaneous topics (units of money, payment of soldiers, the granting and withholding of permits for ships on the river, merchants on the roads). Part VII: Further business topics. Palerme, 25-26 novembre 2003 14 Arab fractions Arab fractions are those used before them by Egyptians. These are unit fractions or capital fractions whose numerator is always 1. In Ancient Egypt, they were indicated by placing an oval over the number representing the denominator. 1/3 is noted : Palerme, 25-26 novembre 2003 15 Arab fractions One half – One third – One fourth – … – One tenth … All computations have to be described by the means of unit fractions. You will not say Five-sixth (5/6) but One third plus one half (1/3 of ½) Palerme, 25-26 novembre 2003 16 Types of active arithmetic 2 . Hisāb as-sittīne (Sexagesimal calculus) • Originated in Babylon 2000 B.C. – adopted by Greeks and Indians. • Base 60 for all fractions • Alphabetical numeration : It uses letters of Arabic alphabet for numbers from 1 to 59 . For example : 1 = ;ا2 = ; ب3 = ; جـ7 = ; ز13 = ; يجـ 27 = كب. • Indian numeration : It uses Arabic numerals from 1 to 59, and also 0 in medial position. Palerme, 25-26 novembre 2003 17 Kushiyar Ibn Labban al-Gili (d.1024) Bagdad Book on fundaments of Indian calculus ... These fundaments are sufficient for all who need to compute in Astronomy, and also for all exchanges between all the people in the world. Palerme, 25-26 novembre 2003 18 Alphabetical numeration 19 صف ـر أ 1 ب 2 جـ 3 د 4 ه 5 و 6 ز 7 ح 8 ط 9 0 ي 10 يـا 11 يب 12 يجـ 13 يـد 14 يـه 15 يـو 16 يـز 17 يـح 18 يـط 19 كـ 20 كــا 21 كب 22 كجـ 23 كـد 24 كـه 25 كـو 26 كـز 27 كـح 28 كـط 29 ل 30 ال 31 لب 32 لجـ 33 لـد 34 لـه 35 لـو 36 لـز 37 لـح 38 لـط 39 م 40 مـا 41 مب 42 مجـ 43 مـد 44 مـه 45 مـو 46 مـز 47 مـح 48 مـط 49 ن 50 نـا 51 نب 52 نجـ 53 نـد 54 نـه 55 نـو 56 نـز 57 نـح 58 نـط 59 Palerme, 25-26 novembre 2003 Types of active arithmetic 3 . Hisāb al-Hind (Hindu arithmetic) • place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. • Only whole positive numbers • It uses a dust board (Takht - Ghubar) You have to continually erase, change and replace parts of the calculation as the computing progresses. Palerme, 25-26 novembre 2003 20 Al-Uqludisi (around 952) Bagdad Book on parts of Indian calculus … Most arithmeticians are obliged to use [Hindu arithmetic] in their work: - it is easy and immediate, - requires little memorisation, - provides quick answers, - demands little thought Palerme, 25-26 novembre 2003 21 Al-Uqludisi (around 952) Bagdad ... Therefore, we say that it is a science and practice that requires a tool, such as a writer, an artisan, a knight needs to conduct their affairs; since if the artisan has difficulty in finding what he needs for his trade, he will never succeed; to grasp it there is no difficulty, impossibility or preparation. Palerme, 25-26 novembre 2003 22 Al-Uqludisi (around 952) Bagdad … Official scribes nevertheless avoid using [the Indian system] because it requires equipment [like a dust board] and they consider that a system that requires nothing but the members of the body is more secure and more fitting to the dignity of a leader. Palerme, 25-26 novembre 2003 23 Palerme, 25-26 novembre 2003 24 How Fractions are represented ? Mathematicians from Baghdad and Egypt have used Hindi’s way of denoting fractions : they place the integral part above the numerator and the numerator above the denominator. The number 26/7 is denoted vertically Integral part 3 Numerator 5 Denominator 7 with no lines separating the vertical numbers. Mathematicians from Andalousia and North Africa have invented the separation line between numerator and denominator. (around the XIIth Century) Palerme, 25-26 novembre 2003 25 An arithmetic textbook in 1300 Ibn al-Banna (1256-1321) Marrakech Lifting the veil on parts of calculus Introduction : Number theory – unity – place-values – signification of fraction as a ratio between two numbers 1. Whole numbers : Addition - summing series – Substraction – Multiplication – Division. Fractions : Different ways of representing and operating on them. Operations. Irrationnals : Operations – Square roots. 2. Proportions : Rule of three – Solving problems by using method of the balance (al-kaff'ayan) 3. Solving problems by using method of algebra. Palerme, 25-26 novembre 2003 26 Arab algebra M. ibn Musa Al-Khwārizmi (780 - 850 Bagdad) Kitab al-Jabr wal muqābala ... what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned. Palerme, 25-26 novembre 2003 27 The name Algebra is a Latin translation of an Arab word : al-Jabr This word is a part of the title of the first textbook presenting equations and treating how to solve them : Kitab al-Jabr wal muqabala written by al-Khwarizmi (780-850). Algorithmus is a Latin transcription of his name Palerme, 25-26 novembre 2003 28 al-Jabr : 31x² - 2x + 40 = 21x then 31x² + 40 = 19x al-Muqābala : 10x² + 3x + 4 = 15x² + 2x + 1 then x + 3 = 5x² Shay : « the thing » or the unknown. Today, it is denoted x Māl : It is «the multiplication of Shay by Shay ». In fact it is the square of the unknown. Today it is denoted x² . Equation x + 3 = 5x² is read in Arabic : Shay plus three equal five Māl Palerme, 25-26 novembre 2003 29 Six classes of equations • • • • 5. 6. Māl equal Shay : 3x² = 5x. Māl equal numbers : 8x² = 127. Shay equal numbers : 89x = 4. Māl and Shay equal numbers : 45x² + 12x = 5. Māl and numbers equal Shay : 3x² + 7 = 2x. Shay and numbers equal Māl : 100x + 2 = x² Palerme, 25-26 novembre 2003 30 Māl and Shay equal numbers x² + px = q • • • • • Take half the roots , that is p/2 , half of p. Multiply it by itself, that is (p/2) x (p/2) Add to it the number, that is q Take the square roots of the result Subtract from it half the roots : It is what you are looking for Palerme, 25-26 novembre 2003 31 x² + 10x = 64 Palerme, 25-26 novembre 2003 32 Development of algebra M. ibn Musa Al-Khwārizmi (780 - 850 Baghdad) : India abu-Kāmil (d.950 Egypt) : al-Khwarizmi + Euclide al-Karāji (born 953 Baghdad - 1029) : al-Khwarizmi + abuKamil + Euclide + Diophante As-Samaw’al (1130 Baghdad - 1180 Iran) : al-Karaji Omar al-Khayyam (1048 - 1131 Iran) : Euclide Sharaf ad-Din at-Tusi (1135 - 1213 Iran) : Khayyam Euclide Ibn al-Banna (1256 – 1321 Marrakech) Ibn al-Hā’im (1352 Cairo – 1412 Jerusalem) Palerme, 25-26 novembre 2003 33 Arab algebra Abu Kāmil (850 - 930 Egypt) Kitab al-Kamil fil Jabr (i) On the solution of quadratic equations, (ii) On applications of algebra to the regular pentagon and decagon, and (iii)On Diophantine equations and problems of recreational mathematics. The content of the work is the application of algebra to geometrical problems. Methods in this book are a combination of the geometric methods developed by the Greeks together with the practical methods developed by al-Khwarizmi mixed with Babylonian methods. Palerme, 25-26 novembre 2003 34 Arab algebra Al-Karāji (953 Bagdad - 1029) Kitab al-Fakhri and Kitab al-Badic fil Jabr He gives rules for the arithmetic operations including (essentially) the multiplication of polynomials. He usually gives a numerical example for his rules but does not give any sort of proof beyond giving geometrical pictures. He explicitely says that he is giving a solution in the style of Diophantus. He does not treat equations above the second degree. The solutions of quadratics are based explicitly on the Euclidean theorems Palerme, 25-26 novembre 2003 35 Arab algebra As-Samaw’al al-Maghribi (1130 Baghdad - 1180 Iran) al-Bāhir fil hisāb (i) Definition of powers x, x2, x3, ... , x-1, x-2, x-3, ... . Addition, subtraction, multiplication and division of polynomials. Extraction of the roots of polynomials. (ii) Theory of linear and quadratic equations, with geometric proofs of all algorithmic solutions. Binomial theorem – Triangle of Pascal. Use of induction. (iii)Arithmetic of the irrationals. n applications of algebra to the regular pentagon and decagon, and (iv)Classification of problems into necessary problems, possible problems and impossible problems . Palerme, 25-26 novembre 2003 36 Arab algebra Omar al-Khayyam (1048 - 1131 Iran) Risala fil Jabr wal muqabala (Treatise on Algebra and muqabala) He starts by showing that the problem : (1) Find a right triangle having the property that the hypothenuse equals the sum of one leg plus the altitude on the hypotenuse. (2) x3 + 200x = 20x2 + 2000 (3) He founds a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. (4) He then gives approximate numerical solution by interpolation in trigonometric tables. Palerme, 25-26 novembre 2003 37 Arab algebra Omar al-Khayyām (1048 - 1131 Iran) Treatise on Algebra and muqabala Complete classification of cubic equations with geometric solutions found by means of intersecting conic sections He demonstrates the existence of cubic equations having two solutions, but unfortunately he does not appear to have found that a cubic can have three solutions. What historians consider as more remarkable is the fact that Omar al-Khayyam has stated that these equations cannot be solved by ruler and compas methods, a result which would not be proved for another 750 years. Palerme, 25-26 novembre 2003 38 Arab algebra Sharaf ad-Din at-Tusi (1135 - 1213 Iran) Treatise on equations In the treatise equations of degree at most three are divided into 25 types : twelve types of equation of degree at most two, eight types of cubic equation which always have a positive solution, then five types which may have no positive solution. The method which al-Tusi used is geometrical. He proves that the cubic equation bx – x3 = a has a positive root if its discriminant D = b3/27 - a2/4 > 0 or = 0. For all cubic equations he approximates the root of the cubic equation. Palerme, 25-26 novembre 2003 39 An algebra textbook in 1387 Ibn al-Hā’im (1352 Cairo – 1412 Jerusalem) Sharh al-Urjuza al-yasminiya fil Jabr Introduction : Terminology 1. The six canonical equations : Definitions – solutions – numerical examples. (All proofs are algebraic with no geometrical arguments.) 2. The arithmetic of polynomials. 3. The arithmetic of irrationnels – Summing series of integers. 4. How to abord a problem and solve it. 5. Solutions of algebraic numerical problems : (1) with rational coefficients (2) with irrational coefficients. Palerme, 25-26 novembre 2003 40 North African Symbols al-Hassār (around 1150 in Andalousia or in Morocco): al-Kitab al-Kamil fi al-hisab. (Complete book of calculus) Ibn al-Yāsamine (d.1204) (Andalousia and Morocco) : His didactical poem (Urjuza) was learned by hart by all pupils up the the XIXth century . al-Qalasādi (born in Andalousia - dead in Tunisia in 1486) He wrote arithmetical and algebra textbooks. Palerme, 25-26 novembre 2003 41 An equation written in symbols "Māl plus seven Shay equal eight " x² + 7x = 8. Palerme, 25-26 novembre 2003 42 Conclusion • Arab mathematics had started by translations of Greek , Indian, Syriac and Persian works. • All this knowledge has been integrated in Arab culture with Arab words and thinking. • Men of different cultures and regions of the world, independently from their races and religions - They worked together in Baghdad, Cairo, Cordoba, Marrakech or in Tunis - They invented new mathematics and wrote treatises and textbooks used elsewhere. - Their contributions to mathematics were known here in Sicilia and transferred to Latin and Italian languages. Palerme, 25-26 novembre 2003 43 References Storia della Scienza Enciclopedia Italiana Vol. III, 2002 on the web In English : www-history.mcs.standrews.ac.uk/history/ HistTopics/Arabic_mathematics.html In French : www.chronomathirem.univ-mrs.fr Palerme, 25-26 novembre 2003 44

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# History of Arab Mathematics