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    Some elements in the history of Arabmathematics

    Mahdi ABDELJAOUAD

    From arithmetic to algebra

    Part 1

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    Summary

    Introduction

    Domains studied by Arabs

    Arithmetic and number theory

    Algebra

    Conclusion

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    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 inAmerica.

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    From 750 up to 900

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    From 900 up to 1000

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    From 1100 up to 1300

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    From 1300 up to 1500

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    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-Khawarizmi4. Decimal numbers : al-Kashi5. Combinatorics : Ibn al-Muncim

    Trigonometry : Nasir ad-Dine at-Tusi Astronomy : al-Biruni Science of music : al-Farabi.

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    Science of numbers

    al arithmtika andcIlm al-hisb

    The first one is speculative or theoretical and isinterested to abstract numbers and to pytagoricaland euclidian arithmetic.

    The second one is active or practical and isinterested to concrete numbers and to the needs

    of merchants.

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    Speculative arithmetic

    Inspired from Aristotle philosophy

    Two approaches :

    1. Euclids ElementsBooks VII VIII and IX

    2. Pythagoras through Nicomachus of Gerases

    Introduction to Arithmetic.

    Thabit ibn Qurra (d.901) Bagdad(a star worshipper)

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    Types of active arithmetic

    1 . Al-his

    b al-hawi

    (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

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    Abu'l-Wafa (d.998) BagdadBook 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 landtaxes and all kinds of business needed in

    administrations, proportions, multiplication,

    division, measurements, land taxes, distribution,

    exchange and all other practices used by variouscategories of men for doing business and which

    are useful to them in their daily life.

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    Abu'l-Wafa (d.998) BagdadPart 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 permitsfor ships on the river, merchants on the roads).

    Part VII: Further business topics.

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    Arab fractions

    Arab fractions are those used before them byEgyptians. These are unit fractions or capitalfractions whose numerator is always 1. In AncientEgypt, they were indicated by placing an oval over

    the number representing the denominator.1/3 is noted :

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    Arab fractions

    One half One third One fourth One tenth

    All computations have to be described by the meansof unit fractions.

    You will not say Five-sixth (5/6)but

    One third plus one half (1/3 of )

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    Types of active arithmetic

    2 . His

    b as-sittne (Sexagesimal calculus)

    Originated in Babylon 2000 B.C. adoptedby Greeks and Indians.

    Base 60 for all fractions Alphabetical numeration : It uses letters of

    Arabic alphabet for numbers from 1 to 59 . Forexample : 1 = = ; 2 ; 3 = ; 7 = ; 13 = ;

    27 = . Indian numeration : It uses Arabic numerals from

    1 to 59, and also 0 in medial position.

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    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.

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    9

    8

    7

    6

    5

    4

    3

    2

    1

    019

    18

    17

    16

    15

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    13

    12

    11

    10

    29

    28

    27

    26

    25

    24

    23

    22

    21

    20

    39

    38

    37

    36

    35

    34

    33

    32

    31

    30

    49

    48

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    43

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    40

    59

    58

    57

    56

    55

    54

    53

    52

    51

    50

    Alphabetical numeration

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    Types of active arithmetic

    3 . His

    b 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 partsof the calculation as the computing progresses.

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    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

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    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 isno difficulty, impossibility or preparation.

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    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 ismore secure and more fitting to the dignity of a

    leader.

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    How Fractions are represented ?

    Mathematicians from Baghdad and Egypt have used

    Hindis way of denoting fractions : they place the integralpart above the numerator and the numerator above thedenominator. 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 haveinvented the separation line between numerator anddenominator. (around the XIIth Century)

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    An arithmetic textbook in 1300Ibn 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 :

    OperationsSquare roots.

    2. Proportions : Rule of three

    Solving problems by usingmethod of the balance (al-kaff'ayan)

    3. Solving problems by using method of algebra.

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    Arab algebra

    M. ibn Musa Al-Khwrizmi (780 - 850 Bagdad)

    Kitab al-Jabr wal muqbala

    ... what is easiest and most useful inarithmetic, such asmen constantly require in cases of inheritance, legacies,partition, lawsuits, and trade, and in all their dealings withone another, or where the measuring of lands, the digging

    of canals, geometrical computations, and other objects ofvarious sorts and kinds are concerned.

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    The nameAlgebra

    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 treatinghow to solve them :Kitab al-Jabr wal muqabala

    written by

    al-Khwarizmi (780-850).

    Algorithmusis a Latin transcription of his name

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    al-Jabr : 31x - 2x + 40 = 21x

    then 31x + 40 = 19x

    al-Muqbala : 10x + 3x + 4 = 15x + 2x + 1then x + 3 = 5x

    Shay : the thing or the unknown. Today, it is denoted xMl: It is the multiplication ofShay by Shay . In fact itis the square of the unknown. Today it is denoted x .

    Equation x + 3 = 5x is read in Arabic :

    Shay plus three equal fiveMl

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    Six classes of equations

    Mlequal Shay : 3x = 5x.

    Mlequal numbers : 8x = 127.

    Shay equal numbers : 89x = 4.

    Mland Shay equal numbers : 45x + 12x = 5.

    5. Mland numbers equal Shay : 3x + 7 = 2x.

    6. Shay and numbers equalMl: 100x + 2 = x

    Ml d Sh l b

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    Mland Shay equal numbersx + 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

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    x + 10x = 64

    D l t f l b

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    Development of algebra

    M. ibn Musa Al-Khwrizmi (780 - 850 Baghdad) : India

    abu-Kmil(d.950 Egypt) : al-Khwarizmi + Euclide

    al-Karji(born 953 Baghdad - 1029) : al-Khwarizmi + abu-Kamil + Euclide + Diophante

    As-Samawal (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-Him (1352 Cairo 1412 Jerusalem)

    Arab algebra

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    Arab algebraAbu Kmil (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 ofalgebra to geometrical problems.

    Methods in this book are a combination of the geometric

    methods developed by the Greeks together with thepractical methods developed by al-Khwarizmi mixedwith Babylonian methods.

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    Arab algebraAl-Karji (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

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    Arab algebraAs-Samawal al-Maghribi (1130 Baghdad - 1180 Iran)

    al-Bhir fil hisb

    (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 .

    Arab algebra

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    Arab algebraOmar 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 onthe 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.

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    Arab algebraOmar al-Khayym (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 cannotbe solved by ruler and compas methods, a result which

    would not be proved for another 750 years.

    Arab algebra

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    Arab algebraSharaf 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 nopositive solution.

    The method which al-Tusi used is geometrical. He proves that

    the cubic equation bxx3 = 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.

    An algebra textbook in 1387

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    An algebra textbook in 1387Ibn al-Him (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.

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    North African Symbols

    al-Hassr (around 1150 in Andalousia or in Morocco):

    al-Kitab al-Kamil fi al-hisab. (Complete book ofcalculus)

    Ibn al-Ysamine(d.1204) (Andalousia and Morocco) :

    His didactical poem (Urjuza) was learned by hart byall pupils up the the XIXth century .

    al-Qalasdi(born in Andalousia - dead in Tunisia in 1486)

    He wrote arithmetical and algebra textbooks.

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    An equation written in symbols

    "Mlplus sevenShayequal eight"

    x + 7x = 8.

    Conclusion

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    Arab mathematics had started by translations ofGreek , Indian, Syriac and Persian works.

    All this knowledge has been integrated in Arab culturewith 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 wrotetreatises and textbooks used elsewhere.

    - Their contributions to mathematics were knownhere in Sicilia and transferred to Latin and Italianlanguages.

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    References

    Storia della ScienzaEnciclopedia 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