Conf Mahdi 1 (1)
-
Upload
omair-anas -
Category
Documents
-
view
226 -
download
0
Transcript of Conf Mahdi 1 (1)
-
8/4/2019 Conf Mahdi 1 (1)
1/44
Some elements in the history of Arabmathematics
Mahdi ABDELJAOUAD
From arithmetic to algebra
Part 1
-
8/4/2019 Conf Mahdi 1 (1)
2/44
Palerme, 25-26 novembre 2003 2
Summary
Introduction
Domains studied by Arabs
Arithmetic and number theory
Algebra
Conclusion
-
8/4/2019 Conf Mahdi 1 (1)
3/44
Palerme, 25-26 novembre 2003 3
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.
-
8/4/2019 Conf Mahdi 1 (1)
4/44
Palerme, 25-26 novembre 2003 4
-
8/4/2019 Conf Mahdi 1 (1)
5/44
Palerme, 25-26 novembre 2003 5
From 750 up to 900
-
8/4/2019 Conf Mahdi 1 (1)
6/44
Palerme, 25-26 novembre 2003 6
From 900 up to 1000
-
8/4/2019 Conf Mahdi 1 (1)
7/44
Palerme, 25-26 novembre 2003 7
From 1100 up to 1300
-
8/4/2019 Conf Mahdi 1 (1)
8/44
Palerme, 25-26 novembre 2003 8
From 1300 up to 1500
-
8/4/2019 Conf Mahdi 1 (1)
9/44
Palerme, 25-26 novembre 2003 9
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.
-
8/4/2019 Conf Mahdi 1 (1)
10/44
Palerme, 25-26 novembre 2003 10
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.
-
8/4/2019 Conf Mahdi 1 (1)
11/44
Palerme, 25-26 novembre 2003 11
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)
-
8/4/2019 Conf Mahdi 1 (1)
12/44
Palerme, 25-26 novembre 2003 12
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
-
8/4/2019 Conf Mahdi 1 (1)
13/44
Palerme, 25-26 novembre 2003 13
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.
-
8/4/2019 Conf Mahdi 1 (1)
14/44
Palerme, 25-26 novembre 2003 14
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.
-
8/4/2019 Conf Mahdi 1 (1)
15/44
Palerme, 25-26 novembre 2003 15
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 :
-
8/4/2019 Conf Mahdi 1 (1)
16/44
Palerme, 25-26 novembre 2003 16
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 )
-
8/4/2019 Conf Mahdi 1 (1)
17/44
Palerme, 25-26 novembre 2003 17
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.
-
8/4/2019 Conf Mahdi 1 (1)
18/44
Palerme, 25-26 novembre 2003 18
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.
-
8/4/2019 Conf Mahdi 1 (1)
19/44
Palerme, 25-26 novembre 2003 19
9
8
7
6
5
4
3
2
1
019
18
17
16
15
14
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
47
46
45
44
43
42
41
40
59
58
57
56
55
54
53
52
51
50
Alphabetical numeration
-
8/4/2019 Conf Mahdi 1 (1)
20/44
Palerme, 25-26 novembre 2003 20
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.
-
8/4/2019 Conf Mahdi 1 (1)
21/44
Palerme, 25-26 novembre 2003 21
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
-
8/4/2019 Conf Mahdi 1 (1)
22/44
Palerme, 25-26 novembre 2003 22
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.
-
8/4/2019 Conf Mahdi 1 (1)
23/44
Palerme, 25-26 novembre 2003 23
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.
-
8/4/2019 Conf Mahdi 1 (1)
24/44
Palerme, 25-26 novembre 2003 24
-
8/4/2019 Conf Mahdi 1 (1)
25/44
Palerme, 25-26 novembre 2003 25
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)
-
8/4/2019 Conf Mahdi 1 (1)
26/44
Palerme, 25-26 novembre 2003 26
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.
-
8/4/2019 Conf Mahdi 1 (1)
27/44
Palerme, 25-26 novembre 2003 27
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.
-
8/4/2019 Conf Mahdi 1 (1)
28/44
Palerme, 25-26 novembre 2003 28
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
-
8/4/2019 Conf Mahdi 1 (1)
29/44
Palerme, 25-26 novembre 2003 29
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
-
8/4/2019 Conf Mahdi 1 (1)
30/44
Palerme, 25-26 novembre 2003 30
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
-
8/4/2019 Conf Mahdi 1 (1)
31/44
Palerme, 25-26 novembre 2003 31
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
-
8/4/2019 Conf Mahdi 1 (1)
32/44
Palerme, 25-26 novembre 2003 32
x + 10x = 64
D l t f l b
-
8/4/2019 Conf Mahdi 1 (1)
33/44
Palerme, 25-26 novembre 2003 33
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
-
8/4/2019 Conf Mahdi 1 (1)
34/44
Palerme, 25-26 novembre 2003 34
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.
-
8/4/2019 Conf Mahdi 1 (1)
35/44
Palerme, 25-26 novembre 2003 35
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
-
8/4/2019 Conf Mahdi 1 (1)
36/44
Palerme, 25-26 novembre 2003 36
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
-
8/4/2019 Conf Mahdi 1 (1)
37/44
Palerme, 25-26 novembre 2003 37
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.
-
8/4/2019 Conf Mahdi 1 (1)
38/44
Palerme, 25-26 novembre 2003 38
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
-
8/4/2019 Conf Mahdi 1 (1)
39/44
Palerme, 25-26 novembre 2003 39
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
-
8/4/2019 Conf Mahdi 1 (1)
40/44
Palerme, 25-26 novembre 2003 40
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.
-
8/4/2019 Conf Mahdi 1 (1)
41/44
Palerme, 25-26 novembre 2003 41
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.
-
8/4/2019 Conf Mahdi 1 (1)
42/44
Palerme, 25-26 novembre 2003 42
An equation written in symbols
"Mlplus sevenShayequal eight"
x + 7x = 8.
Conclusion
-
8/4/2019 Conf Mahdi 1 (1)
43/44
Palerme, 25-26 novembre 2003 43
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.
-
8/4/2019 Conf Mahdi 1 (1)
44/44
Palerme, 25-26 novembre 2003 44
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