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
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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 in
America.
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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-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.
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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.
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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)
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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
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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.
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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.
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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 :
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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)
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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.
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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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‫‪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
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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.
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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 is
no 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 is
more 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
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)
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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.
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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.
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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
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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
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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
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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
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x² + 10x = 64
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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)
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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.
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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
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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 .
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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.
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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.
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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.
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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.
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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.
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An equation written in symbols
"Māl plus seven Shay
equal eight "
x² + 7x = 8.
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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.
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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
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History of Arab Mathematics