some elements in the history of arab mathematics mahdi abdeljaouad from arithmetic to algebra part 1

Some elements in the history of Arab mathematics

Mahdi ABDELJAOUAD

From arithmetic to algebraPart 1

Palerme, 25-26 novembre 2003 2

Summary

Introduction Domains studied by ArabsArithmetic and number theoryAlgebraConclusion


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.


From 750 up to 900


From 900 up to 1000


From 1100 up to 1300


From 1300 up to 1500


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-Kashi5. Combinatorics : Ibn al-Muncim

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


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.


Speculative arithmetic

Inspired from Aristotle philosophyTwo approaches :

1. Euclid’s ElementsBooks VII – VIII and IX

2. Pythagoras through Nicomachus of Gerase’s Introduction to Arithmetic.

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


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


Abu'l-Wafa (d.998) BagdadBook on what Is necessary from the science of

arithmetic for scribes and businessmenThis 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.


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 permits for ships on the river, merchants on the roads).

Part VII: Further business topics.


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 :


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)



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 = يجـ = 13 ; ز = 7 ; جـ = 3 ; ب = 2; ا ;

كب = 27 .• Indian numeration : It uses Arabic numerals

from 1 to 59, and also 0 in medial position.


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.


ط9

ح8

ز7

و6

ه5

د4

جـ3

ب2

أ1

صفـر0

يـط19

يـح18

يـز17

يـو16

يـه15

يـد14

يجـ13

يب12

يـا11

ي10

كـط29

كـح28

كـز27

كـو26

كـه25

كـد24

كجـ23

كب22

كــا21

كـ20

لـط39

لـح38

لـز37

لـو36

لـه35

لـد34

لجـ33

لب32

ال31

ل30

مـط49

مـح48

مـز47

مـو46

مـه45

مـد44

مجـ43

مب42

مـا 41

م40

نـط59

نـح58

نـز57

نـو56

نـه55

نـد54

نجـ53

نب52

نـا51

ن50

Alphabetical numeration



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.


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



... 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.



… 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.


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)


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 : 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.


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.


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


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


Six classes of equations

• 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. 6. Shay and numbers equal Māl : 100x + 2 = x²


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


x² + 10x = 64


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 + abu-Kamil + 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)


Arab algebraAbu 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.


Arab algebraAl-Karāji (953 Bagdad - 1029)

Kitab al-Fakhri and Kitab al-Badic fil JabrHe 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


Arab algebraAs-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 .


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 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.


Arab algebraOmar 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.


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 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.


An algebra textbook in 1387Ibn 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.


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.


An equation written in symbols

"Māl plus seven Shay equal eight "

x² + 7x = 8.


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.


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

some elements in the history of arab mathematics mahdi abdeljaouad from arithmetic to algebra part 1

Documents

science of arithmetic

business arithmetic

arithmetic needs

speculative arithmetic

star worshipper slide

arabs arithmetic

euclidian arithmetic

science of numbers