54 Some elements in the history of Arab mathematics

The roots of al-Khwarizmi's algebra Yes : Indian origin • He has written a treatise on Hindu-Arabic numerals and an astronomical table Sindhind zij No : it is not from Indian origin Rode

Some elements in the history of

Arab mathematics

From arithmetic to algebra

Part 2

Luis Radford 1996

The roots of algebra : arithmetic or geometry ?     

• He discusses some hypotheses on the

origins of Diophantus’s algebraic ideas

• He suggests that the historical conceptual

structure of the concept of unknown and the concept of variable are quite different

• He raises some questions about teaching

algebra in high schools

geometry”

• Interpretation of Radford : The procedure of

solving some types of problems consists of

an arithmetical method of false position,

based on an idea of proportionality

Luis Radford 1996

Unknowns and variables

• While the unknown does not vary, a variable

designate a quantity whose value can change 

• in Diophantus' Arithmetica "the kew concept of

unknown (the arithme) is not represented

geometrically .

• Nichomacus uses empirical set values, that is a

concrete-arithmetical treatment of numbers, 

• Diophantus deals with abstract set values. "The

propositions in his theory of polygonal numbers are supported by a deductive organization He is concerned about variables, not through the

concept of function but through the concept of formula 

Luis Radford 1996

Implications for the teaching of Algebra

1 Introducing cut-and-paste algebra to facilitate

acquisition of basic algebraic concepts.

2 Introducing certain elements of the "false

position" method prior to introducing the concept

of the unknown in solutions of word problems.

3 Use proportional thinking as a useful link to

algebraic thinking.

4 Introduce an appropriate distinction between the

concepts of unknown and variables, using

historical ideas.

5 Try to develop the concept of abstract set value in a 

deductive context as a prerequisite to a deep learning of 

The roots of al-Khwarizmi's algebra

The roots of al-Khwarizmi's algebra

Yes : Indian origin

• He has written a treatise on Hindu-Arabic numerals and an

astronomical table Sindhind zij

No : it is not from Indian origin (Rodet, 1878)

• Al-Khawarizmi does not know negative numbers Hindus

use them abondantly as they use « 0 »

• Al-Khwarizmi "al-jabr" means "completion", it is the

process of removing negative terms from an equation

• There are six canonical forms for Arabic equations, while

Indian had only one canonical equation

The roots of al-Khwarizmi's algebra

Al-Khwarizmi "al-jabr" means "completion", it is the process

of removing negative terms from an equation

50x2 + 300 - 6x = 10x - 100 - x2

Arab method : Complete each side by removing

negative terms

50x2 + 100 + x2 + 100 = 10x + 300 + 6x

Indian method : Sustract from right side the unknown

and from the left side the number even if it is negative,

so all unknowns are on the left and numbers on the right.

50x 2 + 300 - 6x - 300 - 10x - (-x 2 ) = 10x -100 - x 2 - 300 - 10x - (-x 2 )

The roots of al-Khwarizmi's algebra

There are six canonical forms for Arabic equations, while

Indian had only one canonical equation

Arab canonical equation :

with a, b and c strictly positive numbers

Indian canonical equation :

ax2 ± bx = ±cwith a, b and c positive numbers or nil

The roots of al-Khwarizmi's algebra

Yes : Greek origin (Rodet, 1878) : "he is purely and

simply a disciple of the Greek School"

(a)He never uses negative numbers

(b)He gives cut-and-paste geometrical proofs

for solutions of quadratic equations

No : (Gandz,1931) : "Euclid's Elements" in

their spirit and letter are entirely unknown

to al-Khwarizmi who has neither definitions, nor axioms, nor postulates, nor any demonstration of the Euclidean kind.« May be : Rashed (1997) suggested an

intermediary position : al-Khwarizmi's :

"treatment was very probably inspired by

The roots of al-Khwarizmi's algebra

Babylonian tradition : Hoyrup 

• Babylonian algebra "did not deal with known and

unknown numbers represented by words or symbols Strictly speaking it did not deal with numbers at all, but with mesurable line segments

• The operations used to define and solve these

problems were not arithmetical but concrete and

geometrical

• In all cases, the geometry involved can be

characterized as "nạve" This nạve geometry is

fairly similar to the proofs given by al-Khwarizmi

Palerme, 25-26 novembre 2003 13

The roots of al-Khwarizmi's algebra

Epistemological arguments 1

When we look at the structure of al-Jabr wa

al-muqabala, we notice specific features

who are common to Hawa'i arithmetic and

absent from theoretical arithmetic and from Indian arithmetic :

1 He starts by a very short description of

numbers

2 Then he introduces all tools of algebra

3 This textbook is a completely rhetorical

algebra

4 He uses unit fractions and expresses all

fractions in function of them

5 He works in numerical settings : positive

7 But he stays in the realm of geometric

conception of numbers (lines, surfaces and solids)

8 He uses cut-and-paste geometrical

The roots of al-Khwarizmi's algebra

Our conclusion

It is clear from the contents of al-Khwarizmi's algebra that it is in fact part of the usual arithmetic textbooks The author intention is to help people to solve their practical problems He had collected

techniques people used to transmit orally without it

being recorded in any book

The roots of Arabic algebra

1 A numerical origin

2 Babylonian geometrical influences

3 Euclidian geometrical influences

4 A geometric theory of equations

5 An arithmetic of polynomial expressions

6 Algebra becomes a section of textbooks of Indian

arithmetic

The roots of Arabic algebra

The epistemical status of algebra did not stabilise for

centuries

1 The philosopher al-Farabi (d.950), for example,

considered arithmetic as a science having a

theoretical part and a practical one ; but for him

algebra was not a science but one technique

common to arithmetic and geometry

2 Ibn Sina (Avicenna 980-1038) subdivised

arithmetic into Indian calculus and the art of

algebra", that is into two different subjects

The roots of Arabic algebra

A business school origin

No : Abu l-Wafa (d.998)

• For al-Khwarizmi, algebra is a section of business

arithmetic

• However Abu l-Wafa who is the author of the most

read business textbook says : All transactions are

solved by the use of one unique Euclidian proposition, the one requiring finding an unknown placed in a proportion

The roots of Arabic algebra

A business school origin

Yes : al-Karaji (d.1029) : al-Kafi fi al-hisab

• He considers the first one - when the unknown is in

Al-Karaji is also the author of two important books of

algebra al-Fakhri and al-Badii

The roots of Arabic algebra

Babylonian geometrical influences

treatment of numbers, and does not deal with abstract  set values.

• references  to  Euclid  are  non-existent  and  his  direct 

influence never acknowledged. 

• However,  al-Khwarizmi's  innovation  with  respect  to 

Babylonians  consists  in  introducing  the  unknown  into  account through the problem solving-procedure, being 

of the object of calculation

The roots of Arabic algebra

Euclidian geometrical influences : Thabit ibn Qurra (826 – 900)

(a) al-Khwarizmi’s pragmatic proofs which are not based

on Euclid's Elements did not please him.

(b) He then proofs them referring directly to Euclid,

(c) He has a more general and more intellectual character

than that of al-Khwarizmi In fact he deals with abstract set values and he never gives a numerical example

The roots of Arabic algebra

Euclidian geometrical influences :

Abu-Kamil (850 – 930)

(a) al-Khwarizmi’s pragmatic proofs are said « visual »

(b) He introduces proofs referring directly to Euclid

(c) But he has a concrete-arithmetical treatment of numbers, 

and  does  not  deal  with  abstract  set  values.  Generic    examples (implicite induction).

(d) He enriches the toolbox by including in it numerous

algebraic identities proved geometrically and never gives a numerical example

The roots of Arabic algebra

Euclidian geometrical influences : Al-Karaji (d 1029), Omar al-Khayyam (1048-1131) and

as-Samaw'al (1130-1174)

(a) They take up the geometric proofs of al-Khwarizmi

and those of Abu Kamil and extend them systematically

(b) They complete the algebraic toolbox by placing in it all

the arithmetic propositions on whole numbers, on fractions and on quadratic irrationals, adding algebraic identities, all proved geometrically using Books II and

VII to X of the Elements

The roots of Arabic algebra

A geometric theory of equations Omar al-Khayyam

(1048-1131) ,

1 He classify all third degree equations and solve

them

2 For each of the types, he finds a construction of a

positive root by the intersection of two conics

3 He works entirely within a Euclidean framework

Palerme, 25-26 novembre 2003 25

The roots of Arabic algebra

An arithmetic of polynomial expressions

1 The major obstacle encountered in legitimizing the

algebraic reasoning concerns the nature of the

product of numbers

2 Al-Karaji gets around this difficulty by

creating the field of "known numbers" in parallel with the field of "unknown numbers"

“Operating in the field of knowns keeps them in this field no matter what the operation”

3 It is thus no longer a matter of reasoning

on geometric figures but directly on

The roots of Arabic algebra

An arithmetic of polynomial expressionsAl-Karaji (d 1029) and as-Samaw'al (1130-1174)

Representing polynomials by tables

Palerme, 25-26 novembre 2003 27

The roots of Arabic algebra

Algebra becomes a section of textbooks

of Indian arithmetic

The legacy of al-Karaji is found in later arithmetic

textbooks written in North Africa (beginning in the XIIth century) :They combine three heritages :

(1) Indian arithmetic

(2) algebra as an autonomous science,

with no geometrical proofs

(3) Business textbooks heritage how to

find an unknown from known quantities

or numbers :

- the unknown placed in a proportion

- the double false position method

with its own way of reasoning and its proper proofs. (4)This autonomy is illustrated by business textbooks 

of North Africa combining Indian arithmetic with 

algebra