54 Some elements in the history of Arab mathematics

Some elements in the history of
Arab mathematics
From arithmetic to algebra
Part 2


Summary
Introduction
The roots of algebra : arithmetic or geometry
The roots of al-Khwarizmi's algebra
The roots of Arabic algebra
Conclusion

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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.

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3


Luis Radford 1996

• Classical interpretation : Babylonian math know
the general formula for solving quadratic
equations without being able to express it as
such, because they lacked the symbols to do so.
• Interpretation of Hoyrup (1990) : Babylonian
algebra cannot have been arithmetical. Instead it
appears to have been “naïve” , non deductive
geometry, consisting of “cut-and-paste
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.
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4


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
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5


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 concept of variable.
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6


The roots of al-Khwarizmi's algebra
 

•
•
•
•

Indian origin
Greek origin
Babylonian tradition
Our conclusion : Epistemological arguments

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7


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.
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8


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.
50x2 + 300 - 6x - 300 - 10x - (-x2) = 10x -100 - x2 - 300 - 10x - (-x2)
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9


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 :
ax2 = bx
ax2 = c
bx = c

ax2 + bx = c
ax2 + c = bx
x2 = bx + c

with a, b and c strictly positive numbers

Indian canonical equation :
ax2 ± bx = ±c
with a, b and c positive numbers or nil

Palerme, 25-26 nove

10


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
recent knowledge of Euclid's Elements".
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11


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 "naïve"... This naïve geometry is
fairly similar to the proofs given by al-Khwarizmi ..
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12


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.
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13
5. He works in numerical settings : positive


6.
7.

8.
9.

The roots of al-Khwarizmi's algebra
Epistemological arguments 2
He deals with "shay“ as if it was a known
number.
But he stays in the realm of geometric
conception of numbers (lines, surfaces and
solids).
He
uses
cut-and-paste
geometrical
arguments.
He gives applications of algebra techniques to four
kinds of problems :
(1) Numerical ones (for example : division of ten in
two parts).
(2) Commercial problems.
(3) Mensurations of geometrical figures.
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14
(4) Inheritance problems (this section is about the half


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.

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15


The roots of Arabic algebra
1.
2.
3.
4.
5.
6.

A numerical origin
Babylonian geometrical influences
Euclidian geometrical influences
A geometric theory of equations
An arithmetic of polynomial expressions
Algebra becomes a section of textbooks of Indian
arithmetic

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16

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.
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17


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 .
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18

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
a proportion
• He includes a section of algebra in his business
arithmetic textbook.
• He completes by many solved numerical and
practical problems.
Al-Karaji is also the author of two important books of
algebra al-Fakhri and al-Badii .
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19


The roots of Arabic algebra
Babylonian geometrical influences
•
•
•

al-Khwarizmi's approach is a concrete-arithmetical
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.

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20


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.
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21


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.
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22


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.
Palerme, 25-26 nove

23


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

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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
Palerme,
25-26 nove
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on geometric
figures

but directly on