Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi (Arabic: أبو الحسن أحمد بن ابراهيم الإقليدسي) was
a Muslim Arab mathematician, who was active
in Damascus and Baghdad. He wrote the earliest surviving book on
the positional use of the Arabic numerals, Kitab
al-Fusul fi al-Hisab al-Hindi (The Arithemetics of Al-Uqlidisi) around
952. It is especially notable for its treatment of decimal fractions,
and that it showed how to carry out calculations without deletions.
While
the Persian mathematician Jamshīd al-Kāshī claimed to have
discovered decimal fractions himself in the 15th century, J. Lennart Berggrenn
notes that he was mistaken, as decimal fractions were first used five centuries
before him by al-Uqlidisi as early as the 10th century.
A. S. Saidan who
studied al-Uqlidisi's mathematical treatise in detail wrote:
The most remarkable
idea in this work is that of decimal fraction. Al-Uqlidisi uses decimal
fractions as such, appreciates the importance of a decimal sign, and
suggests a good one. Not al-Kashi (d. 1436/7) who treated decimal fractions
in his "Miftah al-Hisab", but al-Uqlidisi, who lived five centuries
earlier, is the first Muslim mathematician so far known to write about decimal
fractions.
Al-Uqlidisi is
a mathematician who is only known to us through two manuscripts on arithmetic, Kitab
al-fusul fi al-hisab al-Hindi and Kitab al-hajari fi al-hisab.
Despite this he is a figure of some importance and has prompted an interesting
scholarly argument among historians of science.
The manuscript of
the Kitab al-fusul fi al-hisab al-Hindi which has survived is
a copy of the original which was made in 1157. An English translation of this
work has been published by Saidan. The manuscript gives al-Uqlidisi's full name
on the front page as well as the information that he composed the text in
Damascus in 952-53. In the introduction al-Uqlidisi writes that he travelled
widely and learnt from all the mathematicians he met on his travels. He also
claimed to have read all the available texts on arithmetic. Other than being
able to deduce a little of al-Uqlidisi's character from his writing, we have no
other information on his life.
The Kitab
al-fusul fi al-hisab al-Hindi of al-Uqlidisi is the earliest surviving
book that presents the Hindu system. In it al-Uqlidisi argues that the system
is of practical value:
Most arithmeticians are
obliged to use it in their work: since 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.
This treatise on arithmetic is in four parts. The aim
of the first part is to introduce the Hindu numerals, to explain a place value
system and to describe addition, multiplication and other arithmetic operations
on integers and fractions in both decimal and sexagesimal notation.
The part second collects arithmetical methods given by earlier mathematicians
and converts them in the Indian system. For example the method of casting out
nines is described.
The third part of the
treatise tries to answer to the standard type of questions that are asked by
students: why do it this way ... ?, how can I ... ?, etc. There is plenty of
evidence here that al-Uqlidisi must have been a teacher, for only a teacher
would know understand the type of problem that a beginning student would
encounter.
The fourth part has
considerable interest for it claims that up to this work by al-Uqlidisi the
Indian methods had been used with a dust board. A dust board was used because
the methods required the moving of numbers around in the calculation and
rubbing some out as the calculation proceeded. The dust board allowed this in
the same sort of way that one can use a blackboard, chalk and a blackboard
eraser. However, al-Uqlidisi showed how to modify the methods for pen and paper
use.
Al-Uqlidisi's work is
historically important as it is the earliest known text offering a direct
treatment of decimal fractions. It is here that the scholarly argument referred
to above arises. At one time it was thought that Stevin was the first
to propose decimal fractions. Further research showed that decimal fractions
appeared in the work of al-Kashi, who was then credited with this
extremely important contribution. When Saidan studied al-Uqlidisi's Kitab
al-fusul fi al-hisab al-Hindi in detail he wrote:
The most remarkable
idea in this work is that of decimal fraction. Al-Uqlidisi uses decimal
fractions as such, appreciates the importance of a decimal sign, and suggests a
good one. Not al-Kashi (d. 1436/7) who
treated decimal fractions in his "Miftah al-Hisab", but al-Uqlidisi,
who lived five centuries earlier, is the first Muslim mathematician so far
known to write about decimal fractions.
Following Saidan's paper, some historians went even
further in attributing to al-Uqlidisi the complete credit for giving the first
complete description and applications of decimal fractions. Rashed, however,
although he does not wish to minimise the importance of al-Uqlidisi's
contribution to decimal fractions, sees it as :
... preliminary to its history, whereas al-Samawal's text already
constitutes the first chapter.
The argument depends on how one interprets the
following passage in al-Uqlidisi's treatise. He explains how to raise a number
by one tenth five times:
... we want to raise a
number by its tenth five times. We write down this number as usual; write it
down again below moved one place to the right; we therefore know its tenth,
which we add to it. So was have added its tenth to this number. We put the
resulting fraction in front of this number and we move it to the unit place
after marking it [with the ' sign he uses for the decimal point] thus.
We add its tenth and so on five times.
Saidan (writing in) sees in this passage that
al-Uqlidisi has fully understood the idea of decimal fractions, saying that
earlier authors:
... rather mechanically
transformed the decimal fraction obtained into the sexagesimal system, without
showing any sign of comprehension of the decimal idea. ... In all operations
where powers of ten are involved in the numerator or the denominator, [al-Uqlidisi] is
well at home.
On the other hand Rashed sees this passage rather
differently:
...
unlike al-Samawal, al-Uqlidisi never formulates the idea of completing the
sequence of powers of ten by that of their inverse after having defined the
zero power. That said, in the passage just quoted, three basic ideas emerge
whose intuitive resonance may have misled historians; what they thought was a
theoretical exposition was merely understood implicitly, and, as a result, they
have overestimated the author's contribution to decimal fractions.
The two points of view are almost impossible to decide
between since what we are looking at is the development of the idea of decimal
fractions by different mathematicians, each contributing to its understanding.
To take a particular text as the one where the idea appears for the first time
in its entirety must always be a somewhat arbitrary decision. There is no
disagreement on the fact that al-Uqlidisi made a major step forward.
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