The Decimal and Islamic Mathematics

obmar · The Decimal and Islamic Mathematics

The Decimal and Islamic Mathematics
12/17/2006 - - Article Ref: IC0612-3192
Number of comments: 8
By: J. L. Berggren
IslamiCity* -

http://www.islamicity.com/articles/Articles.asp?ref=IC0612-3192

Muslim mathematicians were the first people to write
numbers the way we do, and, although we are the heirs
of the Greeks in geometry, part of our legacy from the
Muslim world is our arithmetic. This is true even if
it was Hindu mathematicians in India, probably a few
centuries before the rise of Islamic civilization, who
began using a numeration system with these two
characteristics:

The numbers from one to nine are represented by nine
digits, all easily made by one or two strokes.
The right-most digit of a numeral counts the number of
units, and a unit in any place is ten of that to its
right. Thus the digit in the second place counts the
number of tens, that in the third place the number of
hundreds (which is ten tens), and so on. A special
mark, the zero, is used to indicate that a given place
is empty.

These two properties describe our present system of
writing whole numbers, and we may summarize the above
by saying the Hindus were the first people to use a
cipherized, decimal, positional system, "Cipherized"
means that the first nine numbers are represented by
nine ciphers, or digits, instead of accumulating
strokes as the Egyptians and Babylonians did, and
"decimal" means that it is base 10. However, the
Hindus did not extend this system to represent parts
of the unit by decimal fractions, and since it was the
Muslims who first did so, they were the first people
to represent numbers as we do. Quite properly,
therefore, we call the system "Hindu-Arabic".

As to when the Hindus first began writing whole
numbers according to this system, the available
evidence shows that the system was not used by the
great Indian astronomer Aryabhata (born in A.D. 476),
but it was in use by the time of his pupil, Bhaskara
I, around the year A.D. 520. (See Van der Waerden and
Folkerts for more details.)

News of the discovery spread, for, about 150 years
later, Severus Sebokht, a bishop of the Nestorian
Church ( one of the several Christian faiths existing
in the East at the time), wrote from his residence in
Keneshra on the upper Euphrates river as follows:

I will not say anything now of the science of the
Hindus, who are not even Syrians, of their subtle
discoveries in this science of astronomy, which are
even more ingenious then those of the Greeks and
Babylonians, and of the fluent method of their
calculation, which surpasses words. I want to say only
that it is done with nine signs. If those who believe
that they have arrived at the limit of science because
they speak Greek ad known these things they would
perhaps be convinced, even if a bit late, that there
are others who know something, not only Greeks but
also men of a different language.

It seems, then, that Christian scholars in the Middle
East, writing only a few years after the great series
of Arab conquests had begun, knew of Hindu numerals
through their study of Hindu astronomy. The interest
of Christian scholars in astronomy and calculation
was, in the main, due to their need to be able to
calculate the date of Easter, a problem that
stimulated much of the Christian interest in the exact
sciences during the early Middle Ages. It is not a
trivial problem, because it requires the calculation
of the date of the first new moon following the spring
equinox. Even the great nineteenth-century
mathematician and astronomer C.F. Gauss was not able
to solve the problem completely, so it is no wonder
that Severus Sebokht was delighted to find in Hindu
sources a method of arithmetic that would make
calculation easier.

We can perhaps explain the reference to the "nine
signs" rather then the ten as follows: the zero
(represented by a small circle) was not regarded as
one of the digits of the system but simply a mark put
in a place when it is empty, i.e. when no digit goes
there. The idea that zero represents a number, just as
any other digit does, is a modern notion, foreign to
medieval though.

With this evidence that the Hindu system of numeration
had spread so far by the year A.D. 662, it may be
surprising to learn that the earliest Arabic work we
know of explaining the Hindu system is one written
early in the ninth century whose title may be
translated as The Book of Addition and Subtraction
According to the Hindu Calculation. The author was
Muhammad ibn Musa al-Khwarizmi who, since the was born
around the year A.D. 780, probably wrote his book
after A.D. 800.

We mentioned in Chapter 1 that al-Khwarizmi, who was
one of the earliest important Islamic scientists, came
from Central Asia and was not an Arab. This was not
unusual, for, by and large, in Islamic civilization it
was not a man's place (or people) of origin, his
native language, or (within limits) his religion that
mattered, but his learning and his achievements in his
chosen profession.

The question arises, however, where al-Khwarizmi
learned of the Hindu arithmetic, given that his home
was in a region far from where Bishop Sebokht learned
of Hindu numerals 150 years earlier. In the absence of
printed books and modern methods of communication, the
penetration of a discovery into a given region by no
means implied its spread to adjacent regions. Thus
al-Khwarizmi may have learned of Hindu numeration not
in his native Kharizm but in Baghdad, where, around
780, the visit of a delegation of scholars from Sind
to the court of the Caliph al-Mansur led to the
translation of Sanskrit astronomical works. Extant
writings of al-Khwarizmi on astronomy show he was much
influenced by Hindu methods, and it may be that it was
from his study of Hindu astronomy that he learned of
Hindu numerals.

Whatever the line of transmission to al-Khwarizmi was,
his work helped spread Hindu numeration both in the
Islamic world and in the Latin West. Although this
work has not survived in the Arabic original
(doubtless because it was superseded by superior
treatises later on), we possess a Latin translation,
made in the twelfth century A.D. From the
introduction to this we learn that the work treated
all the arithmetic operations and not only addition
and subtraction as the title might suggest. Evidently
al-Khwarizmi's usage is parallel to ours when we speak
of a child who is studying arithmetic as "learning his
sums".

This article is excerpted from the book "Episodes in
the Mathematics of Medieval Islam" by J. L. Berggren.
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