The Decimal System
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.
This article is excerpted from the
book "Episodes in the Mathematics of Medieval Islam" by J. L. Berggren.
Click on the image above to buy this book.
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
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.
The problem of parallel lines,
posed by Euclid's parallels postulate, received much attention from
Islamic mathematicians throughout the history of medieval Arabic science.
Nasir ad-Din at-Tusi's was probably the most mature treatment of the
problem in Arabic, making sure use of Euclid's definition of parallel
lines as non-secant lines and drawing on the results of his predecessors. TheModernReligion.com
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
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
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
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. Click the Amazon image above to buy this book.