It is worth starting with a quote by Albert Einstein. He quotes

*“We owe a lot to the Indians, who taught us how to count, without which no worthwhile scientific discovery could have been made.” *

We now turn to the second aspect of the Indian number system, namely the fact that it was a place-value system with the numerals standing for different values depending on their position relative to the other numerals. The Babylonians had a place-value system as early as the 19^{th} century BC but the Babylonian systems were to base 60. The Indians were the first to develop a base 10 positional systems.

The mathematical culture of the Babylonians is known to us from texts dating from the Ancient Babylonian (1800-1600 B.C.) and the Seleucidae epoch (305-64 B.C.). A comparison of these texts shows that no radical changes took place in the mathematics of the Babylonians during this time.

When we compare the Babylonian positional system with our modern one we see that the uncertainty in the multiplier 60 is a result of the absence of a character for zero, which we would add the necessary number of times at the end of a whole number or the beginning of a fraction. Another result of the absence of the zero is an even more serious uncertainty in interpreting a numerical notation that in our system requires a zero in an intermediate position. In the Babylonian notation, how can the number 3,601 = 1 . 60^{2} + 0 . 60 + 1 be distinguished from the number 61 = 1. 60 + 1 ? Both of these numbers are represented by two units (ones). Sometimes this kind of uncertainty was eliminated by separating the numbers, leaving an empty place for the missing position. But this method was not used systematically and in many cases a large gap between numbers did not mean anything. In the astronomical tables of the Seleucidae epoch one finds the missing position designated by means of a character resembling our period. We do not find anything of the sort in the Ancient Babylonian epoch. But how were the ancient Babylonians able to avoid confusion?

The solution to the riddle is believed to consist in the following. The early Babylonian mathematical texts which have come down to us are collections of problems and their solutions, unquestionably created as learning aids. Their purpose was to teach practical methods of solving problems. But not one of the texts describes how to perform arithmetic operations, in particular the operations, complex for that time, of multiplication and division. Therefore, we assume that the students knew how to do them. It is improbable that they performed the computations in their head; they probably used some abacus-like calculating device. On the abacus, numbers appear in their natural, spontaneously positioned form and no special character for the zero is needed; the groove corresponding to an empty position simply remains without pebbles. Representation of a number on the abacus was the basic form of assigning a number, and there was no uncertainty in this representation. The numbers given in cuneiform mathematical texts serve as answers to stage-by-stage calculations, so that they could be used to check correctness during the solution. The student made the calculations on the abacus and checked them against the clay tablet. Clearly the absence of a character for empty positions did not hinder such checking at all. When voluminous astronomical tables became widespread and were no longer used for checks but rather as the sole source of data, a separation sign began to be used to represent the empty positions. But the Babylonians never put their own ”zero” at the end of a word; it is obvious that they perceived it only as a separator, and not as a number.

The modern number system was invented by the Indians at the beginning of the sixth century A.D. The system proved to be consistent, economical, not in contradiction with tradition, and extremely convenient for computations.

The Indians passed their system on to the Arabs. The positional number system appeared in Europe in the twelfth century with translations of al-Kwarizmi’s famous Arab arithmetic. It came into bitter conflict with the traditional Roman system and in the end won out. As late as the sixteenth century, however, an arithmetic textbook was published in Germany and went through many editions using exclusively ”German,” which is to say Roman, numerals. It would be better to say ”numbers,” because at that time the word ”numerals” was used only for the characters of the Indian system. Decimal fractions began to be used in Europe with Simon Stevin (1548-1620).

The oldest dated Indian document which contains a number written in the place-value form used today is a legal document dated 346 in the Chhedi calendar which translates to a date in our calendar of 594 AD. This document is a donation charter of Dadda III of Sankheda in the Bharukachcha region.

Many other charters have been found which are dated and use of the place-value system for either the date or some other numbers within the text. These include:

- a donation charter of Dhiniki dated 794 in the Vikrama calendar which translates to a date in our calendar of 737 AD.
- an inscription of Devendravarman dated 675 in the Shaka calendar which translates to a date in our calendar of 753 AD.
- a donation charter of Danidurga dated 675 in the Shaka calendar which translates to a date in our calendar of 737 AD.
- a donation charter of Shankaragana dated 715 in the Shaka calendar which translates to a date in our calendar of 793 AD.
- a donation charter of Nagbhata dated 872 in the Vikrama calendar which translates to a date in our calendar of 815 AD.
- an inscription of Bauka dated 894 in the Vikrama calendar which translates to a date in our calendar of 837 AD.

The first inscription which is dated and is not disputed is the inscription at Gwalior dated 933 in the Vikrama calendar which translates to a date in our calendar of 876 AD. There are other indirect evidences that the

Indians developed a positional number system as early as the first century AD. The evidence is found from inscriptions which, although not in India, have been found in countries which were assimilating Indian culture. Another source is the Bakhshali manuscript which contains numbers written in place-value notation. The problem here is the dating of this manuscript, a topic which is examined in detail in our article on the Bakhshali manuscript.

Some historians believe that the Babylonian base 60 place-value system was transmitted to the Indians via the Greeks. The real theory here is that these ideas were transmitted to the Indians who then combined this with their own base 10 number systems which had existed in India for a very long time.

…. Continues (Bijaganit, The Modern Algebra)

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*Mathematics*

Posted on August 24, 20110