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In this section we discuss the history of the decimal system and the notion of zero as a place-value as well as a digit. We will see that the decimal system, together with zero as a place value were known, not only in post-Vedic India, but also earlier in the Babylonian system and in the contemporary Mayan civilization. In all likelihood, there could have been independent development of this system in all these places. However the notion of zero as an integer is quite clearly Indian in origin and Brahmagupta appears to be the first person to consider zero as an integer.

The importance of zero as a place marker can be understood from the fact that if we did not have it, then there would be no way of distinguishing between the numbers $201$ and $21$ (in any base). However, the Babylonians did not have it for over 1000 years and from all available evidence, it appears that they did not have any problems with the confusion that must have reigned.

A hybrid of the alphabetic system and the positional number systems appears in Babylonian tablets from around 2000 BCE - this system called the Mari system (named after the place where the tablets were found) was later refined to a positional system by the Babylonians by around 700 BCE. However, the system had its limitations, e.g., 36 had to be represented by three tens juxtaposed together followed by six 1's juxtaposed together.

In a tablet, dating from around 700 BCE, excavated at Kish, Iraq, the first evidence of using zero is found. The scribe Bêl-bân-aplu, who wrote the tablet used three hooks to denote zero. It is from tablets written between the sixth and third century BCE we find that the Babylonians used a variety of symbols to denote zero in a number. They used a single wedge or two wedge symbols to denote zero. Thus by around 400 BCE we find instances like $2\mbox{\lq\lq }1$ to distinguish it from $21$. However, the use of zero in the units place of a number (e.g., $21\mbox{\lq\lq }$) or as the `last' digits of a number is surprisingly absent. Probably one relied on the context to understand the missing zeros from the `end' of a number⁶.

Significantly, a tablet found in Susa reads ``20 minus 20 comes to ... you see?" [Ifrah 2000] - this allusion to zero is quite different from the use of zero as a symbol in expressing numbers. However, this notion of zero as a number is not developed and neither is there anything which would suggest that this was not just a passing fancy of a scribe.

The Greek mathematicians did not have a positional number system and as such they did not need zero as a place-value symbol. Their number system, called the Attic system, dating to around 500 BCE had special symbols for each of the numbers 1, 5, 10, 50, 100, 500, 1000, 5000, 10000 and 50000. Thus the number $3202$ was expressed as $XXXHHII$. The connection between the Attic system and the Roman numerals is immediate.

However, the Greek astronomers, who needed large numbers and were clearly hampered by the fact that large numbers were too cumbersome to be expressed in the Attic system, used the Babylonian sexagesimal system. Thus in Ptolemy's treatise on astronomy, Almagest, written around 130 AD, we find the use of a sexagesimal number system together with the symbol 0 to denote the `empty place' in a number. This is probably the first occurrence of the symbol $0$, although other symbols for zero were also used by the Greek astronomers during this period.

The oldest known writing in the Indian sub-continent is from the Indus valley civilization (2500 - 1500 BCE), however until the script is deciphered weare in no position to determine its relationship with the Indian number systems. It is from the Brahmi edicts during Ashoka's reign (273 - 235 BCE) that the earliest numerical notation is observed, that too rather fragmentary (only the numbers 1,2,4 and 6). Some more numbers appear in the 2nd century BCE during the Shunga and Magadha dynasties. However, by the first and second century AD, more complete number systems have been found in many places in India. There is some debate as to the origin of the Brahmi numerals and whether there were any influences from outside, however Ifrah [2000] taking into account the ``universal constants of both psychology and paleography'' demonstrates that the ``Brahmi numerals were autochthonous, that is to say, their formation was not due to any outside influence''⁷.

Since there had been a lot of influence of Greek astronomy in Indian astronomy and astrology (e.g., Indian names of zodiacal signs and various astronomical terms are Greek in origin, as well as the ratio of the length of the longest day to that of the shortest day which is given as 3:2 - a ratio more true of Babylonia than anywhere in India) many historians have concluded that the place-value system of the Indian numerals have their origin in the Babylonian system (via the Greek astronomical texts). Although this hypothesis has not been ruled out, Ifrah [2000] raises a serious objection to it.

In the Surya Siddhanta (600 AD) we see numbers like 488,203 and 232,238 expressed, by just their digits spelt out one after the other, without taking recourse to the magnitude of the digit in the number. Zero finds its place in the place-value system, although not in the `circular' symbol we know of it now. In fact, Lokavibhaga (458 AD), the Jain cosmological text is the earliest known Indian text to have used the place-value system together with zero. Aryabhata (500 AD) constructed an ingenious method of recording numbers based on consonants and vowels of the Devanagari alphabet. Moreover his method of calculating square and cube roots indicate a use of the place value system.

It is from here that the Indian mathematicians take the profound step to consider zero as an integer and carry on mathematical operations with it. While Varahamihira (575 AD) mentions the use of zero in mathematical operations, Brahmagupta (628 AD) elaborates on these operations in Brahmasphutasiddhanta. Brahmagupta defines zero as that quantity which is obtained when a number is subtracted from itself and he goes on to elaborate on the procedure of addition, subtraction, multiplication and division with zero. Thus he writes

``The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero.
A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero.''

However, division by zero was problematic

``Positive or negative numbers when divided by zero is a fraction the zero as denominator. Zero divided by negative or positive numbers is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.''

Mahavira (830 AD) in his work Ganitasangraha elaborates on Brahmagupta's work and realising the obvious inadequacy of Brahmagupta's explanation about division by zero states that ``A number remains unchanged when divided by zero''. Later Bhaskara (1150 AD) tries to correct this mistake of Mahavira in his book Lilavati.

``A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.''

Here though we have a glimpse of the use of the mathematical notion on infinity, Bhaskara still couldn't arrive at the modern concept that division by zero is not allowed⁸. This would continue to plague Arab and subsequently European mathematicians who learnt from various translations of these Indian works the notion of zero as an integer⁹.

Finally, to complete this brief history of zero, and reinforcing our argument against monogenetic origins of ideas in the sciences of ancient times, it should be noted that the Mayan civilization by 665 AD had a base 20 place-value number system with a symbol and use of zero. It is also recorded that they had used zero prior to having a place-value number system. The calendar system of the Mayans and their development of astronomy would seem to suggest that the Mayans must have been quite capable in their mathematics too. Even the wildest Euro/Indo centrist would have to admit that the Mayan civilization developed independently, without any influence from the `old world'.