27.5 ON THE NUMBER AND SYMBOL ZERO IN MATHEMATICS
The number zero is the subtle gift of the Hindus of antiquity to mankind. The concept itself was one of the most significant inventions in the ascent of Man for the growth of culture and civilization. To it must be credited the enormous usefulness of its counterpart, the place value system of expressing all numbers with just ten symbols. And to these two concepts we owe all the arithmetic and mathematics based upon them, the great ease which it has lent to all computations for two millenia and the binary system which now lies at the foundation of communicating with computers. Already in the first three centuries C.E. the Hindu ancients were using a decimal positional system, that is, a system in which numerals in different positions represent different numbers and in which one of the ten symbols used was a fully functional zero. They called it ‘shUnya’. The word and its meaning ‘void’ were obviously borrowed from its use in philosophical literature. Though the Babylonians used a special symbol for zero as early as the 3rd century B.C.E., they used it only as a place holder; they did not have the concept of zero as an actual value. It appears the Maya civilisation of South America had a zero in the first century C.E. but they did not use it in a fixed base system. Before zero was invented, the art of recokoning remained an exclusive and highly skilled profession. It was difficult to distinguish, say, 27, 207, 270, 2007, because the latter three were all written 2 7, with a ‘space’ in between. The positional system is not possible in the Roman numeral system which had no expression or symbol for zero. A number, say, 101,000, would have to be written only by 101 consecutive M’s. The Egyptians had no zero and never reached the idea of expressing all numbers with ten digits. The mathematical climate among the Hindus, however, was congenial for the invention of zero and for its use as the null-value in all facets of calculation, due to four factors:
A notation for powers of 10 upto the power 17 was already in use even from vedic times. Single words have been used to denote the powers of the number 10. The numbers one, ten, hundred, thousand, ten thousand, … are given by the sequence of words in the list: eka, dasha, shata, sahasra, ayuta, laksha, prayuta, koTi, arbuda, abja, kharva, nikharva, mahâ-padma, shankha, jaladhi, antya, mahAshankha, parArdha. Thus the decimal system was in the culture even in the early part of the first millenium B.C.E. The Yajurveda, in its description of rituals and the mantras employed therein, the Mah~bh~rata and the RAmAyaNa in their descriptions of statistics and measurements, used all these words, with total abandon.
Counting boards with columns representing units and tens were in use from very ancient times. The numberless content of an empty column in course of time was symbolized to be ‘nothing’.
The thriving activity in astrology, astronomy, navigation and business during the first few centuries C.E. naturally looked forward to a superior numerical system that lent itself to complicated calculations.
Distinct symbols for the numbers 1 to 9 already existed and the counting system used the base 10 in all its secular, religious and ritual activities. Compare this with the Babylonian numeration which had only three figures, one for 1, one for 10, and one for 100, so that a number, say, 999, would require 27 symbols, namely, nine of each of the symbols.
Of these, the first and fourth factors are probably unique to Hindu culture and contributed most to the thought process that led to the decimal place value notation as well as zero having a value. When exactly the invention of this most modest of all numerals took place, we do not know. The first time it reached Europe was during the Moorish invasion of Spain around 700 C.E. Later, when massive Latin translations of books from Baghdad took place around the close of the first millenium C.E., the concept was found in an arithmetic book dated 820 C.E. by Muhammad Ibn Musa al-Khouarizmi who explained the whole system in great detail. The Arabs themselves had no number system of their own. It was the Hindu system he was explaining. That is why the numerals are called the Indo-Arabic numerals even today. It is however a misnomer to call them so; they are already found on the Rock Edits of Ashoka (256 B.C.E.). In spite of its being so crucial to our living, it took centuries for the western world to appreciate and incorporate this most valuable numeral, zero, in their recording of accounts or in scholarly writings. For by the time ‘Zero’ reached the West, the Dark Ages of the western world had begun. There are traces, however, of its knowledge in Spain in the tenth century C.E. But the final breakthrough of the introduction to the West was by Leonardo of Pisa, through his popular text Liber Abaci, 1202 C.E., The first European book (in French) that used the zero appeared in 1275.
How the ‘shUnya‘ of the Hindus became the Zero of the modern world is interesting. The ‘shUnya’‘ of Sanskrit became the Arabic ‘sifr’ which means empty space. In medieval Latin it manifested as ‘ciphra’ , then in middle English as ‘siphre’, in English as ‘cypher’ and in American as ‘cipher’. In the middle ages, the word ‘ciphra’ evolved to stand for the whole system. In the wake of this general meaning, the Latin ‘zephirum’ came to be used to denote the ‘shUnya’. And that entered English finally as ‘zero’. In medieval Europe some countries banned the positional number system, along with zero, brought by the Arabs whom they considered as heathens. So they took ‘shUnya’ to be a creation of the devil! As a result ‘ciphra’ came to mean a secret code. From this came ‘deciphering’, the resolution of a code.
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