We may now summarise the various evidences regarding the early use of the place-value notation in India:
(1) The earliest palaeographic record of the use of the place-value system belongs to the close of the sixth century A.D.
(2) The earliest use of the place-value principle with the word numerals belongs to the second or the third century A.D. It occurs in the Agni-Purâna, the Bakhshâlî Manuscript and the Pulisa-siddhânta.
(3) The earliest use of the place-value principle with the letter numerals is found in the works of Bhâskara I about the beginning of the sixth century A.D.
(4) The earliest use of the place-value system in a mathematical work occurs in the Bakhshâlî Manuscript about 200 A.D. It occurs in the Âryabhatîya composed in 499 A.D., and in all later works without exception.
(5) References to the place-value system are found in literature from about 100 B.C. Three references ranging from the second to the fourth century A.D. are found in the Purânas.
(6) The use of a symbol for zero is found in Piṅgala's Chandaḥ-sûtra as early as 200 B.C.
The reader will observe that the literary and non-mathematical works give much earlier instances of the use of the place-value system than the mathematical works. This is exactly what one should expect. The system when invented must have for some time been used only for writing big numbers. A long time must have elapsed before the methods of performing arithmetical operations with them were invented. The system cannot be expected to occur in a mathematical Page 87 work before it is in a perfect form. Therefore, the evidences furnished by non-mathematical works should, in fact, be earlier than those of mathematical works.
Mathematical works are not as permanent as religious or literary works. The study of a particular mathematical work is given up as soon as another better work comes into the field. In fact, a new mathematical work is composed with a view to removing the defects of and superseding the older ones. It is quite probable that works employing the place-value notation were written before Âryabhaṭa I, but they were given up and are lost. It will be idle to expect to find copies of such works after a lapse of sixteen hundred years.
In Europe and in Arabia it is still possible to find mss. copies of works using the old numerals or a mixture of the old numerals with the new place-value numerals, but in India absolutely no trace of any such work exists.
In Europe the first definite traces of the place-value numerals are found in the tenth and eleventh centuries, but the numerals came into general use in mathematical text books in the seventeenth century. In India Âryabhaṭa I (499), Bhâskara I (522), Lalla (c. 598), and Brahmagupta (628), all use the place-value numerals. There is no trave of any other system of notation in their works. Following the analogy of Europe, we may conclude, on the evidence furnished by Hindu mathematical works alone, that the place-value system might have been known in India about 200 B.C.
As the literary evidence also takes us to that period, we may be certain that the place-value system was known in India about 200 B.C. Therefore we shall not be much in error, if we fix 200 B.C. as the probable date of invention of the place-value system and Page 88 zero in India. It is possible that further evidence may force us to fix an earlier date.
All content from History of Hindu Mathematics