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History of Hindu Mathematics: A Source Book v1.c7.s8: 8. The Decimal Place-Value System

History of Hindu Mathematics: A Source Book v1.c7.s8

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8. The Decimal Place-Value System

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**Important Features.** The third and most important of the Hindu numeral notations is the decimal place-value notation. In this system there are only ten symbols, those called *aṅka* (literally meaning "mark") for the numbers one to nine, and the zero symbol, ordinarily called *śûnya* (literally, "empty"). With the application of the principle of place-value these are quite sufficient for the writing of all numbers in as simple a way as possible. The scale is, of course, decimal. This system is now commonly used throughout the civilised world. Without the zero and the place-value, the Hindu numerals would have been no better than many others of the same kind, and would not have been adopted by all the civilised peoples of the world. "The importance of the creation of the zero mark," says Professor Halsted, "can never be exaggerated. This giving to airy nothing, not merely a local habitation and a name, a picture, a symbol, but helpful power, is the characteristic of the Hindu race whence it sprang.

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It is like coining the *Nirvāṇa* into dynamos. No single mathematical creation has been more potent for the general on-go of intelligence and power."¹

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¹ G. B. Halsted, *On the foundation and technique of Arithmetic*, Chicago, 1912, p. 20.

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**Forms.** A large number of scripts differing from each other are in use in different parts of India today. The forms of the numerical signs in these scripts are also different. Although all the Hindu scripts are derived from a common source—the Brâhmî Script—yet the differences in the forms of the various modern Indian scripts are so great that it would have been difficult to establish any relation between them, if their previous history had not been known. The above remark applies to the numerical signs also, as will appear from a study of the numerical signs in the various vernaculars of India given in Table XV. The great divergence in the forms of the numerical symbols shows that in India, people already knew the use of the zero and the place-value principle before the different scripts came into being, and that the numeral forms were independently modified in various parts of India, just as the letters of the alphabet were modified. And as the changes in the forms in different localities were independent of each other, so there has come about a great divergence in the modern forms. That this divergence already existed in the eleventh century is testified to by Al-Bîrûnî who says, "As in different parts of India, the letters have different shapes the numerical signs, too, which are called *aṅka*, differ."²

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² *Alberuni's India*, I, p. 74.

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**Nagari Forms.** The most important as well as the most widely used of the different symbols are those belonging to the *Nâgarî* script. The present forms of these symbols are:

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The gradual development of these figures from the Brâhmî numerals is shown in Table XIV.

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**Epigraphic Instances.** The following is a list of inscriptions and grant plates upto the middle of the tenth century, which contain numerals written in the decimal place-value notation. The numerals in the inscriptions and plates after this period, are always given in decimal figures.

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| 1. | 595 A.D. | Gurjara grant plate from Sankheda, (*EI*, II, p. 19). The date Saṁvat 346 is given in the decimal place-value notation. |
| *2. | 646 A.D. | Belhari Inscription, (*JA*, 1863). |
| *3. | 674 A.D. | Kanheri Inscription, (*JA*, 1863, p. 392). |
| 4. | 8th Century | Ragholi plates of Jaivardhaṇa II, (*EI*, IX, p. 41). The number 30 is written in decimal figures. |
| 5. | 725 A.D. | Two Sanskrit Inscriptions in the British Museum, (*IA*, XIII, p. 250). The dates Saṁvat 781 (=723 A.D.) and Saṁvat 783 (=725 A.D.) are given in decimal figures. |
| *6. | 736 A.D. | Dhiniki copper plate grant, (*IA*, XII, p. 155). The date Vikrama Saṁvat 794 is given in decimal figures. |
| 7. | 753 A.D. | Ciacole plates of Devendravarmaṇa, (*EI*, III, p. 133). The number 20 is written in decimal figures. |
| 8. | 754 A.D. | Râṣṭrakûṭa grant of Dantidurga, (*IA*, XI, p. 108). The date Saṁvat 675 is given in decimal figures. |

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| 9. | 791 A.D. | Inscription of Sâmanta Devadatta, (*IA*, XIV, p. 351). The date Vikrama Saṁvat 847 is given in decimal figures. |
| 10. | 793 A.D. | Daulatabad plates of Śaṅkargaṇa, (*EI*, IX, p. 197). The date Śaka 715 is given in decimal figures. |
| *11. | 813 A.D. | Torkhede plates, (*EI*, III, p. 53; also *IA*, XXV, p. 345). The date Śaka Saṁvat 735 is given in decimal figures. |
| 12. | 815 A.D. | Buchkalâ inscription of Nâgbhaṭa, (*EI*, IX, p. 198). The date Saṁvat 872 is given in decimal figures. |
| 13. | 837 A.D. | Inscription of Bâuka (Rajputana Museum, *PLM*, p. 127; *EI*, XVIII, p. 87). The date Vikrama Saṁvat 894 is given in decimal figures. |
| 14. | 843 A.D. | The inscriptions from Kanheri, No. 43 b., (*IA*, VIII, p. 133). The date Saṁvat 765 is given in decimal figures. |
| 15. | 851 A.D. | The inscriptions from Kanheri, No. 15, (*Ibid*). The date¹ Saṁvat 775 is given in decimal figures. |
| 16. | 853 A.D. | Pâṇḍukeśvara Plates of Lalitasuradeva, (*IA*, XXV, p. 177). The date Saṁvat 21 of the King's reign is given in decimal figures. |
| 17. | 860 A.D. | Ghatiyala Inscription of Kakkuka (*EI*, IX, p. 277). The date Vikrama Saṁvat 918 is given in decimal figures. |

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¹ For correction of date see *IA*, XX, p. 421.

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| 18. | 862 A.D. | Deogarh Jaina Inscription of Bhojadeva, (*EI*, IV, p. 309). The dates Vikrama Saṁvat 919 and the corresponding Śaka Saṁvat 784 are both given in decimal figures. |
| 19. | 870 A.D. | Gwalior inscription of the reign of Bhojadeva (*Archaeological Survey of India*, Report, 1903-4, plate 72). Although the date is not given, the ślokas are numbered from 1 to 26 in decimal figures. |
| 20. | 876 A.D. | Gwalior inscription of Allah, of the reign of Bhojadeva (*EI*, I, p. 159). The date Vikrama Saṁvat 933, as well as the numbers 270, 187 and 50 are given in decimal figures. |
| 21. | 877 A.D. | The inscriptions from Kanheri, No. 43a, (*IA*, XIII, p. 133). The date Saṁvat 799 is given in decimal figures. |
| 22. | 882 A.D. | Pehava inscription (*EI*, I, p. 186). The date Saṁvat 276 (Śrî Harṣa Era) is given in decimal figures. |
| 23. | 893 A.D. | Grant plate of Balavarmaṇa, (*EI*, IX, p. 1). The date Vallabhî Saṁvat 574 is given in decimal figures. |
| 24. | 899 A.D. | Grant plate of Avanîvarmaṇa, (*EI*, IX, p. 1). The date Vikrama Saṁvat 956 is given in decimal figures. |
| 25. | 905 A.D. | The Ahar stone inscription (*Journ. United Provinces Hist. Soc.*, 1926, pp. 83 ff) contains several dates written in decimal figures. |
| 26. | 910 A.D. | Râṣṭrakûṭa grant of Krishna II (*EI*, |

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I, p. 53). The date is given in decimal figures.

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27. 917 A.D. Sanskrit and old Canarese inscriptions, No. 170, (*IA*, XVI, p. 174). The date Saṁvat 974 is given in decimal figures. The number 500 also occurs.

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28. 930 A.D. Cambay plates of Govinda IV, (*EI*, VII, p. 26). The date Śaka Saṁvat 852 is given in decimal figures.

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29. 933 A.D. Sangli plates of Râṣṭrakûṭa Govindarâja IV, (*IA*, XII, p. 249). The date Saṁvat 855 is given in decimal figures.

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30. 951 A.D. Sanskrit and old Canarese inscriptions, No. 135, (*IA*, XII, p. 257). The date Saṁvat 873 is given in decimal figures.

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31. 953 A.D. Inscription of Yaśovarmaṇa, (*EI*, I, p. 122). The date Saṁvat 1011 is given in decimal figures.

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32. 968 A.D. Siyadoni stone inscription (*EI*, I, p. 162). The inscription contains a large number of numerals expressed in decimal figures.

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33. 972 A.D. Râṣṭrakûṭa grant of Amoghavarṣa, (*IA*, XII, p. 263). The date Śaka 894 is given in decimal figures.

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Palaeographic evidences of the early use of the decimal place-value system of notation are found in the Hindu colonies of the Far East.¹ The most important ones among these are the three inscriptions of Śrîvijaya, two found at Palembang in Sumatra, and the third in the island of Banka. These contain the dates 605, 606 and 608 of the Śaka Era (corresponding respectively to A.D. 683, 684 and 686) written in numerical figures. Another instance giving the date 605 Śaka is the inscription of Sambor in Cambodia. In an inscription at Po Nagar in Champa, occurs the date 735 Śaka (= 813 A.D.).

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¹ G. Coedès, “A propos de l’origine des chiffres arabes,” *Bull. School of Oriental Studies* (London), VI, 1931, pp. 323-8.

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**Their Supposed Unreliability.** The above list contains more than thirty undoubted epigraphic instances of the use of the place-value notation in India. G.R. Kaye,¹ who believes in the theory of the non-Hindu origin of the place-value notation, states that all the early epigraphic evidences of its use in India are unreliable. On the basis of the existence of a few forged grant names or descriptions substituted. All such forgeries are easily detected, because of the uneven surface of the part of the plate that is tampered with and the difference in the writing. (2) An entirely new document may be forged. Such cases, though rare, are also easily detected, because there is obvious divergence as to the date recorded in the document, and that inferred on the basis of the forms of the characters used in the writing. Such forgeries are also marked by an obvious inferiority in execution, and inaccuracies in the statement of genealogies and other historical facts.

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Epigraphists have so far found little difficulty in eliminating the spurious grant plates. It might be mentioned that the genuineness of the grant plates included in our list has not been questioned by any epigraphist.¹

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¹ If any of them is forged, the forgery is so good that it cannot be detected. The writing in such cases, if any, is so well forged as to be indistinguishable from that used in the period to which the plate is said to belong. Therefore, the evidence of these plates as to the method of writing numbers, cannot be rejected, even if they be proved to be spurious at some future date—a contingency which is very unlikely to use. It may also be noted that the list contains several *stone inscriptions* which cannot be spurious.

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Kaye, in his article quoted above, has given a list of eighteen inscriptions and grant plates and eliminates all but the last two as forgeries. The arguments he has employed and the assertions of facts that he has made are in most cases incorrect and misleading, so that his conclusions cannot be accepted. As an instance of his method, we quote his criticism about the Gurjara grant plate, No. 1 in our list. He writes: "Dr. Bühler quotes this Gurjara inscription of the Chedi year 346 or A.D. 594 as the earliest epigraphic instance of the use of the decimal notation in India. (i) An examination of the plate (*Ep. Ind.*, II., p. 20) suggests the possibility that the figures were added some time after the plate was engraved. The date is engraved in words as well as in figures. It is ‘three hundred years exceeded by forty-six.’ The symbols are right at the end of the inscription from which they are marked off by a double bar in a most unusual manner. (ii) The figures are of the type of the period, but they were also in use much later, and in no other example are such symbols used with place-value. (iii) Also there are nine dates written in the old notation (*Ep. Ind.*, V), *e. g.*, there is another grant of the Gurjara of Bharoch in which the date Saṁvat 391 (i.e., A.D. 640) is given in the old notation. Again, there is no other Chedi date, at least before the eleventh century A.D., given in the modern (place-value) notation. (iv) There cannot be the remotest doubt as to the unsoundness of this particular piece of evidence of the early use of the modern system of notation in India.”

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The following remarks will show to the reader that Kaye’s criticism and his conclusion are unfounded and invalid:

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(i) An examination of the plate, (*EI*, II, p. 19), will convince every one about its genuineness. The writing is bold and clear, the numerical figures occur at the end, as they ought to be, immediately after the words ‘three hundred years exceeded by forty-six.’ They are separated from the written words by bars, just as they ought to be. There is absolutely nothing suspicious about this method of separation, as it is common custom in India to do so and occurs frequently. That it was the practice to write the date at the end of a document is well known.¹ In fact, the numeral figures of the date occasionally mark the end of the document.¹ The double vertical bar, ||, is a sign of inter-punctuation. Although punctuation marks have been in use in India from the earliest known times, yet their use did not become either regular or compulsory till very recent times.² Different writers used the various marks differently. In inscriptions, the double vertical bar has been found at the end of sentences, half verses, verses, larger prose sections and documents. In the Junâr inscriptions it occurs after numerals and once after the name of the donor.³ In manuscripts, the practice of separating numbers by vertical bars is common. It is found in the Bakhshâlî Manuscript⁴ and in several others. Thus the occurrence of the numerals at the end and the inter-punctuation mark of the double vertical bar cannot form valid grounds for suspecting the document. The suggestion that the figures were added some time after the plate was engraved is absurd, as there appears to us no reason why one should take the trouble to add the figures when the date was already written in words.

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¹ Many of the plates mentioned in our list contain the date at the end.

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¹ This is so in the Chârgâon plates of Huviṣka (*Arch. Survey Report*, 1908-9, plate 56), in the Inscription of Rudradamana (*IA*, VIII, p. 42) and in others.

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² There are some copper plate grants which do not contain any punctuation marks; see Bühler, *l.c.*, p. 90.

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³ Bühler, *l.c.*, p. 89.

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⁴ *E.g.*, | 5 |, 21 r; | 2558 |, 2v; | 330 |, 17v; instances such as these: | 1 | 4 | 9 | 16 |, 16v; and | 2 |, | 4 |, etc., 5v. are very common. Very often, isolated numbers are not separated. The double vertical bar also occurs before and after the words *udâ*, *sûtram*, etc.

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(*ii*) Kaye admits that the figures are of the type of the period. His remark that they were in use much later is incorrect. The Tables III-V and XII show that the use of three horizontal bars to represent 3 is not found after the eighth century. The figure for 4 used in our grant plate is not found after the sixth century, and the same is true for the figure for 6. The forms of the numerical signs alone fix the date of the writing to the sixth century and not later.

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(*iii*) The *Chedi Saṁvat* is one of the thirty-four eras, whose use has been discovered in inscriptions and grant plates. The occurrence of nine dates in the Chédi Saṁvat, written in the old notation after this plate, does not prove the unsoundness of this particular piece of evidence, as Kaye would like us to conclude. It simply shows that in India too, the new system had to fight for supremacy over the older one just as in other countries. In Arabia the new system was introduced in the eighth century, but it did not come into common use until five or six hundred years later. In Europe we find that it was exceptional for common people to use the new system before the sixteenth century—a good witness to this fact being the popular almanacs. Calendars of 1557-96 have generally Roman numerals, while Koebel’s *Calendar* of 1578 gives the Hindu numerals as subordinate to the Roman.¹

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¹ Smith and Karpinski, *l.c.*, p. 133.

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We may, therefore, conclude that the Gurjara grant plate offers us a genuine instance of the use of the new system (with place-value) in India.

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Kaye’s criticisms regarding the genuineness of some other plates included in our list (marked with asterisks) have been found to be baseless.

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**Place of Invention of the New System.** It has been already stated that the same numeral forms for the numbers 1 to 9, as were in use in India from the earliest known times, have been used in the new system of notation with the place-value. Another noteworthy fact regarding the new system is the arrangement of the *aṅka* (digits). It will be observed that the arrangement in the old system was that the bigger numbers were written to the left of the smaller ones.¹ This same arrangement continues in the new system with place-value, where the digits to the left, due to their place or position, have bigger values. The gradual change from the old system to the new one using the same numerical signs, is to be found in India alone, and this, in our opinion, is one of the strongest arguments in favour of the Hindu origin of the new system. The earliest epigraphic instance of the use of the new system is 594 A.D. No other country in the world offers such an early instance of its use. Epigraphic evidence alone is, therefore, sufficient to assign a Hindu origin to the modern system of notation.

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¹ Showing thereby that the place assigned to a numeral depended upon its value. This has been incorrectly thought to be a sort of place-value system by some writers.

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**Inventor Unknown.** It is not known who the inventor of the new system was, and whether it was invented by some great scholar, or by a conference of sages or by gradual development due to the use of some form of the abacus. Likewise, it is not known to which place, city, district or seat of learning belongs the honour of the invention and its first use. Epigraphic evidence cannot help us in this direction. For the system was used in inscriptions, a very long time after its invention, in fact, when it had become quite popular all over Northern India.

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**Time of Invention.** The grant plates were legal documents. They were written by professional writers. The existence of such writers is mentioned in the southern Buddhist canons and in the Epics.² They have been called *lekhaka*, *lipikara* and later on *divira*, *karaṇa*, *kāyastha*, etc. According to Kalhaṇa,¹ the Kings of Kashmir employed a special officer for drafting legal documents. He bore the title of *paṭṭopādhyāya*, *i.e.*, the teacher (charged with the preparation) of title deeds. The existence of manuals such as the *Lekhapañcāśikā*, the *Lekhaprakāśa*, which give rules for drafting letters, land grants, treaties, and various kinds of bonds and bills of exchange, show beyond doubt that the writing of grant plates was a specialised art and that the style of writing those documents must always have been centuries behind the times, just as it is even to-day with respect to legal and state documents. The time of invention of the new system must, therefore, be placed several centuries before its first occurrence in a grant plate in the sixth century A.D. The exact period of invention may be roughly deduced from the history of the growth of numerical notations in other countries.

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² Bühler, *l.c.*, p. 5.

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¹ *Rājatarangiṇī*, V, pp. 397f.

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According to Heath,² the Greek alphabetic notation was invented in the 7th century B.C., but it came into general use only in the second century A.D. Thus it took about eight hundred years to get popular. In Arabia the new notation was introduced in the 8th century A.D., but it came into common use about five or six hundred years later. The same was the case in Europe. The Arabs got the complete decimal arithmetic, including the method of performing the various operations, at a period when intellectual activity in Arabia was at its greatest height, but they could not make the decimal system common before about five or six hundred years had elapsed.³ In legal documents and in recording historical dates, the Arabs even now use their old alphabetic notation.

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² Heath, *History of Greek Mathematics*, I, Oxford, 1921, p. 34.

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³ The arithmetic written by Al-Kharki in the eleventh century does not use the decimal system, showing that at the time there were two schools amongst the Arab mathematicians, one favouring

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Epigraphic evidences show that the new system was quite common in India in the eighth century and that the old system ceased to exist in Northern India by the middle of the tenth century. This would, therefore, place the invention of our system in the period between the first century B.C. and the third century A.D.

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The exact date of the invention, however, would be nearer to the 1st century B.C. or even earlier, because for a long time after its invention, the system must have been looked upon as a mere curiosity and used simply for expressing large numbers. A still longer time must have elapsed before the method of performing the operations of

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