Component- (A) – Personal details:

Prof. P. Bhaskar Reddy Sri Venkateswara University, Tirupati.

Prof. P. Bhaskar Reddy & Dr. K. Muniratnam Director i/c, Epigraphy, ASI, Mysore

Dr. Subrata Kumar Acharya Ravenshaw University, Cuttack.

Prof. P. Bhaskar Reddy Sri Venkateswara University, Tirupati.

Component-I (B) – Description of module:

Subject Name Indian Culture

Paper Name Indian Epigraphy

Module Name/Title Ancient Indian Numerals

Module Id IC / IEP / 14

The vast inscriptional literature of preserves the traces of the early notational systems and how they progressed in the direction of forming new Pre requisites symbols. The symbols in the Brahmi and Brahmi- derived scripts have been discussed in the module

The main objective of this module is to get familiar with the numerous systems of notations known to the people living in the past. It is also important to focus how and when the decimal place value Objectives notation that people across the world use today was invented in India and the knowledge flowed out of it and transmitted to other countries in course of time.

Numerals, Brahmi, Numerical System, Decimal Keywords System, Inscriptions, Symbols.

E-text (Quadrant-I) :

1. Introduction

In this module we shall discuss on the ancient Indian numerals. How the people were counting the numbers and putting them into written forms are the major issues to be addressed. We all know that two different scripts were in use in India during the time of Asoka. Some of the Asokan inscriptions located in the north-western part of India were written in , while most of his edicts written in other parts of his empire were written in Brahmi. In both the scripts we find the numerals were used in the forms of different symbols. The Kharosthi script was short lived and was mostly confined to the north-western part of early India. The numerical system used in the script was believed to have been borrowed from the Aramic system with slight modification. But the was widely popularized and it is considered as the parent of almost all the subsequent regional scripts of India. Like the , the numerical symbols in the Brahmi had a long process of evolution. In the following paragraphs we shall chiefly focus on the system of numeration in Brahmi and the other systems that developed afterwards.

2. The Brahmi numerical notational system

The numerical system of notation was in general use in a large part of India in early times and persisted at least up to the tenth century CE, until it was finally displaced by the decimal place value system. It has been also termed as , Ancient Nagari or Old Indian Notation. In this system the first three numerals are expressed by one, two and three horizontal strokes, from four to nine there are special symbols and there is a distinct figure

for each of the orders of numbers such as ten and its multiples up to ninety while it has also separate signs for a hundred and a thousand. Thus, there are twenty basic signs. The intermediate units are expressed by simply adding their signs. To cite an example, if 77 is to be written then at first the symbol for 70 and then the figure for 7 is written. But the figures for 100 and 100 are treated multiplicatively. The multiples of 2 and 3 are indicated by the addition of one or two horizontal strokes respectively, at the right side of the sign for 100 and 1000. However, when the number 777 is to be expressed then the symbol for 100 is written with the figure for 7, then the figure of 70 and finally the figure for 7. Similarly the number 7777 is denoted at first by the symbol for 1000 followed by 7, then 100 followed by 7, then the symbol for 70 and finally the figure for 7. Thus, the system is essentially an additive and multiplicative system where the numbers are either added or multiplied to express the desired numbers. As in the Brahmi script so also in the , the symbols are written from left to right beginning with the symbols of higher values. It is a non-place value notational system and there is no trace of the employment of the zero. The twenty symbols by which the numbers were expressed are given below.

3. Theories on the origin of the Brahmi numerical notational system

The origin of the Brahmi numerals is a subject of wide controversy among scholars and remains to be solved scientifically and satisfactorily. The subject has long been engaged attention of scholars ever since the discovery of the existence of the old Brahmi numerical symbols by in 1838. Since then a large number of theories have been advanced to explain the origin of the Brahmi numerals. Prinsep was the first to believe that the numerical symbols derived from the characters for the initial letter of the Sanskrit word for each number. The theory is obviously unsatisfactory. In 1877, Bhagwanlal Indraji advanced another theory that rests on the assumption that all the numerical symbols excepting the first three were denoted by letters or groups of letters. In his postscript to Bhagwanlal’s article, Buhler accepted the theory, though he admitted that the actual origin of these numerals remained obscure, since no rationale could be discerned for the particular phonetic values attributed to the numerical signs. For this reason the idea that he numerical symbols are essentially derived from Brahmi letter-forms could not be accepted. Some scholars like A. Cunningham inclined to believe that the Brahmi symbols from 5 to 9 resemble the Ariano- (Kharosthi) letters for initial syllables of the words for each number. But this superficial resemblance has long back rejected. . C. Bayley opined that the Brahmi numerical system was derived from various sources like Phoenician, Egyptian, Bactrian and possibly from . Bayley’s assumptions are also not convincing to many scholars. Recently, Falk has proposed a possible influence from the early Chinese system of numerical notation. But that too is unconvincing and therefore, cannot be accepted. Thus, there is no consensus of opinion regarding the origin of the Brahmi

numerals. The foreign origin theory has no basis at all. The system was essentially originated indigenously and was possibly older than the Brahmi script.

4. Early phase of development in Indian Inscriptions

The inscriptions of Asoka furnish the earliest examples of the use of the Brahmi numerals followed by those found in the Nanaghat, Nasik and Karle caves. The inscriptions found from , Kousambi and the Western Ksatrapa coins and inscriptions are also of immense value for understanding the ancient Brahmi numerals. A complete set of figures is found in the series of dated coins of the Western Ksatrapas issued between second and fourth century CE. The subsequent development of the Brahmi numerals exhibited considerable variability leading to different forms in time and space. The ramification of the numerical symbols in the Brahmi-derived scripts of India is evident from the huge corpus of inscriptional literature. Fortunately, in many inscriptions, the numbers are written both in words and numerals, making the process of of the Brahmi numerals more easy and sure.

5. The Brahmi Numerical symbols

The first three numerals such as one, two and three are indicated by one, two and three horizontal strokes respectively. The symbol for four is a simple . Five is represented by a vertical with a stroke or curve attached to its right medially. The symbol for six has two curves opening to the right and placed one above the other with or without having a loop in the middle. Seven is seen as a down facing curve with its right arm elongated. Eight ism again a curve opened to the top with the right arm gently curves down. Nine is a semi-circle opened to the left with a stroke added to its bottom. Ten and its multiples up to ninety have also separate signs for each of them. The symbol for ten is a circle with two horizontal lines joined to it on the right. The symbol of twenty is simple circle. The symbol for thirty is represented by a curve and a vertical joined together by a firm base-line. One of the early varieties of the symbol for forty is denoted by a small curve attached to a slanting stroke in the right and another small curve goes down from the middle of the stroke. The symbol for fifty is a broad curve either facing to the left or right. The symbol for sixty resembles to the English letter V or Y. The symbol for seventy is indicated by a vertical with a small bar attached to it on the either side. Eighty and ninety are represented respectively by a circle with a vertical and a circle with a cross in the middle. The symbol for one hundred is indicated by a cursive left half and angular right half joined together by a mid-line. The symbols for the multiples of one hundred is indicate such as two and three hundred are formed by adding one or two horizontal bars on the top right respectively. From four hundred onwards, the symbols for each of the numerals such as four, five, six etc., are joined to the symbol for one hundred on its right by an additional bar in order to denote the multiples of one hundred. The symbol of one thousand resembles the English letter T. As in the case of hundreds, so also in the case of the multiples of one thousand, the symbols are placed to the right of the symbol for one thousand. It is significant to note that all the latter varieties of the Brahmi numerals, excepting a few where the idiosyncrasies of the scribes or engravers often led to the formation of the new symbols, used in different parts of India subsequent to the 2nd-3rd centuries CE, essentially developed out of the rudimentary symbols for each of the numerals through an evolutionary process.

6. The decimal system of notation

The most fundamental contribution of ancient India to the progress of human civilization is the invention of the decimal system of notation. In this system there are nine unit figures and a symbol for zero to denote all integral numbers by applying the principle of place-value. This is the most convenient system for all purposes of calculation and is now commonly used throughout the civilized world. Prof. Halsted has rightly pointed out that no single mathematical creation has been more potent for general on-go of intelligence and power than the creation of zero. During the 19th and early decades of 20th century, attempts were made to credit the invention of the decimal system of notation to the Arabs. But mathematicians are now convinced that the inventions entirely the work of the , and the Arabs were the people who borrowed the system from India and from there the knowledge was transmitted to Europe. As it was from the Arabs that the Europeans learned this system, the Europeans called them Arabic numerals. But the Arabs refer to their numerals as Indian numerals, However, popularly they are called as the Hindu-Arabic or Indo-Arabic numerals.

At the present state of our knowledge we do not know precisely who was the inventor of this new system and when the system was invented. Scholars believe that Aryabhata and Varahamihira, the two leading astronomers of the Gupta age used this new system in their works. Jinabhadra Gani in his treatise Brihat-ksetra-samasa and Bhaskara in his commentary on Aryabhatiya used the place value numerals with zero. Both the works belonged to the 6th century CE and, therefore, the system was accepted and popularized by many ancient mathematicians and astronomers.

There is a good deal of controversy among scholars regarding the early use of the decimal system in Indian inscriptions. The early palaeographers like G. Buhler and G.H. Ojha opined that the earliest epigraphic instance of the use of the decimal system occurred in the Mankani plates of king Taralasvami of the Kalachuri era 346 (594-95 CE). Although V.V. Mirashi emphatically argued that the inscription in question is a spurious one and that the decimal system of notation began to be used in north and south India as early as the second quarter of the 7th century CE, yet noted epigraphists like G.S. Gai and D.C. Sircar relied on the Mankani plates as genuine and maintained that the decimal system was in vogue in the later part of the 6th century CE. It may be noted here that the system was also in use during the same time by the Eastern Gangas of Kalinga who held sway over the territories in north Andhra Pradesh and south Odisha. In the inscriptions of the Eastern Gangas we come across an interesting phenomenon where the old system was in use side by side with the new system. Their inscriptions exhibit the process of transition from the numerical to the decimal system of notation. The Siddhantam grant of Devendravarman of Ganga Year 195 equivalent to 693 CE is admittedly the earliest to exhibit the correct use of the decimal place- value system. In the subsequent grants of the Eastern Gangas, the system is exclusively used. In Indian context the old system of numerical notation was completely displaced by the new decimal system by the 10th century CE. The subsequent history of decimal numbers in Brahmi-derived scripts is essentially only a matter of development of outward forms of the figures. They ultimately led to the evolution of the regional variations in different linguistic- palaeographic zones.

Outside India, the use of the decimal system of notation is found in a few stone inscription of South-east Asia dated in era. At least three inscriptions of Sri-Vijaya, two found at Palembang in Sumatra and the third in the island of Banka, mention their dates as S. 605

(683 CE), S. 606 (684 CE) and S. 608 (686 CE) in decimal figures. Besides, the date S. 605 (683 CE) of the Khmere inscription at Sambhor in is also written in the new system. The use of Saka era and decimal place-value notation in these inscriptions of South- east Asia not only allude to their import from India but also confirm that the new notational system was well established and was in use in the records of at least some parts of India.

7. Chronogrammatic system of notation

In Sanskrit there is only one word to indicate a number, e.g. eka, dvi, tri, etc. But there are several other words that can be somehow associated with particular numbers. Such words often denote things, beings or concepts and have some association with numbers. They were used by early astronomers and mathematicians in their metrical compositions. Thus, the four is indicated by the words Veda, Yuga, Samudra and so on. Sometimes, numbers beyond ten are also indicated be chronograms. To cite an example, the word Deva is used to indicate the number thirty-three. The application of the principle of place-value is another important characteristic of this system. The chronograms are arranged in accordance with positional value and unlike the decimal system, here the units are mentioned first followed by the chronograms of higher o0rders. In other words, the chronograms are arranged from left to right but in order to obtain the required number they are to be reversed. For example, in order to indicate the number 2017 we may use the words like sindhu-sasi-gagana-nayana, where sindhu stands for 7, sasi for 1, gagana for 0 and nayana for 2. If we reverse the order we can get the desired number 2017.

One of the earliest use this system of chronogram is noticed in Pingala’s Chanda-. The work is assigned to the second century BCE. In this work, the words only denoted certain numbers and the principle of place-value was not yet developed. But in the early centuries of the Christian era the the system gradually gained perfection and the chronograms were used with positional value. In the Pancasiddhantika of Varahamihira and in the mathematical and astronomical works of later writers the chronograms were used to indicate large numbers. In Indian epigraphy, an inscription of Pratihara Vatsaraja dated in S. 717 or 795 CE furnishes the earliest occurrence of the system. The system was fairly popular in the inscriptional literature of India after 11th century CE.

8. Katapayadi system

The Katapayadi system is another system of notation that was prevalent in south India. In this the ten letters from to na had the value respectively of the numbers 1 to 9 and 0. The same was the case with the following ten letters from ta to na. The five letters from to indicated the numbers 1 to 5, and the nine letters from to had the value of the numbers 1 to 9 respectively. In the formation of chronograms, vowels were added to the consonants without affecting the numerical value of the letters. In the case of the conjuncts, only the last consonant of a conjunct had the numerical value. As in the chronogrammatic expression so also in this system the right to left arrangement of the numbers was followed. The consonants with the numerical value are arranged below in a tabular form.

9. Numbers used in paginating Ceylonese and Burmese manuscripts

In the Ceylonese manuscripts another method of numeration was used which was very much akin to the Katapayadi system. According to this system, ka, ka, ki, ki, ku, ku, kr, kr, kl, kl, ke, kai, ko, kau, kam and kah had the value respectively of 1 to 16. The other thirty-three consonants with sixteen aksaras each were similarly numbered continuously up to 544 (16 x 34). Numbers beyond 544 were expressed by beginning again as 2 ka, 2 ka, 2 ki, 2 ki, and so on. In Burma, the same system was used with the omission of r, r, l, and l, so that ka to kah would indicate the numbers from 1 to 12

10. Summary

Thus, the Asokan edicts furnish the earliest evidence of the use of the some notational system in India. The numerical system of notation in the Brahmi with twenty symbols was probably originated in India sometime before the 3rd century BCE when Asoka used the system. This was an additive and multiplicative system. It was later on popularized by many subsequent ruling dynasties like the Satavahanas, the Saka-Ksatrapas, the Kushanas, the Guptas and their contemporaries. The decimal system of notation is a place value notational system which is widely used today was again originated in India in the 6th century CE. It first of all travelled to South-east Asia , more precisely to Cambodia and , then to Ceylon and subsequently to Persia by the Arabs and from there it was borrowed by the Europeans.