Ancient Indian Mathematical Evolution Since Counting

Home , Brahmi numerals, Kharosthi, Kha (Devanagari), Kha (Indic)

Journal of Statistics and Mathematical Engineering e-ISSN: 2581-7647 Volume 5 Issue 3

Ancient Indian Mathematical Evolution since Counting

1 2 Sankar Prasad Mukherjee , Sandip Ghanta* 1Research Guide, 2Research Scholar 1,2Department of Mathematics, Seacom Skills University, Kolkata, West Bengal, India Email: *[email protected] DOI:

Abstract This paper is an endeavor how chronologically since inception and into growth of mathematics occurred in Ancient India with an effort of counting to establish the numeral system through different ages, i.e., Rigveda, Yajurvada, Buddhist, Indo-Bactrian, Bramhi, Gupta and Devanagari Periods. Ancient India’s such contribution was of immense value helped to accelerate the progress of Mathematical development up to modern age as we see today.

Keywords: Brahmi numerals, centesimal scale, devanagari, rigveda, kharosthi numerals, yajurveda

INTRODUCTION main striking feature being counting and This research paper is an endeavor to evolution of numeral system thereby. synchronize all the historical research with essence of pre-historic and post-historic Mathematical Evolution in Vedic Period respectively interwoven into a texture of Decimal Number System in the Rigveda evolution process of Mathematics. The first Numbers are represented in decimal system form of writing human race was not (i.e., base 10) in the Rigveda, in all other literature but Mathematics. Arithmetic Vedic treatises, and in all subsequent Indian what is today was felt as an essential need texts. No other base occurs in ancient for day to day necessity of human race. In Indian texts, except a few instances of base various countries at various point of time, 100 (or higher powers of 10). The Rigveda people used various symbols, notation, and contains the current Sanskrit single-word methodologies as a starter for a numeration terms for the nine primary numbers: eka aspect. In India, Egypt, Babylon, Greek, (1), dvi (2), tri (3), catur (4), panca (5), sat Chinese and other places; people worked on (6), sapta (7), asta (8) and nava (9); the ﬁrst various bases, which would be held as nine multiples of ten (mostly derived from partial and later on integrated and latterly above): dasa (10), vimsati (20), trimsat developed in to the number system which (30), catvarimsat (40), pancasat (50), sasti the modern age is finding. The bases being (60), saptati (70), asiti (80) and navati (90), decimal, sexagesimal, centesimal as used and the ﬁrst four powers of ten: dasa (10), not in totality but in partial form which sata (100), sahasra (1000) and ayuta were their tool by which they used to fulfill (10000)[4]. their requirements. In Rigveda, there was a conceptualization In this research, the brief evolution process about base ten but could not be conceived of Mathematics is compared since Indian as positional. From various instances such Vedic period vis-a-vis Greek period, as Rigveda, it is found reasonable that ten Babylon period, Chinese period and was the basis of numeration during that thereafter. This have been represented as period. Such information was found from sub topic / title in different paragraphs. The Sanskrit literature of Vedic period wherein

Journal of Statistics and Mathematical Engineering e-ISSN: 2581-7647 Volume 5 Issue 3 no other base was found and as such we well-known ‘Buddhist works’ during the 1st may term it has Hindus numerals. During century BC. From Lalitavistara the that period large digital denominations following quote is very interesting. were found in use, even up to 17, 18, 19, 20th etc. digits were of common use. Those The quote was the dialogue between were as perfect as found in modern times. Arjuna, the Mathematician, and Prince Gautama (Bodhisattva). "The Powers of Ten in the Yajurveda mathematician Arjuna asked the The following has been taken from the Bodhisattva, 'O young man, do you know book entitled “History of Hindu the counting which goes beyond the koti on Mathematics” by B Dutta and A Singh [1]; the centesimal scale? we find that during Vedic period, the power Bodhisattva: I know. of 10 was in used. In “Yajurveda Sambita” and “Taittiriya Sambita” it has been Arjuna: How does the counting proceed mentioned as “eka, dasa, sata, sahasra, beyond the koti on the centesimal scale? ayuta,up to anta, parardhaas values 1, 10, Bodhisattva: Hundred kotis are called 102, 103, 104, up to 1011, 1012.” ayuta, hundred ayutas niyuta, hundred “Maitrayani and Katbaka Sambitas” niyutas kankara, hundred kankaras vivara, contain the same list as above with a bit of hundred vivaras ksobbya, hundred alterations. ksobbyas vivaba, hundred vivabas utsanga, The “Pancavimsa Brabmana” has the same hundred utsangas babula, hundred babulas list of “Yajurveda Sambita” up to nyarbuda nagabala, hundred nagabalas titilambha, and further thereafter added more hundred titilambhas vyavastbana-prajnapti, denominations such as nikbarva, vadava, hundred vyavastbana-prajnaptis betubila, aksiti and many more. hundred betubilas karabu, hundred karabus

betvindriya, hundred betvindriyas Alike “Sankhyayana Srauta Sutra” samaptalambba, hundred samdpta-Iambbas continued further after nyarbuda with gananagati, hundred ganandgatis nikbarva, samudra, salila, antya, ananta (= niravadya, hundred niravadyas mudra-bala, 10 billions). hundred mudra-balas sarva-bala, hundred So, from the above we find that each sarva-balas visamjna-gati, hundred denomination 10 times more than the visamjna-gatis sarvajna, hundred sarvajnas preceding and as such in Sanskrit they were vibbutangama, hundred vibbutangamas 53 term “dasagunottara samjna”. Here, lies the tallaksana (10 ) [1]. concept of 10 as base and multiple of ten while deeming big numbers. This entails that the Hindus anticipated Archimedes by several centuries earlier Evolution Process of Mathematics in in the matter of evolving a series of Buddhist Period number names which “are sufficient to Ten was considered as the base still about exceed not only the number of a sand- the 5th century BC, it was found available heap as large as the whole earth, but one and later on ‘Centesimal Scale’ was in use as large as the universe”. as found in the works “Lalitavistara” a

Journal of Statistics and Mathematical Engineering e-ISSN: 2581-7647 Volume 5 Issue 3

Kharosthi Numeral System (In Indo-Bactrian and Aryan)

Table 1: Symbols used in Kharosthi Numeral System.

During the fourth century BCE to the third numeral system appeared in the third century AD, the Kharosthi lipiwere found century BCE. It was the ancestor of other in Bactrian, Indo-Bactrian and Aryan and numeral system. The Ashoka pillars in 300 appearing in ancient Gandhara, now the BCE was the most authentic source for modern eastern Afghanistan and northern Brahmi inscription. In Brahmi numeral system Punjab. From the fact that the Kharosthi lipi not only the symbols from 1 through 9 were in were written from right to left and also use but also the symbols for 10, 20, 30, . . . 90, reading from right to left. This numeral 100, 200 and so on. The positional notations system the base is 4[1]. were not known on Brahmi numeral system. There was no symbol for a zero in Brahmi Brahmi Numeral System in India numeral system but due to lack of the symbol Brahmi numeral system is an indigenous of zero there was no any huddle in using in this Mathematical development of India. This system.

Figure 1: Brahmi Numeral on Ashoka Pillar.

Table 2: Symbols used in Brahmi Numeral System.

Journal of Statistics and Mathematical Engineering e-ISSN: 2581-7647 Volume 5 Issue 3

Gupta Numerals System in the early 6th century AD. This During the 4th century AD, the Gupta numeral system also developed from the numeral system were used in the Brahmi numerals system. There was no Magadha state in northeastern India. This symbol for a zero in Gupta numeral numerals system were spread over large system as in Brahmi numeral system. areas in the Gupta dynasty ruled territory

Table 3: Symbols used in Gupta Numeral System.

Sanskrit–Devanagari Numeral System depends on its place. If symbolς is present (Includes Aryabhata’s and in tens place, its value is 80, if in hundreds Brahmagupta’s conceptualization of place, its value is 800, and so on. Numbers “Shunyo”) are written in left to right in decreasing This system is world’s first place value magnitude of order. decimal system with 0. Value of a symbol

Table 4: Indian Nagari Numeral Symbols in Range 0–9.

There is no limit to largest representable became complete. This zero could number. All arithmetical simple and demarcate positive and negative numbers complex operations of addition, and gave a meaning of cutoff between subtraction, multiplication, division, taking positivity and negativity. This zero was powers, roots, can be done with most ease. termed by various words as “Pujyam” Fractions are very clearly representable symbolized as “dot” and later on termed as with a decimal point. “Shunyam” keeping the same symbol.

The Concept of Zero So, conceptualization of Void/Nonexistent The idea of nullity/zero and its /zero as Shunyam was one of the symbolization has its origin in ancient remarkable contributions of ancient India. India. This invention resulted into a Philosophically it may be correlated with complete concept of numeral system which the Buddhist concept of “Nirvana” and made possible to undertake form operation thereby it enriched the South East Asian viz, addition, subtraction, multiplication culture. Attaining Salvation, i.e., Nirvana and division, so mathematical computation meaning eternal void was the metaphysical

Journal of Statistics and Mathematical Engineering e-ISSN: 2581-7647 Volume 5 Issue 3 aspect may be drawn from the physical was the first to formalise arithmetic concept of zero.[3] For zero to fulfill its operations using Zero. He indicated zero by potential in Mathematics, it is necessary for ‘dot’. each number up to the base figure to have its own symbol. This seems to have been Next, in the middle east, the arabian achieved first in India. It was one monk mathematicians Mahammed Ibn-Musa Al- philosopher mathematician the Aryabhata Khowarizmi developed zero in the base of who properly included zero as void both as Indian numeral system. a symbol and as an idea. He was born in 476 AD at Kusumapura, Pataliputra, present All operations by zero are easy except day Patna. He wrote “Aryabhatiya” in 499 division. What is the value of any number A.D. He was used Vedic Numeral system divided by zero? In the 16th century, and enriched it. He also used alphabet Newton and Leibniz solved this problem letters to denote the numbers. He included independently and opened the word to zero in place value system and zero denoted tremendous possibilities. The answer is by ‘kha’. indeterminate, but working with this concept is the key to calculus. By working The classical age in India, the great with numbers as they approach zero, mathematician Brahmagupta was born in calculus was born without which we 598 AD in Sind (now in Pakistan). His great wouldn’t have physics, engineering and work “Brahma Sphuta Siddhanta”, where many aspects of economics and finance.

Development of Different Numerals from Brahmi Numeral System

Figure 2: Development of Different Numerals from Brahmi Numeral System.

Table 5: The Taxonomy of Development of Indian Numeral System. Kharosthi Numeral System Brahmi Numeral System Gupta Numerals System Sanskrit – Devanagari Numeral System

Journal of Statistics and Mathematical Engineering e-ISSN: 2581-7647 Volume 5 Issue 3

CONCLUSION REFERENCES The Indian system of numerals is adopted 1. B Dutta, A Singh (1938), “History of now by the whole world. This is the system Hindu Mathematics”, Asian Publishing the world admires like anything, for it has House, Volume I, II, Calcutta. all essential features present in most 2. David Eugene Smith, Louis Cherles suitable form. Indian decimal place-value Kerpinski (2004), “The Hindu Arbic system with zero is the only indigenous Numerals”, Dover Publications, Ine, system rich in all features. Today, this Mineola, New York. numeral system gifted to the world by 3. Krishna Maheshwari, Mathematics, Indian civilization is used so commonly and www.hindupedia.com/en/Mathematics. frequently, that its importance in 4. www.wikiwand.com/en/Brahmi_nume simplifying all algebraic operations is rals. ignored without notice. But it is due to this 5. www.britannica.com/topic/Hindu- simplification that we could advance our Arabic-numerals. mathematical knowledge very rapidly to great heights within very short time span. Cite this article as: Had the world remained stuck to any other complicated numeral system in usage, maybe it would have taken centuries more to be at the stage where we are proudly existing today.