Hindu ‘ganita’ and the maximum number

‘Ganita‘ literally means “the science of calculation” and is the Hindu name for mathematics. The term is a very ancient one and occurs copiously in Vedic literature. The Vedanga Jyotisa (about 1200 BCE) gives it the highest place of honour among the sciences which form the Vedanga: “As the crests of peacocks, as the gems on the hoods of snakes, so is ganita at the top of the sciences known as the Vedanga.”

Books such as भारतीय ज्योतिष का इतिहास (Bharatiya Jyotish Ka Itithas) by Gorakh Prasad (published 1956 by the Publications Bureau of the state government of Uttar Pradesh), have been shunted to the sidelines, or entirely out, of curricula. Jyotish (ज्योतिष) and Ganita (गणित) are inseparable.

The appreciation of mathematics, although it comes at a much later time than the time of the Vedic literature, is from Mahavira (CE 850) a towering mathmatician of his time: “In relation to the movements of the sun and other heavenly bodies, in connection with eclipses and conjunctions of planets, and in connection with the triprasna (direction, position and time) and the course of the moon, indeed in all these it is utilised. The number, the diameter and the perimeter of islands, oceans and mountains, the extensive dimensions of the rows of habitations and halls belonging to the inhabitants of the word, of the interspace between the worlds, of the world of light, of the world of the gods and of the dwellers in hell, and other miscellaneous measurements of all sorts, all these are made out by the help of ganita.” (In the Ganita-sara-samgraha of Mahavira, Rangacharya’s edition.)

It is characteristic of India that from a very early date there were long series of number names for very high numerals. While the Greeks had no terminology for denominations above the ‘myriad‘ (10^4), and the Romans above the ‘mille‘ (10^3), the ancient Hindus dealt freely with no less than 18 denominations. In modern times also, the numeral language of no other civilisation is as scientific and perfect as that of the Hindus.

In the Yajurveda Samhita (यजुर्वेद संहिता) (Vajasaneyi वाजसनेयी) the following list of numeral denominations is given: eka (1), dasa (10), sata (100), sahasra (1,000), ayuta (10,000), niyuta (100,000), prayuta (1,000,000), arbuda (10,000,000), nyarbuda (100,000,000), samudra (1,000,000,000), madhya (10,000,000,000), anta (100,000,000,000), parardha (1,000,000,000,000).

The same list occurs at two places in the Taittiriya Samhita (तैत्तिरिय संहिता). The Maitrayani (मैत्रयनि संहिता) and Kathaka Samhitas (कथका संहिता) contain the same list with minor alterations. The Pancavimsa Brahmana has the Yajurveda list up to nyarbuda inclusive, and then follows nikharva, vadava, aksiti. The Sankhyayana Srauta Sutra continues the series after nyarbuda with nikharva, samudra, salila, antya and ananta (10 billion). Each of these denominations is ten times the preceding one, so that they were aptly called dasagunottara samjna.

In this painting called ‘Arithmetica’, by Gregor Reisch (CE 1503), Boethius, a 5th century translator of works of Greek logic and mathematics, and Pythagoras (about about 569-570 BCE) are engaged in a mathematical competition. Pythagoras uses an abacus, while Boethius uses “numerals from India”. Boethius looks very proud, he is ready while Pythagoras is still trying to find the solution.

In later times, about the 5th centry BCE, there is evidence of number names based on the centesimal scale (based on a 100-fold increase). A well-known Buddhist work of the 1st century BCE recounts the dialogue between Arjuna, the mathematician, and prince Gautama (as given in the Lalitavistara, ed. by Rajendra Lal Mitra, Calcutta, 1877).

The mathematician Arjuna asked him: O young man, do you know the counting which goes beyond the koti on the centesimal scale?
Gautama: Yes, I know.
Arjuna: How does the counting proceed beyond the koti on the centesimal scale?
Gautama: Hundred kotis are called ayuta, hundred ayutas a niyuta, hundred niyutas a kankara, hundred kankaras a vivaha, hundred vivahas a utsanga, hundred utsangas a bahula, hundred bahulas a nagabala, hundred nagabalas a titilambha, hundred titilambhas a vyavasthana-prajnapati, hundred vyavasthana-prajnapatis a hetuhila, hundred hetuhilas a karahu, hundred karahus a hetvindriya, hundred hetvindriyas a samapta-lambha, hundred samapta-lambhas a gananagati, hundred gananagatis a niravadya, hundred niravadyas a mudra-bala, hundred mudra-balas a sarva-bala, hundred sarva-balas a visamjna-gati, hundred visamgjna-gatis a sarvajna, hundred sarvajnas a vibhutangama, hundred vibhutangamas a tallaksana.

Thus a tallaksana is 10^53 and this example shows that the Hindus anticipated Archimedes by several centuries in the matter of evolving a series of number names which “are sufficient to exceed not only the number of a sand heap as large as the whole earth, but one as large as the universe.” (Smith and Karpinski, ‘Hindu Arabic Numerals’, 1911).

In the Anuyogadvara-sutra (circa 100 BC), a Jaina canonical work, the total number of human beings in the world is given thus: “a number which when expressed in terms of the denominations, koti-koti, etc, occupies 29 places (sthana), or it is beyond the 24th place and within the 32nd place, or it is a number obtained by multiplying sixth square (of two) by (its) fifth square (2^96) or it is a number which can be divided by two 96 times.”

In most of the mathematical works, the denominations are called ‘names of places’, and 18 of these are generally enumerated. Sridhara (CE 750) gives the folowing names: eka, dasa, sata, shasra, ayuta, laksa, prayuta, koti, arbuda, abja, kharva, nikharva, mahasaroja, sanku, sarita-pati, antya, madhya, parardha and adds that the names proceed even beyond this. Mahavira (CE 850) gives 24 notational places: eka, dasa, sata, shasra, dasa-sahasra, laksa, dasa-laksa, koti, dasa-koti, sata-koti, arbuda, nyarbuda, kharva, mahakharva, padma, mahapadma, ksoni, mahaksoni, sankha, mahasankha, ksiti, mahaksiti, ksobha, mahaksobha.

My reference for this short note on the greatness of Hindu ganita is the ‘History of Hindu Mathematics, A Source Book’, by Bibhutibhushan Datta and Avadhesh Narayan Singh, Lucknow, 1935.

And finally, Pierre-Simon Laplace, the French mathematician, astronomer and physicist (1749-1827) on Hindu mathematics: “The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of antiquity, Archimedes and Apollonius.”

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