History of large numbers

The Mahayana Lalitavistara Sutra is notable for giving a very extensive such list, with terms going up to 10⁴²¹. The context is an account of a contest including writing, arithmetic, wrestling and archery, in which the Buddha was pitted against the great mathematician Arjuna and showed off his numerical skills by citing the names of the powers of ten up to 1 'tallakshana', which equals 10⁵³, but then going on to explain that this is just one of a series of counting systems that can be expanded geometrically.

The Avataṃsaka Sūtra, a text associated with the Lokottaravāda school of Buddhism, has an even more extensive list of names for numbers, and it goes beyond listing mere powers of ten introducing concatenation of exponentiation, the largest number mentioned being nirabhilapya nirabhilapya parivarta (Bukeshuo bukeshuo zhuan 不可說不可說轉), corresponding to ${\displaystyle 10^{7\times 2^{122}}}$.^[1][2] though chapter 30 (the Asamkyeyas) in Thomas Cleary's translation of it we find the definition of the number "untold" as exactly 10^10*2122, expanded in the 2nd verses to 10^4*5*2121 and continuing a similar expansion indeterminately. Examples for other names given in the Avatamsaka Sutra include: asaṃkhyeya (असंख्येय) 10¹⁴⁰.

The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:^[3] enumerable (lowest, intermediate, and highest), innumerable (nearly innumerable, truly innumerable, and innumerably innumerable), and infinite (nearly infinite, truly infinite, infinitely infinite).

In modern India, the terms lakh for 10⁵ and crore for 10⁷ are in common use. Both are vernacular (Hindustani) forms derived from a list of names for powers of ten in Yājñavalkya Smṛti, where 10⁵ and 10⁷ named lakṣa and koṭi, respectively.

In the Western world, specific number names for larger numbers did not come into common use until quite recently. The Ancient Greeks used a system based on the myriad, that is, ten thousand, and their largest named number was a myriad myriad, or one hundred million.

In The Sand Reckoner, Archimedes (c. 287–212 BC) devised a system of naming large numbers reaching up to

${\displaystyle 10^{8\times 10^{16}}}$,

essentially by naming powers of a myriad myriad. This largest number appears because it equals a myriad myriad to the myriad myriadth power, all taken to the myriad myriadth power. This gives a good indication of the notational difficulties encountered by Archimedes, and one can propose that he stopped at this number because he did not devise any new ordinal numbers (larger than 'myriad myriadth') to match his new cardinal numbers. Archimedes only used his system up to 10⁶⁴.

Archimedes' goal was presumably to name large powers of 10 in order to give rough estimates, but shortly thereafter, Apollonius of Perga invented a more practical system of naming large numbers which were not powers of 10, based on naming powers of a myriad, for example, βΜ would be a myriad squared.

Much later, but still in antiquity, the Hellenistic mathematician Diophantus (3rd century) used a similar notation to represent large numbers.

The Romans, who were less interested in theoretical issues, expressed 1,000,000 as decies centena milia, that is, 'ten hundred thousand'; it was only in the 13th century that the (originally French) word 'million' was introduced.