History of large numbers
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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. This is also the largest number named in the Bible, for example, Daniel 7:10.
In The Sand Reckoner, Archimedes devised a system of naming large numbers reaching up to
- <math>10^{8* 10^{16}}<math>,
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 invented a more practical system of naming large numbers which were not powers of 10, based on naming powers of a myriad, for example,
- <math> M^{\!\!\!\!\! {}^\beta}<math> would be a myriad squared.
Much later, but still in Greek Antiquity, Diophantus 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 .
Indians, who invented the positional numeral system and the zero, were also quite advanced in this aspect. By the 7th century AD Indian mathematicians were familiar enough with the notion of infinity as to define it as the quantity whose denominator is zero.
This achievement might be explained by the Indians' passion for high numbers, which is intimately related to their religious thought. For example, in texts belonging to the Vedic literature which are dated around the third century AD we find individual Sanskrit names for each of the powers of 10 up to 1012. (Even today, the words 'lakh' and 'koti', referring to 100,000 and 10,000,000, respectively, are in common use among English-speaking Indians.)
The 'Lalitavistara Sutra' (a Mahayana Buddhist work) recounts 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 last number at which he arrived after going through nine successive counting systems was 10421, that is, a 1 followed by 421 zeros.
It is noteworthy that there is also an analogous system of Sanskrit terms for fractional numbers, which shows the capacity to deal with both very high and very small numbers.
One interesting point in using large numbers is the confusion on the term billion and milliard in many countries, and the use of zillion to denote a very large number where precision is not required.
Further reading
- Georges Ifrah, The Universal History of Numbers, ISBN 186046324X