History of large numbers

Different cultures used different traditional numeral systems for naming large numbers. The extent of large numbers used varied in each culture.

Two interesting points in using large numbers are 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.

Ancient India

Hindu units of time on a logarithmic scale.

The Indians had a passion for high numbers, which is intimately related to their religious thought. For example, in texts belonging to the Vedic literature, we find individual Sanskrit names for each of the powers of 10 up to a trillion and even 10⁶². (Even today, the words 'lakh' and 'crore', referring to 100,000 and 10,000,000, respectively, are in common use among English-speaking Indians.) One of these Vedic texts, the Yajur Veda, even discusses the concept of numeric infinity (purna "fullness"), stating that if you subtract purna from purna, you are still left with purna.

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.

There is also an analogous system of Sanskrit terms for fractional numbers, capable of dealing with both very large and very small numbers.

Larger number in Buddhism works up to Bukeshuo bukeshuo zhuan (不可說不可說轉) $10^{7\times 2^{122}}$ or 10^{37218383881977644441306597687849648128}, which appeared as Bodhisattva's maths in the Avataṃsaka Sūtra., 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 10^45*2121 and continuing a similar expansion indeterminately.

A few large numbers used in India by about 5th century BC (See Georges Ifrah: A Universal History of Numbers, pp 422–423):

• lakṣá (लक्ष) —10⁵
• kōṭi (कोटि) —10⁷
• ayuta (अयुत) —10⁹
• niyuta (नियुत) —10¹³
• pakoti (पकोटि) —10¹⁴
• vivara (विवारा) —10¹⁵
• kshobhya (क्षोभ्या) —10¹⁷
• vivaha (विवाहा) —10¹⁹
• kotippakoti (कोटिपकोटी) —10²¹
• bahula (बहुल) —10²³
• nagabala (नागाबाला) —10²⁵
• nahuta (नाहूटा) —10²⁸
• titlambha (तीतलम्भा) —10²⁹
• vyavasthanapajnapati (व्यवस्थानापज्नापति) —10³¹
• hetuhila (हेतुहीला) —10³³
• ninnahuta (निन्नाहुता) —10³⁵
• hetvindriya (हेत्विन्द्रिय) —10³⁷
• samaptalambha (समाप्तलम्भ) —10³⁹
• gananagati (गनानागती) —10⁴¹
• akkhobini (अक्खोबिनि) —10⁴²
• niravadya (निरावाद्य) —10⁴³
• mudrabala (मुद्राबाला) —10⁴⁵
• sarvabala (सर्वबाला) —10⁴⁷
• bindu (बिंदु or बिन्दु) —10⁴⁹
• sarvajna (सर्वज्ञ) —10⁵¹
• vibhutangama (विभुतन्गमा) —10⁵³
• abbuda (अब्बुद) —10⁵⁶
• nirabbuda (निर्बुद्ध) —10⁶³
• ahaha (अहाहा) —10⁷⁰
• ababa (अबाबा). —10⁷⁷
• atata (अटाटा) —10⁸⁴
• soganghika (सोगान्घीक) —10⁹¹
• uppala (उप्पल) —10⁹⁸
• kumuda (कुमुद) —10¹⁰⁵
• pundarika (पुन्डरीक) —10¹¹²
• paduma (पद्म) —10¹¹⁹
• kathana (कथन) —10¹²⁶
• mahakathana (महाकथन) —10¹³³
• asaṃkhyeya (असंख्येय) —10¹⁴⁰
• dhvajagranishamani (ध्वजाग्रनिशमनी) —10⁴²¹
• bodhisattva (बोधिसत्व or बोधिसत्त) —10^{37218383881977644441306597687849648128}
• lalitavistarautra (ललितातुलनातारासूत्र) —10²⁰⁰infinities
• matsya (मत्स्य) —10⁶⁰⁰infinities
• kurma (कूर्म) —10²⁰⁰⁰infinities
• varaha (वराह) —10³⁶⁰⁰infinities
• narasimha (नरसिम्हा) —10⁴⁸⁰⁰infinities
• vamana (वामन) —10⁵⁸⁰⁰infinities
• parashurama (परशुराम) —10⁶⁰⁰⁰infinities
• rama (राम) —10⁶⁸⁰⁰infinities
• khrishnaraja (कृष्णराज) —10infinities
• kalki (कल्कि) —10⁸⁰⁰⁰infinities
• balarama (बलराम) —10⁹⁸⁰⁰infinities
• dasavatara (दशावतार) —10¹⁰⁰⁰⁰infinities
• bhagavatapurana (भागवतपुराण) —10¹⁸⁰⁰⁰infinities
• avatamsakasutra (अवतांशकासूत्र) —10³⁰⁰⁰⁰infinities
• mahadeva (महादेव) —10⁵⁰⁰⁰⁰infinities
• prajapati (प्रजापति) —10⁶⁰⁰⁰⁰infinities
• jyotiba (ज्योतिबा) —10⁸⁰⁰⁰⁰infinities

Classical antiquity

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

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

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,

$M^{\!\!\!\!\! {}^\beta}$ 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 .

Medieval India

The Indians, who invented the positional numeral system, along with negative numbers and zero, were quite advanced in this aspect. By the 7th century, Indian mathematicians were familiar enough with the notion of infinity as to define it as the quantity whose denominator is zero.

Modern use of large finite numbers

Far larger finite numbers than any of these occur in modern mathematics. See for instance Graham's number which is too large to express using exponentiation or even tetration. For more about modern usage for large numbers see Large numbers.

Infinity

Main articles: Infinity and Transfinite number

The ultimate in large numbers was, until recently, the concept of infinity, a number defined by being greater than any finite number, and used in the mathematical theory of limits.

However, since the 19th century, mathematicians have studied transfinite numbers, numbers which are not only greater than any finite number, but also, from the viewpoint of set theory, larger than the traditional concept of infinity. Of these transfinite numbers, perhaps the most extraordinary, and arguably, if they exist, "largest", are the large cardinals. The concept of transfinite numbers, however, was first considered by Indian Jaina mathematicians as far back as 400 BC.

Further reading

• Georges Ifrah, The Universal History of Numbers, ISBN 1-86046-324-X