History of large numbers
Encyclopedia
Different culture
s used different traditional numeral system
s for naming large numbers. The extent of large numbers used varied in each culture.
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.
The India
ns 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)
or 1037218383881977644441306597687849648128, 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 1010*2122, expanded in the 2nd verses to 1045*2121 and continuing a similar expansion indeterminately.
A few large numbers used in India by about 5th century BCE (See Georges Ifrah: A Universal History of Numbers, pp 422–423):
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
,
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,
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 .
ns, who invented the positional numeral system
, along with negative numbers and zero
, were quite advanced in this aspect. By the 7th century CE Indian mathematicians
were familiar enough with the notion of infinity as to define it as the quantity whose denominator is zero.
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 limit
s.
However, since the 19th century, mathematicians have studied transfinite number
s, 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 cardinal
s. The concept of transfinite numbers, however, was first considered by Indian Jaina mathematicians as far back as 400 BC
.
Culture
Culture is a term that has many different inter-related meanings. For example, in 1952, Alfred Kroeber and Clyde Kluckhohn compiled a list of 164 definitions of "culture" in Culture: A Critical Review of Concepts and Definitions...
s used different traditional numeral system
Numeral system
A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
s for naming large numbers. The extent of large numbers used varied in each culture.
One interesting point in using large numbers is the confusion on the term billion
Long and short scales
The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale...
and milliard in many countries, and the use of zillion to denote a very large number where precision is not required.
Ancient India
India
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
ns had a passion for high numbers, which is intimately related to their religious thought. For example, in texts belonging to the Vedic literature
Vedas
The Vedas are a large body of texts originating in ancient India. Composed in Vedic Sanskrit, the texts constitute the oldest layer of Sanskrit literature and the oldest scriptures of Hinduism....
, we find individual Sanskrit
Sanskrit
Sanskrit , is a historical Indo-Aryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...
names for each of the powers of 10 up to a trillion and even 1062. (Even today, the words 'lakh
Lakh
A lakh is a unit in the Indian numbering system equal to one hundred thousand . It is widely used both in official and other contexts in Pakistan, Bangladesh, India, Maldives, Nepal, Sri Lanka, Myanmar and is often used in Indian English.-Usage:...
' and 'crore
Crore
A crore is a unit in the Indian number system equal to ten million , or 100 lakhs. It is widely used in India, Bangladesh, Nepal, and Pakistan....
', referring to 100,000 and 10,000,000, respectively, are in common use among English-speaking Indians.) One of these Vedic texts
Vedas
The Vedas are a large body of texts originating in ancient India. Composed in Vedic Sanskrit, the texts constitute the oldest layer of Sanskrit literature and the oldest scriptures of Hinduism....
, the Yajur Veda, even discusses the concept of numeric infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
(purna "fullness"), stating that if you subtract purna from purna, you are still left with purna.
The Lalitavistara Sutra
Lalitavistara Sutra
The Lalitavistara Sutra is a Mahayana Buddhist Vaipulya sutra that describes the sports of Gautama Buddha. It is a compilation of various works by no single author and includes some material from the Sarvastivada school. The scholar P. L...
(a Mahayana
Mahayana
Mahāyāna is one of the two main existing branches of Buddhism and a term for classification of Buddhist philosophies and practice...
Buddhist
Buddhism
Buddhism is a religion and philosophy encompassing a variety of traditions, beliefs and practices, largely based on teachings attributed to Siddhartha Gautama, commonly known as the Buddha . The Buddha lived and taught in the northeastern Indian subcontinent some time between the 6th and 4th...
work) recounts a contest including writing, arithmetic, wrestling and archery, in which the Buddha
Gautama Buddha
Siddhārtha Gautama was a spiritual teacher from the Indian subcontinent, on whose teachings Buddhism was founded. In most Buddhist traditions, he is regarded as the Supreme Buddha Siddhārtha Gautama (Sanskrit: सिद्धार्थ गौतम; Pali: Siddhattha Gotama) was a spiritual teacher from the Indian...
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
Sanskrit
Sanskrit , is a historical Indo-Aryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...
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)
or 1037218383881977644441306597687849648128, which appeared as BodhisattvaBodhisattva
In Buddhism, a bodhisattva is either an enlightened existence or an enlightenment-being or, given the variant Sanskrit spelling satva rather than sattva, "heroic-minded one for enlightenment ." The Pali term has sometimes been translated as "wisdom-being," although in modern publications, and...
's maths in the Avataṃsaka Sūtra
Avatamsaka Sutra
The is one of the most influential Mahayana sutras of East Asian Buddhism. The title is rendered in English as Flower Garland Sutra, Flower Adornment Sutra, or Flower Ornament Scripture....
., though chapter 30 (the Asamkyeyas) in Thomas Cleary's translation of it we find the definition of the number "untold" as exactly 1010*2122, expanded in the 2nd verses to 1045*2121 and continuing a similar expansion indeterminately.
A few large numbers used in India by about 5th century BCE (See Georges Ifrah: A Universal History of Numbers, pp 422–423):
- lakṣáLakhA lakh is a unit in the Indian numbering system equal to one hundred thousand . It is widely used both in official and other contexts in Pakistan, Bangladesh, India, Maldives, Nepal, Sri Lanka, Myanmar and is often used in Indian English.-Usage:...
(लक्ष) —105 - kōṭiCroreA crore is a unit in the Indian number system equal to ten million , or 100 lakhs. It is widely used in India, Bangladesh, Nepal, and Pakistan....
(कोटि) —107 - ayuta (अयुता) —109
- niyuta (नियुता) —1013
- pakoti (पकोटि) —1014
- vivara (विवारा) —1015
- kshobhya (क्षोभ्या) —1017
- vivaha (विवाहा) —1019
- kotippakoti (कोटिपकोटी) —1021
- bahula (बहूला) —1023
- nagabala (नागाबाला) —1025
- nahuta (नाहूटा) —1028
- titlambha (तीतलम्भा) —1029
- vyavasthanapajnapati (व्यवस्थानापज्नापति) —1031
- hetuhila (हेतुहीला) —1033
- ninnahuta (निन्नाहुता) —1035
- hetvindriya (हेत्विन्द्रिया) —1037
- samaptalambha (समाप्तलम्भा) —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 (अब्बुदा) —1056
- nirabbuda (निर्बुद्धा) —1063
- ahaha (अहाहा) —1070
- ababa (अबाबा). —1077
- atata (अटाटा) —1084
- soganghika (सोगान्घीका) —1091
- uppala (उप्पाला) —1098
- kumuda (कुमुदा) —10105
- pundarika (पुन्डरीका) —10112
- paduma (पद्मा) —10119
- kathana (कथाना) —10126
- mahakathana (महाकथाना) —10133
- asaṃkhyeya (असंख्येय) —10140
- dhvajagranishamani (ध्वजाग्रनिशमनी) —10421
- bodhisattvaBodhisattvaIn Buddhism, a bodhisattva is either an enlightened existence or an enlightenment-being or, given the variant Sanskrit spelling satva rather than sattva, "heroic-minded one for enlightenment ." The Pali term has sometimes been translated as "wisdom-being," although in modern publications, and...
(बोधिसत्व or बोधिसत्त) —1037218383881977644441306597687849648128 - lalitavistarautra (ललितातुलनातारासूत्र) —10200infinities
- matsyaMatsyaMatsya was the first Avatar of Vishnu in Hinduism. The great flood finds mention in Hindu mythology texts like the Satapatha Brahmana, where in the Matsya Avatar takes place to save the pious and the first man, Manu and advices him to build a giant boat.-The Legend:According to the Matsya...
(मत्स्य) —10600infinities - kurmaKurmaIn Hinduism, Kurma was the second Avatar of Vishnu. Like the Matsya Avatar also belongs to the Satya yuga.-Samudra manthan :...
(कुरमा) —102000infinities - varahaVarahaVaraha is the third Avatar of the Hindu Godhead Vishnu, in the form of a Boar. He appeared in order to defeat Hiranyaksha, a demon who had taken the Earth and carried it to the bottom of what is described as the cosmic ocean in the story. The battle between Varaha and Hiranyaksha is believed to...
(वरहा) —103600infinities - narasimhaNarasimhaNarasimha or Nrusimha , also spelt as Narasingh and Narasingha, whose name literally translates from Sanskrit as "Man-lion", is an avatar of Vishnu described in the Puranas, Upanishads and other ancient religious texts of Hinduism...
(नरसिम्हा) —104800infinities - vamanaVamanaVamana is described in the Puranic texts of Hinduism as the Fifth Avatar of Vishnu, and the first incarnation of the Second Age, or the Treta yuga. Also he is the first Avatar of Vishnu which appears with a completely human form, though it was that of a dwarf brahmin. He is also sometimes known as...
(वामन) —105800infinities - parashuramaParashuramaParashurama , is the sixth avatar of Vishnu and belongs to the treta yuga, and is the son of a Brahmin father Jamadagni and mother Renuka. He is considered one of the seven immortal human. He received an axe after undertaking a terrible penance to please Shiva, from whom he learned the methods of...
(परशुराम) —106000infinities - ramaRamaRama or full name Ramachandra is considered to be the seventh avatar of Vishnu in Hinduism, and a king of Ayodhya in ancient Indian...
(राम) —106800infinities - khrishnaraja (कृष्णराज) —10infinities
- kaikiKaikiKaiki may refer to:*Kaiki Nobuhide, sumo wrestler*Caïque, is a wooden fishing boat usually found among the waters of the Ionian or Aegean Seas....
(काईकी or काइकी) —108000infinities - balaramaBalaramaBalarama , also known as Baladeva, Balabhadra and Halayudha, is the elder brother of the divine being, Krishna in Hinduism. Within Vaishnavism Hindu traditions Balarama is worshipped as an Avatar of Vishnu, and he is also listed as such in the Bhagavata Purana...
(बलराम) —109800infinities - dasavatara (दशावतारा) —1010000infinities
- bhagavatapurana (भागवतपुराण) —1018000infinities
- avatamsakasutra (अवताम्सकासुत्रा) —1030000infinities
- mahadeva (महादेव) —1050000infinities
- prajapatiPrajapatiIn Hinduism, Prajapati "lord of creatures" is a Hindu deity presiding over procreation, and protector of life. He appears as a creator deity or supreme God Viswakarma Vedic deities in RV 10 and in Brahmana literature...
(प्रजापति) —1060000infinities - jyotibaJyotibaJyotiba is a holy site of Hinduism near Wadi Ratnagiri in Kolhapur district of Maharashtra state in western India. The deity of the temple is known by the same name, and is held by the locals to be an incarnation of three gods: Brahma, Vishnu, Mahesha, and Jamadagni...
(ज्योतिबा) —1080000infinities
Classical antiquity
In the Western world, specific number namesNumber names
In linguistics, number names are specific words in a natural language that represent numbers.In writing, numerals are symbols also representing numbers...
for larger numbers
Large numbers
This article is about large numbers in the sense of numbers that are significantly larger than those ordinarily used in everyday life, for instance in simple counting or in monetary transactions...
did not come into common use until quite recently. The Ancient Greeks used a system based on the myriad
Myriad
Myriad , "numberlesscountless, infinite", is a classical Greek word for the number 10,000. In modern English, the word refers to an unspecified large quantity.-History and usage:...
, that is ten thousand; and their largest named number was a myriad myriad, or one hundred million.
In The Sand Reckoner
The Sand Reckoner
The Sand Reckoner is a work by Archimedes in which he set out to determine an upper bound for the number of grains of sand that fit into the universe. In order to do this, he had to estimate the size of the universe according to the then-current model, and invent a way to talk about extremely...
, Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
(c. 287–212 BC) devised a system of naming large numbers reaching up to
,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
Powers of 10
In mathematics, a power of ten is any of the integer powers of the number ten; in other words, ten multiplied by itself a certain number of times. By definition, the number one is a power of ten. The first few powers of ten are:...
in order to give rough estimates, but shortly thereafter,
Apollonius of Perga
Apollonius of Perga
Apollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...
invented a more practical
system of naming large numbers which were not powers of 10, based on naming powers of a myriad,
for example,
would be a myriad squared.Much later, but still in antiquity
Classical antiquity
Classical antiquity is a broad term for a long period of cultural history centered on the Mediterranean Sea, comprising the interlocking civilizations of ancient Greece and ancient Rome, collectively known as the Greco-Roman world...
, the Hellenistic mathematician
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...
Diophantus
Diophantus
Diophantus of Alexandria , sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, many of which are now lost...
(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
Million
One million or one thousand thousand, is the natural number following 999,999 and preceding 1,000,001. The word is derived from the early Italian millione , from mille, "thousand", plus the augmentative suffix -one.In scientific notation, it is written as or just 106...
' was introduced .
Medieval India
The IndiaIndia
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
ns, who invented the positional numeral system
Numeral system
A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
, along with negative numbers and zero
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
, were quite advanced in this aspect. By the 7th century CE Indian mathematicians
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...
were familiar enough with the notion of infinity as to define it as the quantity whose denominator is zero.
Infinity
- Main articles: InfinityInfinityInfinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
and Transfinite numberTransfinite numberTransfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these...
The ultimate in large numbers was, until recently, the concept of infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
, a number defined by being greater than any finite number, and used in the mathematical theory of limit
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
s.
However, since the 19th century, mathematicians have studied transfinite number
Transfinite number
Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these...
s, numbers which are not only greater than any finite number, but also, from the viewpoint of set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, larger than the traditional concept of infinity. Of these transfinite numbers, perhaps the most extraordinary, and arguably, if they exist, "largest", are the large cardinal
Large cardinal property
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large"...
s. The concept of transfinite numbers, however, was first considered by Indian Jaina mathematicians as far back as 400 BC
400 BC
Year 400 BC was a year of the pre-Julian Roman calendar. At the time, it was known as the Year of the Tribunate of Esquilinus, Capitolinus, Vulso, Medullinus, Saccus and Vulscus...
.
Further reading
- Georges Ifrah, The Universal History of Numbers, ISBN 1-86046-324-X