Two naming scales for
large numbers
Large numbers are numbers significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as mathematics, cosmology, cryptography, and statistical m ...
have been used in English and other European languages since the early modern era: the
long and short scale
The long and short scales are two of several naming systems for integer powers of ten which use some of the same terms for different magnitudes.
For whole numbers smaller than 1,000,000,000 (109), such as one thousand or one million, the t ...
s. Most English variants use the short scale today, but the long scale remains dominant in many non-English-speaking areas, including
continental Europe and
Spanish
Spanish might refer to:
* Items from or related to Spain:
**Spaniards are a nation and ethnic group indigenous to Spain
**Spanish language, spoken in Spain and many Latin American countries
**Spanish cuisine
Other places
* Spanish, Ontario, Can ...
-speaking countries in
Latin America
Latin America or
* french: Amérique Latine, link=no
* ht, Amerik Latin, link=no
* pt, América Latina, link=no, name=a, sometimes referred to as LatAm is a large cultural region in the Americas where Romance languages — languages derived f ...
. These naming procedures are based on taking the number ''n'' occurring in 10
3''n''+3 (short scale) or 10
6''n'' (long scale) and concatenating Latin roots for its units, tens, and hundreds place, together with the suffix ''-illion''.
Names of numbers above a trillion are rarely used in practice; such large numbers have practical usage primarily in the scientific domain, where powers of ten are expressed as ''10'' with a numeric superscript.
Indian English does not use millions, but has its own system of large numbers including
lakhs and
crores.
[ English also has many words, such as "zillion", used informally to mean large but unspecified amounts; see ]indefinite and fictitious numbers
Many languages have words expressing indefinite and fictitious numbers—inexact terms of indefinite size, used for comic effect, for exaggeration, as placeholder names, or when precision is unnecessary or undesirable. One technical term for such ...
.
Standard dictionary numbers
Usage:
* Short scale
The long and short scales are two of several naming systems for integer powers of ten which use some of the same terms for different magnitudes.
For whole numbers smaller than 1,000,000,000 (109), such as one thousand or one million, the t ...
: US, English Canada
Canada comprises that part of the population within Canada, whether of British origin or otherwise, that speaks English.
The term ''English Canada'' can also be used for one of the following:
#Describing all the provinces of Canada tha ...
, modern British
British may refer to:
Peoples, culture, and language
* British people, nationals or natives of the United Kingdom, British Overseas Territories, and Crown Dependencies.
** Britishness, the British identity and common culture
* British English, ...
, Australia, and Eastern Europe
Eastern Europe is a subregion of the European continent. As a largely ambiguous term, it has a wide range of geopolitical, geographical, ethnic, cultural, and socio-economic connotations. The vast majority of the region is covered by Russia, whic ...
* Long scale: French Canada
French Canadians (referred to as Canadiens mainly before the twentieth century; french: Canadiens français, ; feminine form: , ), or Franco-Canadians (french: Franco-Canadiens), refers to either an ethnic group who trace their ancestry to Fre ...
, older British
British may refer to:
Peoples, culture, and language
* British people, nationals or natives of the United Kingdom, British Overseas Territories, and Crown Dependencies.
** Britishness, the British identity and common culture
* British English, ...
, Western
Western may refer to:
Places
*Western, Nebraska, a village in the US
*Western, New York, a town in the US
*Western Creek, Tasmania, a locality in Australia
*Western Junction, Tasmania, a locality in Australia
*Western world, countries that id ...
& Central Europe
Central Europe is an area of Europe between Western Europe and Eastern Europe, based on a common historical, social and cultural identity. The Thirty Years' War (1618–1648) between Catholicism and Protestantism significantly shaped the a ...
Apart from ''million'', the words in this list ending with -''illion'' are all derived by adding prefixes (''bi''-, ''tri''-, etc., derived from Latin) to the stem -''illion''.[ ''Centillion''][ appears to be the highest name ending in -"illion" that is included in these dictionaries. ''Trigintillion'', often cited as a word in discussions of names of large numbers, is not included in any of them, nor are any of the names that can easily be created by extending the naming pattern (''unvigintillion'', ''duovigintillion'', ''duoquinquagintillion'', etc.).
All of the dictionaries included ''googol'' and ''googolplex'', generally crediting it to the Kasner and Newman book and to Kasner's nephew. None include any higher names in the googol family (googolduplex, etc.). The ''Oxford English Dictionary'' comments that ''googol'' and ''googolplex'' are "not in formal mathematical use".
]
Usage of names of large numbers
Some names of large numbers, such as ''million'', ''billion'', and ''trillion'', have real referents in human experience, and are encountered in many contexts. At times, the names of large numbers have been forced into common usage as a result of hyperinflation
In economics, hyperinflation is a very high and typically accelerating inflation. It quickly erodes the real value of the local currency, as the prices of all goods increase. This causes people to minimize their holdings in that currency as t ...
. The highest numerical value banknote ever printed was a note for 1 sextillion pengő (1021 or 1 milliard bilpengő as printed) printed in Hungary
Hungary ( hu, Magyarország ) is a landlocked country in Central Europe. Spanning of the Pannonian Basin, Carpathian Basin, it is bordered by Slovakia to the north, Ukraine to the northeast, Romania to the east and southeast, Serbia to the ...
in 1946. In 2009, Zimbabwe
Zimbabwe (), officially the Republic of Zimbabwe, is a landlocked country located in Southeast Africa, between the Zambezi and Limpopo Rivers, bordered by South Africa to the south, Botswana to the south-west, Zambia to the north, and ...
printed a 100 trillion (1014) Zimbabwean dollar
The Zimbabwean dollar (sign: $, or Z$ to distinguish it from other dollar-denominated currencies) was the name of four official currencies of Zimbabwe from 1980 to 12 April 2009. During this time, it was subject to periods of extreme inflat ...
note, which at the time of printing was worth about US$30.[
Names of larger numbers, however, have a tenuous, artificial existence, rarely found outside definitions, lists, and discussions of how large numbers are named. Even well-established names like ''sextillion'' are rarely used, since in the context of science, including astronomy, where such large numbers often occur, they are nearly always written using ]scientific notation
Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, o ...
. In this notation, powers of ten are expressed as ''10'' with a numeric superscript, e.g. "The X-ray emission of the radio galaxy is ." When a number such as 1045 needs to be referred to in words, it is simply read out as "ten to the forty-fifth". This is easier to say and less ambiguous than "quattuordecillion", which means something different in the long scale and the short scale.
When a number represents a quantity rather than a count, SI prefixes can be used—thus " femtosecond", not "one quadrillionth of a second"—although often powers of ten are used instead of some of the very high and very low prefixes. In some cases, specialized units are used, such as the astronomer's parsec
The parsec (symbol: pc) is a unit of length used to measure the large distances to astronomical objects outside the Solar System, approximately equal to or (au), i.e. . The parsec unit is obtained by the use of parallax and trigonometry, an ...
and light year or the particle physicist's barn
A barn is an agricultural building usually on farms and used for various purposes. In North America, a barn refers to structures that house livestock, including cattle and horses, as well as equipment and fodder, and often grain.Alle ...
.
Nevertheless, large numbers have an intellectual fascination and are of mathematical interest, and giving them names is one way people try to conceptualize and understand them.
One of the earliest examples of this is ''The Sand Reckoner
''The Sand Reckoner'' ( el, Ψαμμίτης, ''Psammites'') is a work by Archimedes, an Ancient Greek mathematician of the 3rd century BC, in which he set out to determine an upper bound for the number of grains of sand that fit into the unive ...
'', in which Archimedes gave a system for naming large numbers. To do this, he called the numbers up to a myriad myriad (108) "first numbers" and called 108 itself the "unit of the second numbers". Multiples of this unit then became the second numbers, up to this unit taken a myriad myriad times, 108·108=1016. This became the "unit of the third numbers", whose multiples were the third numbers, and so on. Archimedes continued naming numbers in this way up to a myriad myriad times the unit of the 108-th numbers, i.e. and embedded this construction within another copy of itself to produce names for numbers up to Archimedes then estimated the number of grains of sand that would be required to fill the known universe, and found that it was no more than "one thousand myriad of the eighth numbers" (1063).
Since then, many others have engaged in the pursuit of conceptualizing and naming numbers that have no existence outside the imagination. One motivation for such a pursuit is that attributed to the inventor of the word '' googol'', who was certain that any finite number "had to have a name". Another possible motivation is competition between students in computer programming courses, where a common exercise is that of writing a program to output numbers in the form of English words.
Most names proposed for large numbers belong to systematic schemes which are extensible. Thus, many names for large numbers are simply the result of following a naming system to its logical conclusion—or extending it further.
Origins of the "standard dictionary numbers"
The words ''bymillion'' and ''trimillion'' were first recorded in 1475 in a manuscript of Jehan Adam. Subsequently, Nicolas Chuquet
Nicolas Chuquet (; born ; died ) was a French mathematician. He invented his own notation for algebraic concepts and exponentiation. He may have been the first mathematician to recognize zero and negative numbers as exponents.
In 1475, Jehan A ...
wrote a book ''Triparty en la science des nombres'' which was not published during Chuquet's lifetime. However, most of it was copied by Estienne de La Roche for a portion of his 1520 book, '' L'arismetique''. Chuquet's book contains a passage in which he shows a large number marked off into groups of six digits, with the comment:
Ou qui veult le premier point peult signiffier million Le second point byllion Le tiers point tryllion Le quart quadrillion Le cinqe quyllion Le sixe sixlion Le sept.e septyllion Le huyte ottyllion Le neufe nonyllion et ainsi des ault's se plus oultre on vouloit preceder
(Or if you prefer the first mark can signify million, the second mark byllion, the third mark tryllion, the fourth quadrillion, the fifth quyillion, the sixth sixlion, the seventh septyllion, the eighth ottyllion, the ninth nonyllion and so on with others as far as you wish to go).
Adam and Chuquet used the long scale of powers of a million; that is, Adam's ''bymillion'' (Chuquet's ''byllion'') denoted 1012, and Adam's ''trimillion'' (Chuquet's ''tryllion'') denoted 1018.
The googol family
The names '' googol'' and ''googolplex
A googolplex is the number 10, or equivalently, 10 or 1010,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 . Written out in ordinary decimal notation, it is 1 fol ...
'' were invented by Edward Kasner
Edward Kasner (April 2, 1878 – January 7, 1955) was an American mathematician who was appointed Tutor on Mathematics in the Columbia University Mathematics Department. Kasner was the first Jewish person appointed to a faculty position in t ...
's nephew Milton Sirotta and introduced in Kasner and Newman's 1940 book ''Mathematics and the Imagination
''Mathematics and the Imagination'' is a book published in New York by Simon & Schuster in 1940. The authors are Edward Kasner and James R. Newman. The illustrator Rufus Isaacs provided 169 figures. It rapidly became a best-seller and received s ...
''[ in the following passage:
]John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
and Richard K. Guy[ have suggested that ''N-plex'' be used as a name for 10N. This gives rise to the name ''googolplexplex'' for 10googolplex = 10. Conway and Guy][ have proposed that ''N-minex'' be used as a name for 10−N, giving rise to the name ''googolminex'' for the ]reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
of a googolplex. None of these names are in wide use, nor are any currently found in dictionaries.
The names ''googol'' and ''googolplex'' inspired the name of the Internet company Google
Google LLC () is an American Multinational corporation, multinational technology company focusing on Search Engine, search engine technology, online advertising, cloud computing, software, computer software, quantum computing, e-commerce, ar ...
and its corporate headquarters, the Googleplex, respectively.
Extensions of the standard dictionary numbers
This section illustrates several systems for naming large numbers, and shows how they can be extended past ''vigintillion''.
Traditional British usage assigned new names for each power of one million (the long scale): ; ; ; and so on. It was adapted from French usage, and is similar to the system that was documented or invented by Chuquet.
Traditional American usage (which was also adapted from French usage but at a later date), Canadian, and modern British usage assign new names for each power of one thousand (the short scale
The long and short scales are two of several naming systems for integer powers of ten which use some of the same terms for different magnitudes.
For whole numbers smaller than 1,000,000,000 (109), such as one thousand or one million, the t ...
.) Thus, a ''billion'' is 1000 × 10002 = 109; a ''trillion'' is 1000 × 10003 = 1012; and so forth. Due to its dominance in the financial world (and by the US dollar
The United States dollar ( symbol: $; code: USD; also abbreviated US$ or U.S. Dollar, to distinguish it from other dollar-denominated currencies; referred to as the dollar, U.S. dollar, American dollar, or colloquially buck) is the officia ...
), this was adopted for official United Nations
The United Nations (UN) is an intergovernmental organization whose stated purposes are to maintain international peace and security, develop friendly relations among nations, achieve international cooperation, and be a centre for harmoniz ...
documents.
Traditional French usage has varied; in 1948, France, which had originally popularized the short scale worldwide, reverted to the long scale.
The term ''milliard'' is unambiguous and always means 109. It is seldom seen in American usage and rarely in British usage, but frequently in continental European usage. The term is sometimes attributed to French mathematician Jacques Peletier du Mans circa 1550 (for this reason, the long scale is also known as the ''Chuquet-Peletier'' system), but the Oxford English Dictionary
The ''Oxford English Dictionary'' (''OED'') is the first and foundational historical dictionary of the English language, published by Oxford University Press (OUP). It traces the historical development of the English language, providing a co ...
states that the term derives from post-Classical Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
term ''milliartum'', which became ''milliare'' and then ''milliart'' and finally our modern term.
Concerning names ending in -illiard for numbers 106''n''+3, ''milliard'' is certainly in widespread use in languages other than English, but the degree of actual use of the larger terms is questionable. The terms "Milliarde" in German, "miljard" in Dutch, "milyar" in Turkish, and "миллиард," milliard (transliterated) in Russian, are standard usage when discussing financial topics.
For additional details, see billion
Billion is a word for a large number, and it has two distinct definitions:
*1,000,000,000, i.e. one thousand million, or (ten to the ninth power), as defined on the short scale. This is its only current meaning in English.
* 1,000,000,000,000, i. ...
and long and short scale
The long and short scales are two of several naming systems for integer powers of ten which use some of the same terms for different magnitudes.
For whole numbers smaller than 1,000,000,000 (109), such as one thousand or one million, the t ...
.
The naming procedure for large numbers is based on taking the number ''n'' occurring in 103''n''+3 (short scale) or 106''n'' (long scale) and concatenating Latin roots for its units, tens, and hundreds place, together with the suffix ''-illion''. In this way, numbers up to 103·999+3 = 103000 (short scale) or 106·999 = 105994 (long scale) may be named. The choice of roots and the concatenation procedure is that of the standard dictionary numbers if ''n'' is 9 or smaller. For larger ''n'' (between 10 and 999), prefixes can be constructed based on a system described by Conway and Guy.[ Today, sexdecillion and novemdecillion are standard dictionary numbers and, using the same reasoning as Conway and Guy did for the numbers up to nonillion, could probably be used to form acceptable prefixes. The Conway–Guy system for forming prefixes:
:(*) When preceding a component marked S or X, "tre" changes to "tres" and "se" to "ses" or "sex"; similarly, when preceding a component marked M or N, "septe" and "nove" change to "septem" and "novem" or "septen" and "noven".
Since the system of using Latin prefixes will become ambiguous for numbers with exponents of a size which the Romans rarely counted to, like 106,000,258, Conway and Guy co-devised with Allan Wechsler the following set of consistent conventions that permit, in principle, the extension of this system indefinitely to provide English short-scale names for any integer whatsoever.][ The name of a number 103''n''+3, where ''n'' is greater than or equal to 1000, is formed by concatenating the names of the numbers of the form 103''m''+3, where ''m'' represents each group of comma-separated digits of ''n'', with each but the last "-illion" trimmed to "-illi-", or, in the case of ''m'' = 0, either "-nilli-" or "-nillion".][ For example, 103,000,012, the 1,000,003rd "-illion" number, equals one "millinillitrillion"; 1033,002,010,111, the 11,000,670,036th "-illion" number, equals one "undecillinilliseptuagintasescentillisestrigintillion"; and 1029,629,629,633, the 9,876,543,210th "-illion" number, equals one "nonilliseseptuagintaoctingentillitresquadragintaquingentillideciducentillion".][
The following table shows number names generated by the system described by Conway and Guy for the short and long scales.
: Googolplex's short scale name is derived from it equal to ten of the 3,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,332nd "-illion"s (This is the value of n when 10 X 10 = 10)
: Googolplex's long scale name (both traditional British and traditional European) is derived from it being equal to ten thousand of the 1,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666th "-illion"s (This is the value of n when 10,000 X 10 = 10).
]
Binary prefixes
The International System of Quantities
The International System of Quantities (ISQ) consists of the quantities used in physics and in modern science in general, starting with basic quantities such as length and mass, and the relationships between those quantities. This system underli ...
(ISQ) defines a series of prefixes denoting integer powers of 1024 between 10241 and 10248.[
]
Other large numbers used in mathematics and physics
* Avogadro number
The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining co ...
* Graham's number
* Skewes' number
In number theory, Skewes's number is any of several large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number x for which
:\pi(x) > \operatorname(x),
where is the prime-counting function ...
* Steinhaus–Moser notation
* TREE(3)
* Rayo's number
See also
* -yllion
* Asaṃkhyeya
* Chinese numerals
Chinese numerals are words and characters used to denote numbers in Chinese.
Today, speakers of Chinese use three written numeral systems: the system of Arabic numerals used worldwide, and two indigenous systems. The more familiar indigenous s ...
* 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 ma ...
* Indefinite and fictitious numbers
Many languages have words expressing indefinite and fictitious numbers—inexact terms of indefinite size, used for comic effect, for exaggeration, as placeholder names, or when precision is unnecessary or undesirable. One technical term for such ...
* Indian numbering system
* Knuth's up-arrow notation
In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976.
In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called ''hyperoperati ...
* Law of large numbers
* List of numbers
This is a list of notable numbers and articles about Notability, notable numbers. The list does not contain all numbers in existence as most of the Set (mathematics), number sets are infinite. Numbers may be included in the list based on their ...
* Long and short scale
The long and short scales are two of several naming systems for integer powers of ten which use some of the same terms for different magnitudes.
For whole numbers smaller than 1,000,000,000 (109), such as one thousand or one million, the t ...
* Metric prefix
A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic. Each prefix has a unique symbol that is prepended to any unit symbol. The pr ...
* Names of small numbers
* Number names
* Number prefix
Numeral or number prefixes are prefixes derived from numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example:
* unicycle, bicycle, tricycle (1-cycle, 2-cycle, 3-cy ...
* Orders of magnitude
An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic dis ...
* Orders of magnitude (data)
An order of magnitude is usually a factor of ten. Thus, four orders of magnitude is a factor of 10,000 or 104.
This article presents a list of multiples, sorted by orders of magnitude, for units of information measured in bits and bytes.
The byt ...
* Orders of magnitude (numbers)
This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantities and probabilities. Each number is given a name in the short scale, which is used in English-speaking countries, as well as a ...
* Power of 10
A power of 10 is any of the integer powers of the number ten; in other words, ten multiplied by itself a certain number of times (when the power is a positive integer). By definition, the number one is a power (the zeroth power) of ten. The f ...
References
{{DEFAULTSORT:Names Of Large Numbers
*
Numeral systems
Numerals
Numbers
Integers