Names of large numbers
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This article lists and discusses the usage and derivation of names of large numbers, together with their possible extensions.
The following table lists those names of large numbers which are found in many English dictionaries and thus have a special claim to being "real words". The "Traditional British" values shown are unused in American English and are obsolete^{[citation needed]} in British English, but are dominant in many nonEnglishspeaking areas, including continental Europe and Spanishspeaking countries in Latin America; see Long and short scales.
[edit] The "standard dictionary numbers"
Name  Short scale (USA and Modern British) 
Long scale (Continental Europe and/or Traditional British) 
Authorities  

AHD4^{[1]}  COD^{[2]}  OED2^{[3]}  OEDnew^{[4]}  RHD2^{[5]}  SOED3^{[6]}  W3^{[7]}  UM^{[8]}  
million  10^{6}  10^{6}  ✓  ✓  ✓  ✓  ✓  ✓  ✓  
milliard  10^{9}  ✓  ✓  ✓  ✓  
billion  10^{9}  10^{12}  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓ 
billiard  10^{15}  ^{[9]}  ^{[9]}  ✓  
trillion  10^{12}  10^{18}  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓ 
trilliard  10^{21}  ^{[9]}  ^{[9]}  ^{[9]}  ✓  
quadrillion  10^{15}  10^{24}  ✓  ✓  ✓  ✓  ✓  ✓  ✓  
quintillion  10^{18}  10^{30}  ✓  ✓  ✓  ✓  ✓  ✓  ✓  
sextillion  10^{21}  10^{36}  ✓  ✓  ✓  ✓  ✓  ✓  ✓  
septillion  10^{24}  10^{42}  ✓  ✓  ✓  ✓  ✓  ✓  ✓  
octillion  10^{27}  10^{48}  ✓  ✓  ✓  ✓  ✓  ✓  ✓  
nonillion  10^{30}  10^{54}  ✓  ✓  ✓  ✓  ✓  ✓  ✓  
decillion  10^{33}  10^{60}  ✓  ✓  ✓  ✓  ✓  ✓  ✓  
undecillion  10^{36}  10^{66}  ✓  ✓  ✓  ✓  
duodecillion  10^{39}  10^{72}  ✓  ✓  ✓  ✓  
tredecillion  10^{42}  10^{78}  ✓  ✓  ✓  ✓  
quattuordecillion  10^{45}  10^{84}  ✓  ✓  ✓  ✓  
quindecillion (quinquadecillion)  10^{48}  10^{90}  ✓  ✓  ✓  ✓  
sexdecillion (sedecillion)  10^{51}  10^{96}  ✓  ✓  ✓  ✓  
septendecillion  10^{54}  10^{102}  ✓  ✓  ✓  ✓  
octodecillion  10^{57}  10^{108}  ✓  ✓  ✓  ✓  
novemdecillion (novendecillion)  10^{60}  10^{114}  ✓  ✓  ✓  ✓  
vigintillion  10^{63}  10^{120}  ✓  ✓  ✓  ✓  ✓  ✓  ✓  
centillion  10^{303}  10^{600}  ✓  ✓  ✓  ✓ 
Name  Value  Authorities  

AHD4  COD  OED2  OEDnew  RHD2  SOED3  W3  UM  
googol  10^{100}  ✓  ✓  ✓  ✓  ✓  ✓  ✓  
googolplex  10^{googol}  ✓  ✓  ✓  ✓  ✓  ✓ 
✓ 
Apart from million, the words in this list ending with illion are all derived by adding prefixes (bi, tri, etc.) to the stem illion.^{[10]} centillion^{[11]} 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".
 In the book Fast Food Nation, author Eric Schlosser claims a Geographic Information System named "Quintillion" is used by McDonald's to analyze data to help predict a new location for one of its restaurants. According to Schlosser, Quintillion uses data such as satellite photos, income, new housing plans, and road layouts to predict future incomes and population patterns.^{[12]}
 The term Vigintillion was used by H. P. Lovecraft in his short stories The Call of Cthulhu and The Dunwich Horror.
 In his book Postsingular, Rudy Rucker jokingly suggested the term "umptisquiddlyzillion" as a substitute for duodecillion (10^{39}).
[edit] 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 excessive inflation. The highest numerical value banknote ever printed was a note for 1 sextillion pengő (10^{21} or 1 milliard bilpengő as printed) printed in Hungary in 1946. In 2009, Zimbabwe printed a 100 trillion Zimbabwean dollar note, which at the time of printing was only worth about 30 US dollars.^{[13]}
Names of larger numbers, however, have a tenuous, artificial existence. Although they may be found in dictionaries, these names are rarely found outside definitions, lists, and discussions of the ways in which large numbers are named. Even wellestablished names like sextillion are rarely used, since in the contexts of science, astronomy, and engineering, where large numbers often occur, numbers are usually written using scientific notation. In this notation, used since the 1800s, powers of ten are expressed as 10 with a numeric superscript, e.g., "The Xray emission of the radio galaxy is 1.3·10^{45} ergs." When a number such as 10^{45} needs to be referred to in words, it is simply read out: "ten to the fortyfifth." This is just as easy to say, easier to understand, 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; one says "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 and light year or the particle physicist's barn.
Nevertheless, large numbers have an intellectual fascination and are of mathematical interest, and giving them names is one of the ways in which people try to conceptualize and understand them.
One of the first examples of this is The Sand Reckoner, in which Archimedes gave a system for naming large numbers. To do this, he called the numbers up to a myriad myriad (10^{8}) "first numbers" and called 10^{8} 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, 10^{8}·10^{8}=10^{16}. 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 10^{8}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" (10^{63}.)
Since then, many others have engaged in the pursuit of conceptualizing and naming numbers that really have no existence outside of 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.
It should be noted, too, that 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.
[edit] Adam, Chuquet and the 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 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 poit tryllion Le quart quadrillion Le cinq^{e} quyllion Le six^{e} sixlion Le sept.^{e} septyllion Le huyt^{e} ottyllion Le neuf^{e} 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).
Chuquet is sometimes credited with inventing the names million, billion, trillion, quadrillion, and so forth. This is an oversimplification.
 Million was certainly not invented by Adam or Chuquet. Milion is an Old French word thought to derive from Old Italian milione, an intensification of mille, a thousand. That is, a million is a big thousand, much as 1728 is a great gross.
 From the way in which Adam and Chuquet use the words, it can be inferred that they were recording usage rather than inventing it. One obvious possibility is that words similar to billion and trillion were already in use and wellknown, but that Chuquet, an expert in exponentiation, extended the naming scheme and invented the names for the higher powers.
 Notice that Chuquet's names are only similar to, not identical to, the modern ones.
Adam and Chuquet used the long scale of powers of a million; that is, Adam's bymillion (Chuquet's byllion) denoted 10^{12}, and Adam's trimillion (Chuquet's tryllion) denoted 10^{18}.
[edit] An aidememoire
An easy way to find the value of the above numbers in the short scale is to take the number indicated by the prefix (such as 2 in billion, 4 in quadrillion, 18 in octodecillion, etc.), add one to it, and multiply that result by 3. For example, in a trillion, the prefix is tri, meaning 3. Adding 1 to it gives 4. Now multiplying 4 by 3 gives us 12, which is the power to which 10 is to be raised to express a shortscale trillion in scientific notation: one trillion = 10^{12}.
In the long scales, this is done simply by multiplying the number from the prefix by 6. For example, in a billion, the prefix is bi, meaning 2. Multiplying 2 by 6 gives us 12, which is the power to which 10 is to be raised to express a longscale billion in scientific notation: one billion = 10^{12}. The intermediate values (billiard, trilliard, etc.) can be converted in a similar fashion, by adding ½ to the number from the prefix and then multiplying by six. For example, in a septilliard, the prefix is sept meaning 7. Multiplying 7½ by 6 yields 45, and one septilliard equals 10^{45}. Doubling the prefix and adding one then multiplying the result by three would give the same result.
These mechanisms are illustrated in the table in long and short scales.
[edit] The googol family
The names googol and googolplex were invented by Edward Kasner's nephew, Milton Sirotta, and introduced in Kasner and Newman's 1940 book, Mathematics and the Imagination,^{[14]} in the following passage:
Words of wisdom are spoken by children at least as often as by scientists. The name "googol" was invented by a child (Dr. Kasner's nineyearold nephew) who was asked to think up a name for a very big number, namely 1 with a hundred zeroes after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested "googol" he gave a name for a still larger number: "Googolplex". A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out. It was first suggested that a googolplex should be 1, followed by writing zeros until you got tired. This is a description of what would actually happen if one actually tried to write a googolplex, but different people get tired at different times and it would never do to have Carnera a better mathematician than Dr. Einstein, simply because he had more endurance. The googolplex is, then, a specific finite number, with so many zeros after the 1 that the number of zeros is a googol.
Value  Name  Authority 

10^{100}  Googol  Kasner and Newman, dictionaries (see above) 
10^{googol} =  Googolplex  Kasner and Newman, dictionaries (see above) 
Conway and Guy ^{[15]} have suggested that Nplex be used as a name for 10^{N}. This gives rise to the name googolplexplex for 10^{googolplex}; however, the word googolplexian is given by one site. In addition, the terms googolduplex, googoltriplex, etc. have been coined by various persons for the numbers 10^{googolplex}, 10^{googolduplex}, etc.^{[citation needed]} Conway and Guy ^{[15]} have proposed that Nminex be used as a name for 10^{N}, giving rise to the name googolminex for the reciprocal of a googolplex. None of these names are in wide use, nor are any currently found in dictionaries.^{[citation needed]}
[edit] Extensions of the standard dictionary numbers
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This table 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): 1,000,000 = 1 million; 1,000,000^{2} = 1 billion; 1,000,000^{3} = 1 trillion; 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, oddly enough, was also adapted from French usage but at a later date), and modern British usage, assigns new names for each power of one thousand (the short scale.) Thus, a billion is 1000 × 1000^{2} = 10^{9}; a trillion is 1000 × 1000^{3} = 10^{12}; and so forth. Due to its dominance in the financial world (and by the USdollar) this was adopted for official United Nations documents.
Traditional French usage has varied; in 1948, France, which had been using the short scale, reverted to the long scale.
The term milliard is unambiguous and always means 10^{9}. It is almost never seen in American usage, rarely in British usage, and frequently in European usage. The term is sometimes attributed to a French mathematician named Jacques Peletier du Mans circa 1550 (for this reason, the long scale is also known as the ChuquetPeletier system), but the Oxford English Dictionary states that the term derives from postClassical Latin term milliartum, which became milliare and then milliart and finally our modern term.
With regard to names ending in illiard for numbers 10^{6·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. For example, as of 2004^{[update]}, Google searches on Frenchlanguage pages for trillion, quadrillion, and quintillion return 6630, 312, and 127 hits respectively, whilst searches for trilliard and quadrilliard return only 102 and 7 hits respectively. However, one has to take into account that these large numbers are not often needed and that scientists almost always use scientific notation. In German the term "Milliarde" and in Dutch "miljard" are standard usage when discussing financial topics.
The naming procedure for large numbers is based on taking the number n occurring in 10^{3n+3} (short scale) or 10^{6n} (long scale) and concatenating Latin roots for its units, tens, and hundreds place, together with the suffix illion. In this way, numbers up to 10^{3·999+3} = 10^{3000} (short scale) or 10^{6·999} = 10^{5994} (long scale) may be named. The choice of roots and the concatenation procedure is that of the standard dictionary numbers if n is 20 or smaller, and, for larger n (between 21 and 999), is due to John Horton Conway and Richard Guy. Since the system of using Latin prefixes will become ambiguous for numbers with exponents of a size which the Romans rarely counted to, like 10^{6,000,258}, Conway and Guy have also proposed a consistent set of conventions which permit, in principle, the extension of this system to provide English names for any integer whatsoever.^{[15]}
Names of reciprocals of large numbers do not need to be listed here, because they are regularly formed by adding th, e.g. quattuordecillionth, centillionth, etc.
For additional details, see Billion (disambiguation) and long and short scales.
Base illion (short scale)  Value  USA and Modern British (short scale) 
Traditional British (long scale) 
Traditional European (Peletier) (long scale) 

1  10^{6}  Million  Million  Million 
2  10^{9}  Billion  Thousand million  Milliard 
3  10^{12}  Trillion  Billion  Billion 
4  10^{15}  Quadrillion  Thousand billion  Billiard 
5  10^{18}  Quintillion  Trillion  Trillion 
6  10^{21}  Sextillion  Thousand trillion  Trilliard 
7  10^{24}  Septillion  Quadrillion  Quadrillion 
8  10^{27}  Octillion  Thousand quadrillion  Quadrilliard 
9  10^{30}  Nonillion  Quintillion  Quintillion 
10  10^{33}  Decillion  Thousand quintillion  Quintilliard 
11  10^{36}  Undecillion  Sextillion  Sextillion 
12  10^{39}  Duodecillion  Thousand sextillion  Sextilliard 
13  10^{42}  Tredecillion  Septillion  Septillion 
14  10^{45}  Quattuordecillion  Thousand septillion  Septilliard 
15  10^{48}  Quindecillion  Octillion  Octillion 
16  10^{51}  Sexdecillion  Thousand octillion  Octilliard 
17  10^{54}  Septendecillion  Nonillion  Nonillion 
18  10^{57}  Octodecillion  Thousand nonillion  Nonilliard 
19  10^{60}  Novemdecillion  Decillion  Decillion 
20  10^{63}  Vigintillion  Thousand decillion  Decilliard 
21  10^{66}  Unvigintillion  Undecillion  Undecillion 
22  10^{69}  Duovigintillion  Thousand undecillion  Undecilliard 
23  10^{72}  Tresvigintillion  Duodecillion  Duodecillion 
24  10^{75}  Quattuorvigintillion  Thousand duodecillion  Duodecilliard 
25  10^{78}  Quinquavigintillion  Tredecillion  Tredecillion 
26  10^{81}  Sesvigintillion  Thousand tredecillion  Tredecilliard 
27  10^{84}  Septemvigintillion  Quattuordecillion  Quattuordecillion 
28  10^{87}  Octovigintillion  Thousand quattuordecillion  Quattuordecilliard 
29  10^{90}  Novemvigintillion  Quindecillion  Quindecillion 
30  10^{93}  Trigintillion  Thousand quindecillion  Quindecilliard 
31  10^{96}  Untrigintillion  Sexdecillion  Sexdecillion 
32  10^{99}  Duotrigintillion  Thousand sexdecillion  Sexdecilliard 
33  10^{102}  Trestrigintillion  Septendecillion  Septendecillion 
34  10^{105}  Quattuortrigintillion  Thousand septendecillion  Septendecilliard 
35  10^{108}  Quinquatrigintillion  Octodecillion  Octodecillion 
36  10^{111}  Sestrigintillion  Thousand octodecillion  Octodecilliard 
37  10^{114}  Septentrigintillion  Novemdecillion  Novemdecillion 
38  10^{117}  Octotrigintillion  Thousand novemdecillion  Novemdecilliard 
39  10^{120}  Noventrigintillion  Vigintillion  Vigintillion 
40  10^{123}  Quadragintillion^{[16]}  Thousand vigintillion  Vigintilliard 
50  10^{153}  Quinquagintillion  Thousand quinquavigintillion  Quinquavigintilliard 
60  10^{183}  Sexagintillion  Thousand trigintillion  Trigintilliard 
70  10^{213}  Septuagintillion  Thousand quinquatrigintillion  Quinquatrigintilliard 
80  10^{243}  Octogintillion  Thousand quadragintillion  Quadragintilliard 
90  10^{273}  Nonagintillion  Thousand quinquaquadragintillion  Quinquaquadragintilliard 
100  10^{303}  Centillion  Thousand quinquagintillion  Quinquagintilliard 
101  10^{306}  Uncentillion  Unquinquagintillion  Unquinquagintillion 
102  10^{309}  Duocentillion  Thousand unquinquagintillion  Unquinquagintilliard 
103  10^{312}  Trescentillion  Duoquinquagintillion  Duoquinquagintillion 
110  10^{333}  Decicentillion  Thousand quinquaquinquagintillion  Quinquaquinquagintilliard 
121  10^{366}  Unviginticentillion  Unsexagintillion  Unsexagintillion 
130  10^{393}  Trigintacentillion  Thousand quinquasexagintillion  Quinquasexagintilliard 
140  10^{423}  Quadragintacentillion  Thousand septuagintillion  Septuagintilliard 
150  10^{453}  Quinquagintacentillion  Thousand quinquaseptuagintillion  Quinquaseptuagintilliard 
160  10^{483}  Sexagintacentillion  Thousand octogintillion  Octogintilliard 
170  10^{513}  Septuagintacentillion  Thousand quinquaoctogintillion  Quinquaoctogintilliard 
180  10^{543}  Octogintacentillion  Thousand nonagintillion  Nonagintilliard 
190  10^{573}  Nonagintacentillion  Thousand quinquanonagintillion  Quinquanonagintilliard 
200  10^{603}  Ducentillion  Thousand centillion  Centilliard 
300  10^{903}  Trecentillion  Thousand quinquagintacentillion  Quinquagintacentilliard 
400  10^{1203}  Quadringentillion  Thousand ducentillion  Ducentilliard 
500  10^{1503}  Quingentillion  Thousand quinquagintaducentillion  Quinquagintaducentilliard 
600  10^{1803}  Sescentillion  Thousand trecentillion  Trecentilliard 
700  10^{2103}  Septingentillion  Thousand quinquagintatrecentillion  Quinquagintatrecentilliard 
800  10^{2403}  Octingentillion  Thousand quadringentillion  Quadringentilliard 
900  10^{2703}  Nongentillion  Thousand quinquagintaquadringentillion  Quinquagintaquadringentilliard 
Value  USA and Modern British (short scale) 
Traditional British (long scale) 
Traditional European (Peletier) (long scale) 

10^{100}  Googol (Ten duotrigintillion)  Googol (Ten thousand sexdecillion)  Googol (Ten sexdecilliard) 
Googolplex  Googolplex  Googolplex 
[edit] Other large numbers used in mathematics and physics
[edit] See also
[edit] References
 ^ American Heritage Dictionary, 4th edition, ISBN 0395825172. [1]
 ^ Cambridge Dictionaries Online, Cambridge, UK: Cambridge University Press.
 ^ Oxford English Dictionary, 2nd edition, Oxford, UK: Oxford University Press. ISBN 0198611862 (and addendums since publication in 1989.)
 ^ Oxford English Dictionary, New Edition, Oxford, UK: Oxford University Press. [2] (subscription required), checked April 2007
 ^ The Random House Dictionary, 2nd Unabridged Edition, 1987, Random House.
 ^ Shorter Oxford English Dictionary, 3rd edition, 1993, Oxford: Clarendon Press.
 ^ Webster's Third New International Dictionary, Unabridged, 1993, MerriamWebster.
 ^ "How Many? A Dictionary of Units of Measures". Russ Rowlett and the University of North Carolina at Chapel Hill. http://www.unc.edu/~rowlett/units/index.html. Retrieved on 20071101. "billiard unit of quantity equal to 10^{15}, which is one quadrillion in American terminology or 1000 billion in traditional British terminology. The name is coined to parallel milliard, which has long been a name for 1000 million.
trilliard a unit of quantity equal to 10^{21}, which is one sextillion in American terminology or 1000 trillion in traditional British terminology. The name is coined to parallel milliard, which has long been a name for 1000 million."  ^ ^{a} ^{b} ^{c} ^{d} ^{e} Not verified whether this term is mentioned in this work of reference
 ^ p. 316, The History of the English Language, Oliver Farrar Emerson, New York, London: Macmillan and Co., 1894.
 ^ Entry for centillion in the American Heritage Dictionary
 ^ Schlosser, Eric (2002) [2001]. Fast Food Nation. London, England: Penguin Books. pp. 66. ISBN 0141006870.
 ^ "Zimbabwe rolls out Z$100tr note". BBC News. 01162009. http://news.bbc.co.uk/2/hi/africa/7832601.stm.
 ^ Kasner, Edward and James Newman, Mathematics and the Imagination, 1940, Simon and Schuster, New York.
 ^ ^{a} ^{b} ^{c} The Book of Numbers, J. H. Conway and R. K. Guy, New York: SpringerVerlag, 1996, pp. 15–16. ISBN 038797993X.
 ^ Often misspelled quardragintillion.
[edit] External links
 Robert Munafo's Large Numbers
 How high can you count? by Landon Curt Noll.
 Full list of large number names list sorted by 10^{n} and by word length
 Big numbers Educational site, which can name any numbers put into it (up to centillion)
