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 that 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 becoming rare in British English, but their otherlanguage variants are dominant in many nonEnglishspeaking areas, including continental Europe and Spanishspeaking countries in Latin America; see Long and short scales.
English also has many words, such as "zillion", used informally to mean large but unspecified amounts; see indefinite and fictitious numbers.
Contents
 1 Standard dictionary numbers
 2 Usage of names of large numbers
 3 Origins of the "standard dictionary numbers"
 4 The googol family
 5 Extensions of the standard dictionary numbers
 6 Binary prefixes
 7 Proposals for new naming system
 8 Other large numbers used in mathematics and physics
 9 See also
 10 References
Standard dictionary numbers
Name 
Short scale (US, English Canadian, Australian, and modern British) 
Long scale (continental Europe, older British, and French Canadian) 
Authorities  

AHD4^{[1]}  CED^{[2]}  COD^{[3]}  OED2^{[4]}  OEDnew^{[5]}  RHD2^{[6]}  SOED3^{[7]}  W3^{[8]}  UM^{[9]}  
Million  10^{6}  10^{6}  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓ 
Milliard  10^{9}  ✓  ✓  ✓  ✓  ✓  ✓  
Billion  10^{9}  10^{12}  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓ 
Billiard  10^{15}  ✓  ✓  ✓  ✓  ✓  ✓  
Trillion  10^{12}  10^{18}  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓ 
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  10^{48}  10^{90}  ✓  ✓  ✓  ✓  ✓  
Sexdecillion  10^{51}  10^{96}  ✓  ✓  ✓  ✓  ✓  
Septendecillion  10^{54}  10^{102}  ✓  ✓  ✓  ✓  ✓  
Octodecillion  10^{57}  10^{108}  ✓  ✓  ✓  ✓  ✓  
Novemdecillion  10^{60}  10^{114}  ✓  ✓  ✓  ✓  ✓  
Vigintillion  10^{63}  10^{120}  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓  
Centillion  10^{303}  10^{600}  ✓  ✓  ✓  ✓  ✓  ✓ 
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.^{[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.).
Name  Value  Authorities  

AHD4  CED  COD  OED2  OEDnew  RHD2  SOED3  W3  UM  
Googol  10^{100}  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓ 
Googolplex  10^{googol} (10^{10100})  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓ 
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. 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 (10^{14}) Zimbabwean dollar note, which at the time of printing was worth about US$30.^{[12]}
Names of larger numbers, however, have a tenuous, artificial existence, 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 context of science, including astronomy, where such large numbers often occur, they are nearly always written using scientific notation. In this notation, powers of ten are expressed as 10 with a numeric superscript, e.g. "The Xray emission of the radio galaxy is ×10^{45} joules." When a number such as 10^{45} needs to be referred to in words, it is simply read out as "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. 1.3
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 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 earliest 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 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"
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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 point 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, but 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 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.^{[original research?]}
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}.
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,^{[13]} in the following passage:
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 one 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, equal to 1 with a googol zeros after it.
Value  Name  Authority 

10^{100}  Googol  Kasner and Newman, dictionaries (see above) 
10^{googol} = 10^{10100}  Googolplex  Kasner and Newman, dictionaries (see above) 
Conway and Guy^{[14]} have suggested that Nplex be used as a name for 10^{N}. This gives rise to the name googolplexplex for 10^{googolplex} = 10^{1010100}. This number (ten to the power of a googolplex) is also known as a googolduplex and googolplexian.^{[15]} Conway and Guy^{[14]} 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.
The names googol and googolplex have inspired the name of the Internet company Google 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): 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 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.) 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 US dollar), 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 French mathematician 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^{6n+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" in Russian are standard usage when discussing financial topics.
For additional details, see billion and long and short scales.
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. For larger n (between 21 and 999), prefixes can be constructed based on a system described by John Horton Conway and Richard K. Guy:^{[14]}
Units  Tens  Hundreds  

1  Un  ^{N} Deci  ^{NX} Centi 
2  Duo  ^{MS} Viginti  ^{N} Ducenti 
3  Tre ^{(*)}  ^{NS} Triginta  ^{NS} Trecenti 
4  Quattuor  ^{NS} Quadraginta  ^{NS} Quadringenti 
5  Quinqua  ^{NS} Quinquaginta  ^{NS} Quingenti 
6  Se ^{(*)}  ^{N} Sexaginta  ^{N} Sescenti 
7  Septe ^{(*)}  ^{N} Septuaginta  ^{N} Septingenti 
8  Octo  ^{MX} Octoginta  ^{MX} Octingenti 
9  Nove ^{(*)}  Nonaginta  Nongenti 
 ^{(*)} ^ 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 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.^{[14]}
The following table shows number names generated by the system described by Conway and Guy for the short and long scales.
Names of reciprocals of large numbers are not listed, as they are regularly formed by adding th, e.g. quattuordecillionth, centillionth, etc.
Base illion (short scale) 
Base illion (long scale) 
Value  US, Canada and modern British (short scale) 
Traditional British (long scale) 
Traditional European (Peletier) (long scale) 
SI Symbol 
SI Prefix 

1  1  10^{6}  Million  Million  Million  M  Mega 
2  1  10^{9}  Billion  Thousand million  Milliard  G  Giga 
3  2  10^{12}  Trillion  Billion  Billion  T  Tera 
4  2  10^{15}  Quadrillion  Thousand billion  Billiard  P  Peta 
5  3  10^{18}  Quintillion  Trillion  Trillion  E  Exa 
6  3  10^{21}  Sextillion  Thousand trillion  Trilliard  Z  Zetta 
7  4  10^{24}  Septillion  Quadrillion  Quadrillion  Y  Yotta 
8  4  10^{27}  Octillion  Thousand quadrillion  Quadrilliard  
9  5  10^{30}  Nonillion  Quintillion  Quintillion  
10  5  10^{33}  Decillion  Thousand quintillion  Quintilliard  
11  6  10^{36}  Undecillion  Sextillion  Sextillion  
12  6  10^{39}  Duodecillion  Thousand sextillion  Sextilliard  
13  7  10^{42}  Tredecillion  Septillion  Septillion  
14  7  10^{45}  Quattuordecillion  Thousand septillion  Septilliard  
15  8  10^{48}  Quinquadecillion  Octillion  Octillion  
16  8  10^{51}  Sedecillion  Thousand octillion  Octilliard  
17  9  10^{54}  Septendecillion  Nonillion  Nonillion  
18  9  10^{57}  Octodecillion  Thousand nonillion  Nonilliard  
19  10  10^{60}  Novendecillion  Decillion  Decillion  
20  10  10^{63}  Vigintillion  Thousand decillion  Decilliard  
21  11  10^{66}  Unvigintillion  Undecillion  Undecillion  
22  11  10^{69}  Duovigintillion  Thousand undecillion  Undecilliard  
23  12  10^{72}  Tresvigintillion  Duodecillion  Duodecillion  
24  12  10^{75}  Quattuorvigintillion  Thousand duodecillion  Duodecilliard  
25  13  10^{78}  Quinquavigintillion  Tredecillion  Tredecillion  
26  13  10^{81}  Sesvigintillion  Thousand tredecillion  Tredecilliard  
27  14  10^{84}  Septemvigintillion  Quattuordecillion  Quattuordecillion  
28  14  10^{87}  Octovigintillion  Thousand quattuordecillion  Quattuordecilliard  
29  15  10^{90}  Novemvigintillion  Quindecillion  Quindecillion  
30  15  10^{93}  Trigintillion  Thousand quindecillion  Quindecilliard  
31  16  10^{96}  Untrigintillion  Sedecillion  Sedecillion  
32  16  10^{99}  Duotrigintillion  Thousand sedecillion  Sedecilliard  
33  17  10^{102}  Trestrigintillion  Septendecillion  Septendecillion  
34  17  10^{105}  Quattuortrigintillion  Thousand septendecillion  Septendecilliard  
35  18  10^{108}  Quinquatrigintillion  Octodecillion  Octodecillion  
36  18  10^{111}  Sestrigintillion  Thousand octodecillion  Octodecilliard  
37  19  10^{114}  Septentrigintillion  Novendecillion  Novendecillion  
38  19  10^{117}  Octotrigintillion  Thousand novendecillion  Novendecilliard  
39  20  10^{120}  Noventrigintillion  Vigintillion  Vigintillion  
40  20  10^{123}  Quadragintillion  Thousand vigintillion  Vigintilliard  
50  25  10^{153}  Quinquagintillion  Thousand quinquavigintillion  Quinquavigintilliard  
60  30  10^{183}  Sexagintillion  Thousand trigintillion  Trigintilliard  
70  35  10^{213}  Septuagintillion  Thousand quinquatrigintillion  Quinquatrigintilliard  
80  40  10^{243}  Octogintillion  Thousand quadragintillion  Quadragintilliard  
90  45  10^{273}  Nonagintillion  Thousand quinquaquadragintillion  Quinquaquadragintilliard  
100  50  10^{303}  Centillion  Thousand quinquagintillion  Quinquagintilliard  
101  51  10^{306}  Uncentillion  Unquinquagintillion  Unquinquagintillion  
102  51  10^{309}  Duocentillion  Thousand unquinquagintillion  Unquinquagintilliard  
103  52  10^{312}  Trescentillion  Duoquinquagintillion  Duoquinquagintillion  
110  55  10^{333}  Decicentillion  Thousand quinquaquinquagintillion  Quinquaquinquagintilliard  
111  56  10^{336}  Undecicentillion  Sesquinquagintillion  Sesquinquagintillion  
120  60  10^{363}  Viginticentillion  Thousand sexagintillion  Sexagintilliard  
121  61  10^{366}  Unviginticentillion  Unsexagintillion  Unsexagintillion  
130  65  10^{393}  Trigintacentillion  Thousand quinquasexagintillion  Quinquasexagintilliard  
140  70  10^{423}  Quadragintacentillion  Thousand septuagintillion  Septuagintilliard  
150  75  10^{453}  Quinquagintacentillion  Thousand quinquaseptuagintillion  Quinquaseptuagintilliard  
160  80  10^{483}  Sexagintacentillion  Thousand octogintillion  Octogintilliard  
170  85  10^{513}  Septuagintacentillion  Thousand quinquaoctogintillion  Quinquaoctogintilliard  
180  90  10^{543}  Octogintacentillion  Thousand nonagintillion  Nonagintilliard  
190  95  10^{573}  Nonagintacentillion  Thousand quinquanonagintillion  Quinquanonagintilliard  
200  100  10^{603}  Ducentillion  Thousand centillion  Centilliard  
300  150  10^{903}  Trecentillion  Thousand quinquagintacentillion  Quinquagintacentilliard  
400  200  10^{1203}  Quadringentillion  Thousand ducentillion  Ducentilliard  
500  250  10^{1503}  Quingentillion  Thousand quinquagintaducentillion  Quinquagintaducentilliard  
600  300  10^{1803}  Sescentillion  Thousand trecentillion  Trecentilliard  
700  350  10^{2103}  Septingentillion  Thousand quinquagintatrecentillion  Quinquagintatrecentilliard  
800  400  10^{2403}  Octingentillion  Thousand quadringentillion  Quadringentilliard  
900  450  10^{2703}  Nongentillion  Thousand quinquagintaquadringentillion  Quinquagintaquadringentilliard  
1000  500  10^{3003}  Millinillion  Thousand quingentillion  Quingentilliard 
Value  Name  Equivalent  

US, Canadian and modern British (short scale) 
Traditional British (long scale) 
Traditional European (Peletier) (long scale) 

10^{100}  Googol  Ten duotrigintillion  Ten thousand sedecillion  Ten sedecilliard 
10^{10100}  Googolplex  n/a  n/a  n/a 
Binary prefixes
The International System of Quantities (ISQ) defines a series of prefixes denoting integer powers of 1024 between 1024^{1} and 1024^{8}.^{[16]}
Power  Value  ISQ symbol 
ISQ prefix 

1  1024^{1}  Ki  Kibi 
2  1024^{2}  Mi  Mebi 
3  1024^{3}  Gi  Gibi 
4  1024^{4}  Ti  Tebi 
5  1024^{5}  Pi  Pebi 
6  1024^{6}  Ei  Exbi 
7  1024^{7}  Zi  Zebi 
8  1024^{8}  Yi  Yobi 
Proposals for new naming system
In 2001, Russ Rowlett, Director of the Center for Mathematics and Science Education at the University of North Carolina at Chapel Hill proposed that, to avoid confusion, the Latinbased short scale and long scale systems should be replaced by an unambiguous Greekbased system for naming large numbers that would be based on powers of one thousand.^{[17]}



Other large numbers used in mathematics and physics
See also
 Chinese numerals
 Indefinite and fictitious numbers
 Indian numbering system
 Knuth's uparrow notation
 Law of large numbers
 List of numbers
 Long and short scales
 Asaṃkhyeya
 History of large numbers
 Names of small numbers
 Nicolas Chuquet
 Number names
 Number prefix
 Orders of magnitude
 Orders of magnitude (data)
 Orders of magnitude (numbers)
 Powers of 10
References
 ^ The American Heritage® Dictionary of the English Language. American Heritage Dictionary (4th ed.). ISBN 0395825172. Archived from the original on 20070112.
 ^ Collins English Dictionary, 11th Edition, HarperCollins Publishers.
 ^ Cambridge Dictionaries Online, Cambridge, UK: Cambridge University Press.
 ^ Oxford 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. [1] (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. Retrieved 20090815.
 ^ 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
 ^ "Zimbabwe rolls out Z$100tr note". BBC News. 16 January 2009. Retrieved 20090116.
 ^ Kasner, Edward and James Newman, Mathematics and the Imagination, 1940, Simon and Schuster, New York.
 ^ ^{a} ^{b} ^{c} ^{d} The Book of Numbers, J. H. Conway and R. K. Guy, New York: SpringerVerlag, 1996, pp. 15–16. ISBN 038797993X.
 ^ Bowers, Jonathan. "Infinity Scrapers". Polytope, 2010.
 ^ "IEC 8000013:2008". International Organization for Standardization. Retrieved 21 July 2013.
 ^ Rowlett, Russ (20011101). "Names for large numbers". University of North Carolina. Retrieved 20080131.