Names Of 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 becoming rare in British English, but their other language 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.
Standard dictionary numbers
Name  Short scale (U.S. and modern British) 
Long scale (continental Europe, older British) 
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}  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓  ✓ 
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 (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  CED  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., 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.).
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
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.
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 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. This is an oversimplification.
Million was certainly not invented by Adam or Chuquet. Milion is an Old French word thought to derive from Italian milione, an intensification of mille, a thousand. That is, a million is a big thousand.
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.
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}.
An aidememoire
It can be a problem to find the values for large numbers, either in scientific notation or in sheer digits. Every number listed in this article larger than a million has two values: one in the short scale, where successive names differ by a factor of one thousand, and another in the long scale, where successive names differ by a factor of one million.
An easy way to find the value of the above numbers in the short scale (as well as the number of zeroes needed to write them) 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 scale, 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 the article on long and short scales.
Note that when writing out large numbers using this system, one should place a comma or space after every three digits, starting from the right and moving left.
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} = 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}. This number (ten to the power of a googolplex) is also known as a googolduplex.^{[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 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} =  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}. This number (ten to the power of a googolplex) is also known as a googolduplex.^{[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.
Extensions of the standard dictionary numbers
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 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 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^{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.
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^{[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” increases to “tres” and “se” to “ses” or “sex”; similarly, when preceding a component marked ^{m} or ^{n}, “septe” and “nove” increase 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]}
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 and long and short scales.
Base illion (short scale) 
Value  U.S. and modern British (short scale) 
Traditional British (long scale) 
Traditional European (Peletier) (long scale) 
SI Symbol 
SI Prefix 

1  10^{6}  Million  Million  Million  M  mega 
2  10^{9}  Billion  Thousand million  Milliard  G  giga 
3  10^{12}  Trillion  Billion  Billion  T  tera 
4  10^{15}  Quadrillion  Thousand billion  Billiard  P  peta 
5  10^{18}  Quintillion  Trillion  Trillion  E  exa 
6  10^{21}  Sextillion  Thousand trillion  Trilliard  Z  zetta 
7  10^{24}  Septillion  Quadrillion  Quadrillion  Y  yotta 
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}  Sedecillion  Thousand octillion  Octilliard  
17  10^{54}  Septendecillion  Nonillion  Nonillion  
18  10^{57}  Octodecillion  Thousand nonillion  Nonilliard  
19  10^{60}  Novendecillion  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  Sedecillion  Sedecillion  
32  10^{99}  Duotrigintillion  Thousand sedecillion  Sedecilliard  
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  Novendecillion  Novendecillion  
38  10^{117}  Octotrigintillion  Thousand novendecillion  Novendecilliard  
39  10^{120}  Noventrigintillion  Vigintillion  Vigintillion  
40  10^{123}  Quadragintillion  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  
111  10^{336}  Undecicentillion  Sesquinquagintillion  Sesquinquagintillion  
120  10^{363}  Viginticentillion  Thousand sexagintillion  Sexagintilliard  
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  
1000  10^{3003}  Millinillion  Thousand quingentillion  Quingentilliard 
Value  U.S. and modern British (short scale) 
Traditional British (long scale) 
Traditional European (Peletier) (long scale) 

10^{100}  Googol (Ten duotrigintillion)  Googol (Ten thousand sedecillion)  Googol (Ten sedecilliard) 
Googolplex  Googolplex  Googolplex 
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.^{[16]}


