Names of large numbers

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 non-English-speaking areas, including continental Europe and Spanish-speaking 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]	COD[2]	OED2[3]	OEDnew[4]	RHD2[5]	SOED3[6]	W3[7]	UM[8]
Million	106	106	✓	✓	✓	✓	✓	✓	✓	✓
Milliard		109	✓		✓		✓			✓
Billion	109	1012	✓	✓	✓	✓	✓	✓	✓	✓
Billiard		1015	✓		✓		✓			✓
Trillion	1012	1018	✓	✓	✓	✓	✓	✓	✓	✓
Quadrillion	1015	1024	✓		✓	✓	✓	✓	✓	✓
Quintillion	1018	1030	✓		✓	✓	✓	✓	✓	✓
Sextillion	1021	1036	✓		✓	✓	✓	✓	✓	✓
Septillion	1024	1042	✓		✓	✓	✓	✓	✓	✓
Octillion	1027	1048	✓		✓	✓	✓	✓	✓	✓
Nonillion	1030	1054	✓		✓	✓	✓	✓	✓	✓
Decillion	1033	1060	✓		✓	✓	✓	✓	✓	✓
Undecillion	1036	1066	✓				✓		✓	✓
Duodecillion	1039	1072	✓				✓		✓	✓
Tredecillion	1042	1078	✓				✓		✓	✓
Quattuordecillion	1045	1084	✓				✓		✓	✓
Quindecillion (Quinquadecillion)	1048	1090	✓				✓		✓	✓
Sexdecillion (Sedecillion)	1051	1096	✓				✓		✓	✓
Septendecillion	1054	10102	✓				✓		✓	✓
Octodecillion	1057	10108	✓				✓		✓	✓
Novemdecillion (Novendecillion)	1060	10114	✓				✓		✓	✓
Vigintillion	1063	10120	✓		✓	✓	✓	✓	✓	✓
Centillion	10303	10600	✓		✓	✓				✓

Name	Value	Authorities
		AHD4	COD	OED2	OEDnew	RHD2	SOED3	W3	UM
Googol	10100	✓		✓	✓	✓	✓	✓	✓
Googolplex	10Googol	✓		✓	✓	✓	✓	✓	✓

Apart from million, the words in this list ending with -illion are all derived by adding prefixes (bi-, tri-, etc.) to the stem -illion.^[9] Centillion^[10] 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 excessive inflation. The highest numerical value banknote ever printed was a note for 1 sextillion pengő (10²¹ or 1 milliard bilpengő as printed) printed in Hungary in 1946. In 2009, Zimbabwe printed a 100 trillion (10¹⁴) Zimbabwean dollar note, which at the time of printing was only worth about US$30.^[11]

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 well-established 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, powers of ten are expressed as 10 with a numeric superscript, e.g., "The X-ray emission of the radio galaxy is 1.3×10⁴⁵ ergs." When a number such as 10⁴⁵ needs to be referred to in words, it is simply read out: "ten to the forty-fifth". 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—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 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 (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., $(10^8)^{(10^8)}=10^{8\cdot 10^8},$ and embedded this construction within another copy of itself to produce names for numbers up to $\left((10^8)^{(10^8)}\right)^{(10^8)}=10^{8\cdot 10^{16}}.$ 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 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 well-known, 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 10¹², and Adam's trimillion (Chuquet's tryllion) denoted 10¹⁸.

An aide-memoire

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 short-scale trillion in scientific notation: one trillion = 10¹².

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 long-scale billion in scientific notation: one billion = 10¹². 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 10⁴⁵. 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,^[12] in the following passage:

The name "googol" was invented by a child (Dr. Kasner's nine-year-old 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
10100	Googol	Kasner and Newman, dictionaries (see above)
10googol = $\,\!10^{10^{100}}$	Googolplex	Kasner and Newman, dictionaries (see above)

Conway and Guy ^[13] have suggested that N-plex 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. ^[14] Conway and Guy ^[13] have proposed that N-minex 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; 1,000,000² = 1 billion; 1,000,000³ = 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 × 1000² = 10⁹; a trillion is 1000 × 1000³ = 10¹²; 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 10⁹. 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 Chuquet-Peletier system), but the Oxford English Dictionary states that the term derives from post-Classical Latin term milliartum, which became milliare and then milliart and finally our modern term.

With regard to names ending in -illiard for numbers 10⁶ⁿ⁺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 10³ⁿ⁺³ (short scale) or 10⁶ⁿ (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 = 10³⁰⁰⁰ (short scale) or 10^6·999 = 10⁵⁹⁹⁴ (long scale) may be named. The choice of roots and the concatenation procedure is that of the standard dictionary numbers if n is 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.^[13]

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	106	Million	Million	Million	M	mega-
2	109	Billion	Thousand million	Milliard	G	giga-
3	1012	Trillion	Billion	Billion	T	tera-
4	1015	Quadrillion	Thousand billion	Billiard	P	peta-
5	1018	Quintillion	Trillion	Trillion	E	exa-
6	1021	Sextillion	Thousand trillion	Trilliard	Z	zetta-
7	1024	Septillion	Quadrillion	Quadrillion	Y	yotta-
8	1027	Octillion	Thousand quadrillion	Quadrilliard
9	1030	Nonillion	Quintillion	Quintillion
10	1033	Decillion	Thousand quintillion	Quintilliard
11	1036	Undecillion	Sextillion	Sextillion
12	1039	Duodecillion	Thousand sextillion	Sextilliard
13	1042	Tredecillion	Septillion	Septillion
14	1045	Quattuordecillion	Thousand septillion	Septilliard
15	1048	Quindecillion	Octillion	Octillion
16	1051	Sexdecillion	Thousand octillion	Octilliard
17	1054	Septendecillion	Nonillion	Nonillion
18	1057	Octodecillion	Thousand nonillion	Nonilliard
19	1060	Novemdecillion	Decillion	Decillion
20	1063	Vigintillion	Thousand decillion	Decilliard
21	1066	Unvigintillion	Undecillion	Undecillion
22	1069	Duovigintillion	Thousand undecillion	Undecilliard
23	1072	Tresvigintillion	Duodecillion	Duodecillion
24	1075	Quattuorvigintillion	Thousand duodecillion	Duodecilliard
25	1078	Quinquavigintillion	Tredecillion	Tredecillion
26	1081	Sesvigintillion	Thousand tredecillion	Tredecilliard
27	1084	Septemvigintillion	Quattuordecillion	Quattuordecillion
28	1087	Octovigintillion	Thousand quattuordecillion	Quattuordecilliard
29	1090	Novemvigintillion	Quindecillion	Quindecillion
30	1093	Trigintillion	Thousand quindecillion	Quindecilliard
31	1096	Untrigintillion	Sexdecillion	Sexdecillion
32	1099	Duotrigintillion	Thousand sexdecillion	Sexdecilliard
33	10102	Trestrigintillion	Septendecillion	Septendecillion
34	10105	Quattuortrigintillion	Thousand septendecillion	Septendecilliard
35	10108	Quinquatrigintillion	Octodecillion	Octodecillion
36	10111	Sestrigintillion	Thousand octodecillion	Octodecilliard
37	10114	Septentrigintillion	Novemdecillion	Novemdecillion
38	10117	Octotrigintillion	Thousand novemdecillion	Novemdecilliard
39	10120	Novemtrigintillion	Vigintillion	Vigintillion
40	10123	Quadragintillion	Thousand vigintillion	Vigintilliard
50	10153	Quinquagintillion	Thousand quinquavigintillion	Quinquavigintilliard
60	10183	Sexagintillion	Thousand trigintillion	Trigintilliard
70	10213	Septuagintillion	Thousand quinquatrigintillion	Quinquatrigintilliard
80	10243	Octogintillion	Thousand quadragintillion	Quadragintilliard
90	10273	Nonagintillion	Thousand quinquaquadragintillion	Quinquaquadragintilliard
100	10303	Centillion	Thousand quinquagintillion	Quinquagintilliard
101	10306	Uncentillion	Unquinquagintillion	Unquinquagintillion
102	10309	Duocentillion	Thousand unquinquagintillion	Unquinquagintilliard
103	10312	Trescentillion	Duoquinquagintillion	Duoquinquagintillion
110	10333	Decicentillion	Thousand quinquaquinquagintillion	Quinquaquinquagintilliard
111	10336	Undecicentillion	Sexaquinquagintillion	Sexaquinquagintillion
120	10363	Viginticentillion	Thousand sexagintillion	Sexagintilliard
121	10366	Unviginticentillion	Unsexagintillion	Unsexagintillion
130	10393	Trigintacentillion	Thousand quinquasexagintillion	Quinquasexagintilliard
140	10423	Quadragintacentillion	Thousand septuagintillion	Septuagintilliard
150	10453	Quinquagintacentillion	Thousand quinquaseptuagintillion	Quinquaseptuagintilliard
160	10483	Sexagintacentillion	Thousand octogintillion	Octogintilliard
170	10513	Septuagintacentillion	Thousand quinquaoctogintillion	Quinquaoctogintilliard
180	10543	Octogintacentillion	Thousand nonagintillion	Nonagintilliard
190	10573	Nonagintacentillion	Thousand quinquanonagintillion	Quinquanonagintilliard
200	10603	Ducentillion	Thousand centillion	Centilliard
300	10903	Trecentillion	Thousand quinquagintacentillion	Quinquagintacentilliard
400	101203	Quadringentillion	Thousand ducentillion	Ducentilliard
500	101503	Quingentillion	Thousand quinquagintaducentillion	Quinquagintaducentilliard
600	101803	Sescentillion	Thousand trecentillion	Trecentilliard
700	102103	Septingentillion	Thousand quinquagintatrecentillion	Quinquagintatrecentilliard
800	102403	Octingentillion	Thousand quadringentillion	Quadringentilliard
900	102703	Nongentillion	Thousand quinquagintaquadringentillion	Quinquagintaquadringentilliard
1000	103003	Millinillion	Thousand quingentillion	Quingentilliard

Value	U.S. and modern British (short scale)	Traditional British (long scale)	Traditional European (Peletier) (long scale)
10100	Googol (Ten duotrigintillion)	Googol (Ten thousand sexdecillion)	Googol (Ten sexdecilliard)
$10^{10^{100}}\,\!$	Googolplex	Googolplex	Googolplex

Proposals for new naming system

See also: -yllion

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 Latin-based short scale and long scale systems should be replaced by an unambiguous Greek-based system for naming large numbers that would be based on powers of one thousand.^[15]

Value Name 103 Thousand 106 Million 109 Gillion 1012 Tetrillion 1015 Pentillion 1018 Hexillion 1021 Heptillion 1024 Oktillion 1027 Ennillion 1030 Dekillion

Value Name 1033 Hendekillion 1036 Dodekillion 1039 Trisdekillion 1042 Tetradekillion 1045 Pentadekillion 1048 Hexadekillion 1051 Heptadekillion 1054 Oktadekillion 1057 Enneadekillion 1060 Icosillion

Value Name 1063 Icosihenillion 1066 Icosidillion 1069 Icositrillion 1072 Icositetrillion 1075 Icosipentillion 1078 Icosihexillion 1081 Icosiheptillion 1084 Icosioktillion 1087 Icosiennillion 1090 Triacontillion

Other large numbers used in mathematics and physics

• Avogadro's number
• Graham's number
• Skewes' number
• Steinhaus–Moser notation

See also

Chinese numbers Indefinite and fictitious numbers Indian numbering system Knuth's up-arrow notation Law of large numbers List of numbers

Mathematics portal Names of small numbers Nicolas Chuquet Number names Number prefix Orders of magnitude Orders of magnitude (numbers)

References

1. ^ American Heritage Dictionary, 4th edition, ISBN 0-395-82517-2. [1]
2. ^ Cambridge Dictionaries Online, Cambridge, UK: Cambridge University Press.
3. ^ Oxford English Dictionary, 2nd edition, Oxford, UK: Oxford University Press. ISBN 0198611862 (and addendums since publication in 1989.)
4. ^ Oxford English Dictionary, New Edition, Oxford, UK: Oxford University Press. [2] (subscription required), checked April 2007
5. ^ The Random House Dictionary, 2nd Unabridged Edition, 1987, Random House.
6. ^ Shorter Oxford English Dictionary, 3rd edition, 1993, Oxford: Clarendon Press.
7. ^ Webster's Third New International Dictionary, Unabridged, 1993, Merriam-Webster.
8. ^ "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 2009-08-15. 
9. ^ p. 316, The History of the English Language, Oliver Farrar Emerson, New York, London: Macmillan and Co., 1894.
10. ^ Entry for centillion in the American Heritage Dictionary
11. ^ "Zimbabwe rolls out Z$100tr note". BBC News. January 16, 2009. http://news.bbc.co.uk/2/hi/africa/7832601.stm. Retrieved 2009-01-16. 
12. ^ Kasner, Edward and James Newman, Mathematics and the Imagination, 1940, Simon and Schuster, New York.
13. ^ ^{a b c} The Book of Numbers, J. H. Conway and R. K. Guy, New York: Springer-Verlag, 1996, pp. 15–16. ISBN 0-387-97993-X.
14. ^ Bowers, Jonathan. "Infinity Scrapers". Polytope, 2010.
15. ^ Rowlett, Russ (2001-11-01). "Names for large numbers". University of North Carolina. http://www.unc.edu/~rowlett/units/large.html. Retrieved 2008-01-31. 

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 10ⁿ and by word length
• Big numbers Educational site, which can name any numbers put into it (up to centillion)
• The English name of a number An online tool that prints names of numbers of any size