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 that are found in many English dictionaries and thus have a claim to being "real words." The "Traditional British" values shown are unused in American English and are obsolete 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.

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.

Standard dictionary numbers

Name	Short scale (US, English Canada, modern British, Australia, and Eastern Europe)	Long scale (French Canada, older British, Western & Central Europe)	Authorities
Name			AHD4^[1]	CED^[2]	COD^[3]	OED2^[4]	OEDnew^[5]	RHD2^[6]	SOED3^[7]	W3^[8]	UM^[9]
Million	10⁶	10⁶	✓	✓	✓	✓	✓	✓	✓	✓	✓
Milliard	Not used	10⁹	✓	✓		✓	✓	✓			✓
Billion	10⁹	10¹²	✓	✓	✓	✓	✓	✓	✓	✓	✓
Billiard	Not used	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	10⁴⁸	10⁹⁰	✓	✓				✓		✓	✓
Sexdecillion	10⁵¹	10⁹⁶	✓	✓				✓		✓	✓
Septendecillion	10⁵⁴	10¹⁰²	✓	✓				✓		✓	✓
Octodecillion	10⁵⁷	10¹⁰⁸	✓	✓				✓		✓	✓
Novemdecillion	10⁶⁰	10¹¹⁴	✓	✓				✓		✓	✓
Vigintillion	10⁶³	10¹²⁰	✓	✓		✓	✓	✓	✓	✓	✓
Centillion	10³⁰³	10⁶⁰⁰	✓	✓		✓	✓	✓			✓

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
Name	Value	AHD4	CED	COD	OED2	OEDnew	RHD2	SOED3	W3	UM
Googol	10¹⁰⁰	✓	✓	✓	✓	✓	✓	✓	✓	✓
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 (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 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 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. 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 joules." 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 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 (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 $ {\displaystyle ((10^{8})^{(10^{8})})^{(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 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"
Chuquet.gif

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 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 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 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)
10^googol = 10^10¹⁰⁰	Googolplex	Kasner and Newman, dictionaries (see above)

John Horton Conway and Richard K. Guy^[14] have suggested that N-plex be used as a name for 10^N. This gives rise to the name googolplexplex for 10^googolplex = 10^{10^10¹⁰⁰}. 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 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.

The names googol and googolplex 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; 1,000,000² = 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 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 × 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 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 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," milliard (transliterated) 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 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 9 or smaller. For larger n (between 10 and 999), prefixes can be constructed based on a system described by Conway and Guy.^[14] 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:

	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	Quin	^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 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.^[14] The name of a number 10³ⁿ⁺³, where n is greater than or equal to 1000, is formed by concatenating the names of the numbers of the form 10^3m+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".^[14] For example, 10^3,000,012, the 1,000,003rd "-illion" number, equals one "millinillitrillion"; 10^{33,002,010,111}, the 11,000,670,036th "-illion" number, equals one "undecillinilliseptuagintasescentillisestrigintillion"; and 10^{29,629,629,633}, the 9,876,543,210th "-illion" number, equals one "nonilliseseptuagintaoctingentillitresquadragintaquingentillideciducentillion".^[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	10⁶	Million	Million	Million	M	Mega-
2	1	10⁹	Billion	Thousand million	Milliard	G	Giga-
3	2	10¹²	Trillion	Billion	Billion	T	Tera-
4	2	10¹⁵	Quadrillion	Thousand billion	Billiard	P	Peta-
5	3	10¹⁸	Quintillion	Trillion	Trillion	E	Exa-
6	3	10²¹	Sextillion	Thousand trillion	Trilliard	Z	Zetta-
7	4	10²⁴	Septillion	Quadrillion	Quadrillion	Y	Yotta-
8	4	10²⁷	Octillion	Thousand quadrillion	Quadrilliard
9	5	10³⁰	Nonillion	Quintillion	Quintillion
10	5	10³³	Decillion	Thousand quintillion	Quintilliard
11	6	10³⁶	Undecillion	Sextillion	Sextillion
12	6	10³⁹	Duodecillion	Thousand sextillion	Sextilliard
13	7	10⁴²	Tredecillion	Septillion	Septillion
14	7	10⁴⁵	Quattuordecillion	Thousand septillion	Septilliard
15	8	10⁴⁸	Quindecillion	Octillion	Octillion
16	8	10⁵¹	Sedecillion	Thousand octillion	Octilliard
17	9	10⁵⁴	Septendecillion	Nonillion	Nonillion
18	9	10⁵⁷	Octodecillion	Thousand nonillion	Nonilliard
19	10	10⁶⁰	Novendecillion	Decillion	Decillion
20	10	10⁶³	Vigintillion	Thousand decillion	Decilliard
21	11	10⁶⁶	Unvigintillion	Undecillion	Undecillion
22	11	10⁶⁹	Duovigintillion	Thousand undecillion	Undecilliard
23	12	10⁷²	Tresvigintillion	Duodecillion	Duodecillion
24	12	10⁷⁵	Quattuorvigintillion	Thousand duodecillion	Duodecilliard
25	13	10⁷⁸	Quinvigintillion	Tredecillion	Tredecillion
26	13	10⁸¹	Sesvigintillion	Thousand tredecillion	Tredecilliard
27	14	10⁸⁴	Septemvigintillion	Quattuordecillion	Quattuordecillion
28	14	10⁸⁷	Octovigintillion	Thousand quattuordecillion	Quattuordecilliard
29	15	10⁹⁰	Novemvigintillion	Quindecillion	Quindecillion
30	15	10⁹³	Trigintillion	Thousand quindecillion	Quindecilliard
31	16	10⁹⁶	Untrigintillion	Sedecillion	Sedecillion
32	16	10⁹⁹	Duotrigintillion	Thousand sedecillion	Sedecilliard
33	17	10¹⁰²	Trestrigintillion	Septendecillion	Septendecillion
34	17	10¹⁰⁵	Quattuortrigintillion	Thousand septendecillion	Septendecilliard
35	18	10¹⁰⁸	Quintrigintillion	Octodecillion	Octodecillion
36	18	10¹¹¹	Sestrigintillion	Thousand octodecillion	Octodecilliard
37	19	10¹¹⁴	Septentrigintillion	Novendecillion	Novendecillion
38	19	10¹¹⁷	Octotrigintillion	Thousand novendecillion	Novendecilliard
39	20	10¹²⁰	Noventrigintillion	Vigintillion	Vigintillion
40	20	10¹²³	Quadragintillion	Thousand vigintillion	Vigintilliard
50	25	10¹⁵³	Quinquagintillion	Thousand quinvigintillion	Quinvigintilliard
60	30	10¹⁸³	Sexagintillion	Thousand trigintillion	Trigintilliard
70	35	10²¹³	Septuagintillion	Thousand quintrigintillion	Quintrigintilliard
80	40	10²⁴³	Octogintillion	Thousand quadragintillion	Quadragintilliard
90	45	10²⁷³	Nonagintillion	Thousand quinquadragintillion	Quinquadragintilliard
100	50	10³⁰³	Centillion	Thousand quinquagintillion	Quinquagintilliard
101	51	10³⁰⁶	Uncentillion	Unquinquagintillion	Unquinquagintillion
110	55	10³³³	Decicentillion	Thousand quinquinquagintillion	Quinquinquagintilliard
111	56	10³³⁶	Undecicentillion	Sesquinquagintillion	Sesquinquagintillion
120	60	10³⁶³	Viginticentillion	Thousand sexagintillion	Sexagintilliard
121	61	10³⁶⁶	Unviginticentillion	Unsexagintillion	Unsexagintillion
130	65	10³⁹³	Trigintacentillion	Thousand quinsexagintillion	Quinsexagintilliard
140	70	10⁴²³	Quadragintacentillion	Thousand septuagintillion	Septuagintilliard
150	75	10⁴⁵³	Quinquagintacentillion	Thousand quinseptuagintillion	Quinseptuagintilliard
160	80	10⁴⁸³	Sexagintacentillion	Thousand octogintillion	Octogintilliard
170	85	10⁵¹³	Septuagintacentillion	Thousand quinoctogintillion	Quinoctogintilliard
180	90	10⁵⁴³	Octogintacentillion	Thousand nonagintillion	Nonagintilliard
190	95	10⁵⁷³	Nonagintacentillion	Thousand quinnonagintillion	Quinnonagintilliard
200	100	10⁶⁰³	Ducentillion	Thousand centillion	Centilliard
300	150	10⁹⁰³	Trecentillion	Thousand quinquagintacentillion	Quinquagintacentilliard
400	200	10¹²⁰³	Quadringentillion	Thousand ducentillion	Ducentilliard
500	250	10¹⁵⁰³	Quingentillion	Thousand quinquagintaducentillion	Quinquagintaducentilliard
600	300	10¹⁸⁰³	Sescentillion	Thousand trecentillion	Trecentilliard
700	350	10²¹⁰³	Septingentillion	Thousand quinquagintatrecentillion	Quinquagintatrecentilliard
800	400	10²⁴⁰³	Octingentillion	Thousand quadringentillion	Quadringentilliard
900	450	10²⁷⁰³	Nongentillion	Thousand quinquagintaquadringentillion	Quinquagintaquadringentilliard
1000	500	10³⁰⁰³	Millinillion^[16]	Thousand quingentillion	Quingentilliard

Value	Name	Equivalent
Value	Name	US, Canadian and modern British (short scale)	Traditional British (long scale)	Traditional European (Peletier) (long scale)
10¹⁰⁰	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 10241 and 10248.[17]

Power	Value	ISQ symbol	ISQ prefix
1	1024¹	Ki	Kibi-
2	1024²	Mi	Mebi-
3	1024³	Gi	Gibi-
4	1024⁴	Ti	Tebi-
5	1024⁵	Pi	Pebi-
6	1024⁶	Ei	Exbi-
7	1024⁷	Zi	Zebi-
8	1024⁸	Yi	Yobi-

Other large numbers used in mathematics and physics

Avogadro number
Graham's number
Skewes' number
Steinhaus–Moser notation
TREE(3)