Names of large numbers explained

Two naming scales for large numbers have been used in English and other European languages since the early modern era: the long and short scales. Most English variants use the short scale today, but the long scale remains dominant in many non-English-speaking areas, including continental Europe and Spanish-speaking countries in Latin America. These naming procedures are 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.

Names of numbers above a trillion are rarely used in practice; such large numbers have practical usage primarily in the scientific domain, where powers of ten are expressed as 10 with a numeric superscript. However, these somewhat rare names are considered acceptable for approximate statements. For example, the statement "There are approximately 7.1 octillion atoms in an adult human body" is understood to be in short scale of the table below (and is only accurate if referring to short scale rather than long scale).

Indian English do not use millions, but have their own system of large numbers including lakhs (Anglicised as lacs) 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

x	Name (SS/LS, LS)	SS (10^3x+3)	LS (10^6x, 10^6x+3)	Authorities
				AHD4	CED	COD	OED2	OEDweb	RHD2	SOED3	W3	HM
1	Million	10^6	10^6	✓	✓	✓	✓	✓	✓	✓	✓	✓
	Milliard		10^9	✓	✓	✓	✓	✓	✓			✓
2	Billion	10^9	10^12	✓	✓	✓	✓	✓	✓	✓	✓	✓
3	Trillion	10^12	10^18	✓	✓	✓	✓	✓	✓	✓	✓	✓
4	Quadrillion	10^15	10^24	✓	✓	✓	✓	✓	✓	✓	✓	✓
5	Quintillion	10^18	10^30	✓	✓	✓	✓	✓	✓	✓	✓	✓
6	Sextillion	10^21	10^36	✓	✓		✓	✓	✓	✓	✓	✓
AHD4	CED	COD	Dcom	OED2	OEDnew	RHD2	SOED3	W3-->	7	Septillion	10^24	10^42	✓	✓		✓	✓	✓	✓	✓	✓
8	Octillion	10^27	10^48	✓	✓		✓	✓	✓	✓	✓	✓
9	Nonillion	10^30	10^54	✓	✓		✓	✓	✓	✓	✓	✓
10	Decillion	10^33	10^60	✓	✓		✓	✓	✓	✓	✓	✓
11	Undecillion	10^36	10^66	✓	✓				✓		✓	✓
12	Duodecillion	10^39	10^72	✓	✓	✓			✓		✓	✓
13	Tredecillion	10^42	10^78	✓	✓				✓		✓	✓
14	Quattuordecillion	10^45	10^84	✓	✓				✓		✓	✓
15	Quindecillion	10^48	10^90	✓	✓				✓		✓	✓
16	Sexdecillion	10^51	10^96	✓	✓				✓		✓	✓
17	Septendecillion	10^54	10^102	✓	✓				✓		✓	✓
18	Octodecillion	10^57	10^108	✓	✓				✓		✓	✓
19	Novemdecillion	10^60	10^114	✓	✓				✓		✓	✓
20	Vigintillion	10^63	10^120	✓	✓		✓	✓	✓	✓	✓	✓
100	Centillion	10^303	10^600	✓	✓		✓	✓	✓			✓

Usage:

• Short scale: US, English Canada, modern British, Australia, and Eastern Europe
• Long scale: French Canada, older British, Western & Central Europe

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. Centillion 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	OEDweb	RHD2	SOED3	W3	HM
Googol	10^100	✓	✓	✓	✓	✓	✓	✓	✓	✓
	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 (see below). 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.

Names of larger numbers, however, have a tenuous, artificial existence, rarely found outside definitions, lists, and discussions of how 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 ." When a number such as 10⁴⁵ needs to be referred to in words, it is simply read out as "ten to the forty-fifth" or "ten to the forty-five". 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 way 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 ⋅ 10 ^{8}} , and embedded this construction within another copy of itself to produce names for numbers up to (( 10 ^{8)} ^{( 10 ^{8)}} ) ^{( 10 ^{8)}} =10 ^{8 ⋅ 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 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"

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).

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 in the following passage:

width=200	Value	Name	Authority
10^100		Kasner and Newman, dictionaries (see above)
10^googol = 10		Kasner and Newman, dictionaries (see above)

John Horton Conway and Richard K. Guy have suggested that N-plex be used as a name for 10^N. This gives rise to the name googolplexplex for 10^googolplex = 10. Conway and Guy 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, which is written as 10. None of these names are in wide use.

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): ; ; ; 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 originally popularized the short scale worldwide, reverted to the long scale.

The term milliard is unambiguous and always means 10⁹. It is seldom 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 (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.

Concerning 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 "milliardo" in Italian, "Milliarde" in German, "miljard" in Dutch, "milyar" in Turkish, and "миллиард," milliard (transliterated) 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 9 or smaller. For larger n (between 10 and 999), prefixes can be constructed based on a system described by Conway and Guy. 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. 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". 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".

The following table shows number names generated by the system described by Conway and Guy for the short and long scales.^[1]

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	R	Ronna-
9	5	10^30	Nonillion	Quintillion	Quintillion	Q	Quetta-
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	Quindecillion	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	Quinvigintillion	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	Quintrigintillion	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 quinvigintillion	Quinvigintilliard
60	30	10^183	Sexagintillion	Thousand trigintillion	Trigintilliard
70	35	10^213	Septuagintillion	Thousand quintrigintillion	Quintrigintilliard
80	40	10^243	Octogintillion	Thousand quadragintillion	Quadragintilliard
90	45	10^273	Nonagintillion	Thousand quinquadragintillion	Quinquadragintilliard
100	50	10^303	Centillion	Thousand quinquagintillion	Quinquagintilliard
101	51	10^306	Uncentillion	Unquinquagintillion	Unquinquagintillion
110	55	10^333	Decicentillion	Thousand quinquinquagintillion	Quinquinquagintilliard
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 quinsexagintillion	Quinsexagintilliard
140	70	10^423	Quadragintacentillion	Thousand septuagintillion	Septuagintilliard
150	75	10^453	Quinquagintacentillion	Thousand quinseptuagintillion	Quinseptuagintilliard
160	80	10^483	Sexagintacentillion	Thousand octogintillion	Octogintilliard
170	85	10^513	Septuagintacentillion	Thousand quinoctogintillion	Quinoctogintilliard
180	90	10^543	Octogintacentillion	Thousand nonagintillion	Nonagintilliard
190	95	10^573	Nonagintacentillion	Thousand quinnonagintillion	Quinnonagintilliard
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	Googolplex	Ten trillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentilliduotrigintatrecentillion	Ten thousand millisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillion	Ten millisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentilliard

Googolplex's short scale name is derived from it equal to ten of the 3,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,332nd "-illion"s (This is the value of n when 10 × 10 = 10)

Googolplex's long scale name (both traditional British and traditional European) is derived from it being equal to ten thousand of the 1,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666th "-illion"s (This is the value of n when 10,000 × 10 = 10).

Binary prefixes

The International System of Quantities (ISQ) defines a series of prefixes denoting integer powers of 1024 between 1024¹ and 1024⁸.

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-

Other named large numbers used in mathematics, physics and chemistry

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

Notes and References

1. Web site: Fish. Conway's illion converter. March 1, 2023.