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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.

Two naming scales 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.

Indian English does not use millions, but has its own system of large numbers including lakhs and crores.[1] 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
(103x+3) 		LS
(106x, 106x+3) 		Authorities
AHD4[2] CED[3] COD[4] OED2[5] OEDnew[6] RHD2[7] SOED3[8] W3[9] UM[10]
1 Million, Milliard 106 106, 109 ✓✓ ✓✓ ✓. ✓✓ ✓✓ ✓✓ ✓. ✓. ✓✓
2 Billion, Billiard 109 1012, 1015 ✓✓ ✓✓ ✓. ✓✓ ✓✓ ✓✓ ✓. ✓. ✓✓
3 Trillion 1012 1018 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
4 Quadrillion 1015 1024 ✓ ✓   ✓ ✓ ✓ ✓ ✓ ✓
5 Quintillion 1018 1030 ✓ ✓   ✓ ✓ ✓ ✓ ✓ ✓
6 Sextillion 1021 1036 ✓ ✓   ✓ ✓ ✓ ✓ ✓ ✓
7 Septillion 1024 1042 ✓ ✓   ✓ ✓ ✓ ✓ ✓ ✓
8 Octillion 1027 1048 ✓ ✓   ✓ ✓ ✓ ✓ ✓ ✓
9 Nonillion 1030 1054 ✓ ✓   ✓ ✓ ✓ ✓ ✓ ✓
10 Decillion 1033 1060 ✓ ✓   ✓ ✓ ✓ ✓ ✓ ✓
11 Undecillion 1036 1066 ✓ ✓       ✓   ✓ ✓
12 Duodecillion 1039 1072 ✓ ✓       ✓   ✓ ✓
13 Tredecillion 1042 1078 ✓ ✓       ✓   ✓ ✓
14 Quattuordecillion 1045 1084 ✓ ✓       ✓   ✓ ✓
15 Quindecillion 1048 1090 ✓ ✓       ✓   ✓ ✓
16 Sexdecillion 1051 1096 ✓ ✓       ✓   ✓ ✓
17 Septendecillion 1054 10102 ✓ ✓       ✓   ✓ ✓
18 Octodecillion 1057 10108 ✓ ✓       ✓   ✓ ✓
19 Novemdecillion 1060 10114 ✓ ✓       ✓   ✓ ✓
20 Vigintillion 1063 10120 ✓ ✓   ✓ ✓ ✓ ✓ ✓ ✓
100 Centillion 10303 10600 ✓ ✓   ✓ ✓ ✓     ✓

Usage:

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.[11] Centillion[12] 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 10100 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
Googolplex 10googol (1010100) ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

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.[13]

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. 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" (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"

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[14] 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 = 1010100 Googolplex Kasner and Newman, dictionaries (see above)

John Horton Conway and Richard K. Guy[15] have suggested that N-plex be used as a name for 10N. This gives rise to the name googolplexplex for 10googolplex = 101010100. This number (ten to the power of a googolplex) is also known as a googolduplex and googolplexian.[16] Conway and Guy[15] 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 are called googol and googolplex which 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 originally popularized the short scale worldwide, 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.

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 103·999+3 = 103000 (short scale) or 106·999 = 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.[15] 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 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 106,000,258, 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.[15] 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 103m+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".[15] For example, 103,000,012, the 1,000,003rd "-illion" number, equals one "millinillitrillion"; 1033,002,010,111, the 11,000,670,036th "-illion" number, equals one "undecillinilliseptuagintasescentillisestrigintillion"; and 1029,629,629,633, the 9,876,543,210th "-illion" number, equals one "nonilliseseptuagintaoctingentillitresquadragintaquingentillideciducentillion".[15]

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[17] Thousand quingentillion Quingentilliard
Value Name Equivalent
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
1010100 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.[18]

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

See also

References

  1. ^ Bellos, Alex (2011). Alex's Adventures in Numberland (illustrated ed.). A&C Black. p. 114. ISBN 978-1-4088-0959-4. Extract of page 114
  2. ^ The American Heritage® Dictionary of the English Language. American Heritage Dictionary (4th ed.). 2000. ISBN 0-395-82517-2.
  3. ^ Collins English Dictionary, 11th Edition, HarperCollins Publishers.
  4. ^ Cambridge Dictionaries Online, Cambridge, UK: Cambridge University Press.
  5. ^ Oxford English Dictionary, 2nd edition, Oxford, UK: Oxford University Press. ISBN 0-19-861186-2 (and addendums since publication in 1989.)
  6. ^ Oxford English Dictionary, New Edition, Oxford, UK: Oxford University Press. [1] (subscription required), checked April 2007
  7. ^ The Random House Dictionary, 2nd Unabridged Edition, 1987, Random House.
  8. ^ Shorter Oxford English Dictionary, 3rd edition, 1993, Oxford: Clarendon Press.
  9. ^ Webster's Third New International Dictionary, Unabridged, 1993, Merriam-Webster.
  10. ^ "How Many? A Dictionary of Units of Measures". Russ Rowlett and the University of North Carolina at Chapel Hill. Retrieved 15 August 2009.
  11. ^ p. 316, The History of the English Language, Oliver Farrar Emerson, New York, London: Macmillan and Co., 1894.
  12. ^ Entry for centillion in the American Heritage Dictionary
  13. ^ "Zimbabwe rolls out Z$100tr note". BBC News. 16 January 2009. Retrieved 16 January 2009.
  14. ^ Kasner, Edward, and James Newman, Mathematics and the Imagination, 1940, Simon and Schuster, New York.
  15. ^ a b c d e f The Book of Numbers, J. H. Conway and R. K. Guy, New York: Springer-Verlag, 1996, pp. 15–16. ISBN 0-387-97993-X.
  16. ^ Bowers, Jonathan. "Infinity Scrapers". Polytope, 2010.
  17. ^ Stewart, Ian (2017). Infinity: A Very Short Introduction (illustrated ed.). Oxford University Press. p. 20. ISBN 978-0-19-875523-4. Extract of page 20
  18. ^ "IEC 80000-13:2008". International Organization for Standardization. Retrieved 21 July 2013.

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