Names of large numbers

From Wikipedia, the free encyclopedia

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., Canada and modern British)	Long scale (continental Europe, older British)	Authorities
Name	Short scale (U.S., Canada and modern British)	Long scale (continental Europe, older British)	AHD4^[1]	CED^[2]	COD^[3]	OED2^[4]	OEDnew^[5]	RHD2^[6]	SOED3^[7]	W3^[8]	UM^[9]
Million	10⁶	10⁶	✓	✓	✓	✓	✓	✓	✓	✓	✓
Milliard		10⁹	✓	✓		✓	✓	✓			✓
Billion	10⁹	10¹²	✓	✓	✓	✓	✓	✓	✓	✓	✓
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
Name	Value	AHD4	CED	COD	OED2	OEDnew	RHD2	SOED3	W3	UM
Googol	10¹⁰⁰	✓	✓		✓	✓	✓	✓	✓	✓
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

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 only 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 contexts of science, astronomy, and engineering, 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 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,^[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 = $\,\!10^{{10^{{100}}}}$	Googolplex	Kasner and Newman, dictionaries (see above)

Conway and 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. 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.

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), Canadian 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:^[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., Canada 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	Quinquadecillion	Octillion	Octillion
16	10⁵¹	Sedecillion	Thousand octillion	Octilliard
17	10⁵⁴	Septendecillion	Nonillion	Nonillion
18	10⁵⁷	Octodecillion	Thousand nonillion	Nonilliard
19	10⁶⁰	Novendecillion	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	Sedecillion	Sedecillion
32	10⁹⁹	Duotrigintillion	Thousand sedecillion	Sedecilliard
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	Novendecillion	Novendecillion
38	10¹¹⁷	Octotrigintillion	Thousand novendecillion	Novendecilliard
39	10¹²⁰	Noventrigintillion	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	Sesquinquagintillion	Sesquinquagintillion
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., Canada and modern British (short scale)	Traditional British (long scale)	Traditional European (Peletier) (long scale)
10¹⁰⁰	Googol (Ten duotrigintillion)	Googol (Ten thousand sedecillion)	Googol (Ten sedecilliard)
$10^{{10^{{100}}}}\,\!$	Googolplex	Googolplex	Googolplex

Proposals for new naming system

Other large numbers used in mathematics and physics

References

↑ American Heritage Dictionary, 4th edition, ISBN 0-395-82517-2.
↑ Collins English Dictionary, 11th Edition, HarperCollins Publishers.
↑ Cambridge Dictionaries Online, Cambridge, UK: Cambridge University Press.
↑ Oxford English Dictionary, 2nd edition, Oxford, UK: Oxford University Press. ISBN 0-19-861186-2 (and addendums since publication in 1989.)
↑ Oxford English Dictionary, New Edition, Oxford, UK: Oxford University Press. (subscription required), checked April 2007
↑ The Random House Dictionary, 2nd Unabridged Edition, 1987, Random House.
↑ Shorter Oxford English Dictionary, 3rd edition, 1993, Oxford: Clarendon Press.
↑ Webster's Third New International Dictionary, Unabridged, 1993, Merriam-Webster.
↑ "How Many? A Dictionary of Units of Measures". Russ Rowlett and the University of North Carolina at Chapel Hill. Retrieved 2009-08-15.
↑ p. 316, The History of the English Language, Oliver Farrar Emerson, New York, London: Macmillan and Co., 1894.
↑ Entry for centillion in the American Heritage Dictionary
↑ "Zimbabwe rolls out Z$100tr note". BBC News. January 16, 2009. Retrieved 2009-01-16.
↑ Kasner, Edward and James Newman, Mathematics and the Imagination, 1940, Simon and Schuster, New York.
↑ 14.0 14.1 14.2 14.3 The Book of Numbers, J. H. Conway and R. K. Guy, New York: Springer-Verlag, 1996, pp. 15–16. ISBN 0-387-97993-X.
↑ Bowers, Jonathan. "Infinity Scrapers". Polytope, 2010.
↑ Rowlett, Russ (2001-11-01). "Names for large numbers". University of North Carolina. 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

Large numbers

Examples in numerical order

Million
Billion
Trillion
Quadrillion
Googol
Centillion
Googolplex
Googolplexplex
Skewes' number
Moser's number
Graham's number
Rayo's number
Transfinite numbers
Infinity

Expression methods

Notations	Knuth's up-arrow notation Conway chained arrow notation Steinhaus–Moser notation

Operators	Hyper operators Tetration Ackermann function

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