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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.

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 obsolete[citation needed]
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.




Contents

[hide]

* The "standard dictionary numbers"
* Usage of names of large numbers
* Origins of the "standard dictionary numbers"
* An aide-memoire
* The googol family
* Extensions of the standard dictionary numbers
* Proposals for new naming system
* Other large numbers used in mathematics and physics
* See also
* References
* External links



[edit] The "standard dictionary numbers"


Name Short scale
(USA and
Modern British) Long scale
(Continental Europe,
Archaic British, and India) Authorities
AHD4] COD] OED2] OEDnew] RHD2] SOED3] W3] UM]
million 106 106 ✓ ✓ ✓   ✓ ✓ ✓ ✓
milliard   109 ✓   ✓   ✓     ✓
billion 109 1012 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
billiard   1015 ✓   ✓   ✓     ✓
trillion 1012 1018 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
trilliard   1021                
quadrillion 1015 1024 ✓   ✓ ✓ ✓ ✓ ✓ ✓
quadrilliard   1027                
quintillion 1018 1030 ✓   ✓ ✓ ✓ ✓ ✓ ✓
quintilliard   1033                
sextillion 1021 1036 ✓   ✓ ✓ ✓ ✓ ✓ ✓
sextilliard   1039                
septillion 1024 1042 ✓   ✓ ✓ ✓ ✓ ✓ ✓
septilliard   1045                
octillion 1027 1048 ✓   ✓ ✓ ✓ ✓ ✓ ✓
octilliard   1051                
nonillion 1030 1054 ✓   ✓ ✓ ✓ ✓ ✓ ✓
nonilliard   1057                
decillion 1033 1060 ✓   ✓ ✓ ✓ ✓ ✓ ✓
decilliard   1063                
undecillion 1036 1066 ✓       ✓   ✓ ✓
undecilliard   1069                
duodecillion 1039 1072 ✓       ✓   ✓ ✓
duodecilliard   1075                
tredecillion 1042 1078 ✓       ✓   ✓ ✓
tredecilliard   1081                
quattuordecillion 1045 1084 ✓       ✓   ✓ ✓
quattuordecilliard   1087                
quindecillion (quinquadecillion) 1048 1090 ✓       ✓   ✓ ✓
quindecilliard (quinquadecilliard)   1093                
sexdecillion (sedecillion) 1051 1096 ✓       ✓   ✓ ✓
sexdecilliard (sedecilliard)   1099                
septendecillion 1054 10102 ✓       ✓   ✓ ✓
septendecilliard   10105                
octodecillion 1057 10108 ✓       ✓   ✓ ✓
octodecilliard   10111                
novemdecillion (novendecillion) 1060 10114 ✓       ✓   ✓ ✓
novemdecilliard (novendecilliard)   10117                
vigintillion 1063 10120 ✓   ✓ ✓ ✓ ✓ ✓ ✓
vigintilliard   10123                
...
centillion 10303 10600 ✓   ✓ ✓       ✓
centilliard   10603                

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.] 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.).

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".


[edit] 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 30 US dollars.]

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., 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 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.


[edit] 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).

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 Old Italian milione, an intensification of mille, a thousand.
That is, a million is a big thousand, much as 1728 is a great gross.

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.


[edit] An aide-memoire


An easy way to find the value of the above numbers in the short scale 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 scales, 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 long and short scales.


[edit] 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:

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, with so many zeros after
the 1 that the number of zeros is a googol.

Value Name Authority
10100 Googol Kasner and Newman, dictionaries (see above)
10googol = Googolplex Kasner and Newman, dictionaries (see above)

Conway and Guy ] have suggested that N-plex be used as a name for 10N. This gives rise
to the name googolplexplex for 10googolplex. 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. None of these names are in wide use, nor are any currently found in
dictionaries.


[edit] Extensions of the standard dictionary numbers




 This section needs additional citations for verification.
Please help improve this article by adding reliable references. Unsourced material may
be challenged and removed. (June 2006)

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 106·n+3, milliard is certainly in
widespread use in languages other than English, but the degree of actual use of the
larger terms is questionable. For example, as of 2004[update], Google searches on
French-language pages for trillion, quadrillion, and quintillion return 6630, 312, and
127 hits respectively, whilst searches for trilliard and quadrilliard return only 102
and 7 hits respectively. 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
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 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
106,000,258, 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.]

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 USA and Modern British
(short scale) Traditional British
(long scale) Traditional European (Peletier)
(long scale)
1 106 Million Million Million
2 109 Billion Thousand million Milliard
3 1012 Trillion Billion Billion
4 1015 Quadrillion Thousand billion Billiard
5 1018 Quintillion Trillion Trillion
6 1021 Sextillion Thousand trillion Trilliard
7 1024 Septillion Quadrillion Quadrillion
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 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 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 Quinquagintaquadringentill
iard

Value USA 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)
 Googolplex Googolplex Googolplex


[edit] 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. ]


[edit] Other large numbers used in mathematics and physics


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


[edit] See also



* Number prefix
* Knuth's up-arrow notation
* Large numbers
* List of numbers
* Long and short scales
* Nicolas Chuquet
* Number names
 

 Mathematics portal


* Indefinite and fictitious numbers
* Indian numbering system
* Orders of magnitude
* Orders of magnitude (numbers)
* SI prefix



[edit] 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. ^ ". 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. ^ 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. ^ Often misspelled quardragintillion.
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. 



[edit] 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)


[hide]

v • d • e

Large numbers

Subarticles 
Names of large numbers · History of large numbers


Examples (numerical order)
Standardized list · Name list 
million · googol · googolplex · Skewes' number · Moser's number · Graham's number
· Transfinite numbers


Expression methods 


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


Operators 
Hyper operators (Tetration)  · Ackermann function



Related articles 
Number systems · Number names · Orders of magnitude (numbers) · List of numbers ·
Indefinite and fictitious numbers



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Categories: Large numbers | Numeration | Numerals

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