Names of large numbers
Encyclopedia : N : NA : NAM : Names of large numbers
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Throughout this article, exponential or scientific notation is used. 106 could also be written as the number 1 followed by six 0s, 1 000 000; 109 could be written as 1 000 000 000; and so on.
Names of numbers larger than a quadrillion are almost never used, for reasons discussed further below. It is debatable which of them should be considered real working English vocabulary and which are merely trivia, curiosities, or coinages. The following table lists those names of numbers which are found in many English dictionaries and thus have a special claim to being "real words". The "Traditional British" values shown are becoming obsolete in English, but are still prevalent in continental Europe; see Long and short scales.
The entry "approximate frequency of use" shows how often the word is used compared to the word "trillion".[#endnote_usage] Million is used thirty-nine times as often as trillion, while septillion is used only 1/200 as often.
Centillion [AHD] 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 Shorter Oxford English Dictionary comments that googol and googolplex are "not in formal mathematical use".
Some large numbers have real referents in human experience. Their names are real words, encountered in many contexts. For example, on one day in 2004, Google News showed 78 600 hits on "billion", starting with "Turkey Repays USD 1.6 Billion In Foreign Debt". It shows 9870 hits on "trillion", and 56 on "quadrillion": for example, "The US Department of Energy reports that in 2002, the United States economy consumed 97.6 quadrillion BTUs (quad BTUs)."
References to names of quantities larger than a quadrillion, however, are rare, and increasingly artificial; they tend to be limited to discussions of names of numbers, or mathematical concepts. For example, the first hit on "quintillion" is about a man who is trying to preserve the Etsako language by codifying it, and "went as far as providing numerals from one to quintillion". "Septillion" appears—as the name of a racing yacht.
Nevertheless, large numbers have an intellectual fascination and are of mathematical interest. Giving them names is one of the ways in which people try to conceptualize and understand them.
In The Sand Reckoner, Archimedes estimated the number of grains of sand that would be required to fill the known universe. To do this, he called a myriad myriad (108) a "first number"; a myriad myriad first numbers as "second numbers" (1016), and so on up to "eighth numbers" (1064). He concluded that it would take less than "one thousand myriad myriad eighth numbers" (1075) grains of sand to pack the universe solid with sand. This much sand would fill a volume larger than our galaxy, the Milky Way, but smaller than the group of galaxies it is part of, the Local Group.
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.
Names of numbers larger than a quadrillion, even if found in dictionaries, have a tenuous existence. Just as one can debate whether floccinaucinihilipilification is really an English word—it is in the Oxford English Dictionary, but no other—it is questionable how real the word "trigintillion" is. "Trigintillion" is only encountered in definitions, lists of names of large numbers, and "Sand Reckoner"-like discourses on the meaning of very large numbers.
Even well-established names like "sextillion" are rarely used, even in those contexts where they have a real meaning—science, astronomy, and engineering. In science, since the 1800s, numbers have been written using the familiar "scientific notation", in which powers of ten are expressed as a ten 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 measurement rather than a count, SI prefixes are used; one says "femtosecond", not "one quadrillionth of a second." In some cases, specialized very large units are used, such as the astronomer's parsec and light year.
It should be noted, too, that 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.
Here we present some names that have been given to large numbers, and the context and authority for the names. These numbers are almost pure mathematical abstractions, not physical realities. The names for such numbers are very rarely used. They may have a claim staked out for them in reference books, but they remain more in the nature of curiosities, trivia, or mathematical recreation than genuine working English vocabulary.
Nicolas Chuquet's book Triparty en la science des nombres was not published during his lifetime, but most of it was copied by Estienne de la Roche for a portion of his 1520 book, Larismetique. Chuquet's book contains a passage in which he shows a large number marked off into groups of six digits, with the comment:
Chuquet is sometimes credited with "inventing" the names million, billion, trillion, quadrillion, and so forth. This is an oversimplification.
This is done in the long scale by simply multiplying the Latin numeral by 6. For example, in a billion, the Latin numeral is 'bi', or 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.
Conway and Guy The Book of Numbers, J. H. Conway and R. K. Guy, New York: Springer-Verlag, 1996, pp. 15–16
have suggested that N-plex be used as a name for 10N. This gives rise to the name googolplexplex for
10googolplex; this number has also been called googolplexian. In addition,
the terms googolduplex, googoltriplex,
etc. have been coined by various persons for the numbers 10googolplex, 10googolduplex, etc.
None of these names are
in wide use, nor are any currently found in dictionaries.
This table illustrates several systems for naming large numbers, and shows how they can be extended past decillion.
Traditional British usage assigned new names for each power of one million (the long scale). 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 109, a "trillion" is 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 Pelletier du Mans circa 1550 (therefore the long scale is also known as 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 to our modern term.
With regard to names ending in -illiard, 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, Google searches on French-language pages for "trillion", "trilliard", "quadrillion", "quadrilliard", and "quintillion" return 6630, 102, 312, 7, and 127 hits, respectively. However, one has to take into account that these large numbers are not needed too often and scientists almost always use exponents.
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.
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. These names are not in wide use.
For additional details, see billion and long scale.
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''Steps to :
The \"standard dictionary numbers\"
Name
American
("short scale")
Traditional British value
("long scale")
Authorities
Approx. relative frequency of use
AHD4
COD
Dcom
OED2
OEDnew
RHD2
SOED3
W3
million
106
106
39
milliard
109
1/94
billion
109
1012
14.0
trillion
1012
1018
1.0
quadrillion
1015
1024
1/44
quintillion
1018
1030
1/150
sextillion
1021
1036
1/190
COD
Dcom
OED2
OEDnew
RHD2
SOED3
W3-->
septillion
1024
1042
1/200
octillion
1027
1048
1/360
nonillion
1030
1054
1/1200
decillion
1033
1060
1/590
undecillion
1036
1066
1/1400
duodecillion
1039
1072
1/17000
tredecillion
1042
1078
1/20000
quattuordecillion
1045
1084
1/32000
quindecillion (quinquadecillion)
1048
1090
1/32000
COD
Dcom
OED2
OEDnew
RHD2
SOED3
W3-->
sexdecillion (sedecillion)
1051
1096
1/1700
septendecillion
1054
10102
1/1600
octodecillion
1057
10108
1/27000
novemdecillion (novendecillion)
1060
10114
1/21000
vigintillion
1063
10120
1/20000
googol
10100
10100
1/69
centillion
10303
10600
1/2000
googolplex
[\,\!10^}]
[\,\!10^}]
1/600
Dictionaries cited
✓ means the word is included in the dictionaryUsage of names of large numbers
Chuquet and the origins of the \"standard dictionary numbers\"
Ou qui veult le premier point peult signiffier million Le second point byllion Le tiers poit 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, a bimillion (Adam) 1012, and a trimillion was 1018.An Aide Memoire
An easy way to find out scientific notation for one of the above numbers in the short scale is to take the Latin cardinal number indicated in the name (Such as 2 in 'bi'llion, 4 in 'quadri'llion, 18 in 'octodec'illion etc), then add one to it, and finally multiply the whole thing by 3. For example, in a trillion, the Latin numeral is 'tri', or 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.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,Kasner, Edward and James Newman, Mathematics and the Imagination, 1940, Simon and Schuster, New York.
in the following passage:
Words of wisdom are spoken by children at least as often as by scientists. 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 a 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 = [\,\!10^}]
Googolplex
Kasner and Newman, dictionaries (see above)
Extensions of the standard dictionary numbers
You can [WikiProject Fact and Reference Checkhelp] Wikipedia by introducing appropriate citations.''
Value
USA and Modern British
("short scale")
Traditional British
("long scale")
Traditional European (Pelletier)
("long scale")
109
Billion
Thousand million
Milliard
1012
Trillion
Billion
Billion
1015
Quadrillion
Thousand billion
Billiard
1018
Quintillion
Trillion
Trillion
1021
Sextillion
Thousand trillion
Trilliard
1024
Septillion
Quadrillion
Quadrillion
1027
Octillion
Thousand quadrillion
Quadrilliard
1030
Nonillion
Quintillion
Quintillion
1033
Decillion
Thousand quintillion
Quintilliard
1036
Undecillion
Sextillion
Sextillion
1039
Duodecillion
Thousand sextillion
Sextilliard
1042
Tredecillion
Septillion
Septillion
1045
Quattuordecillion
Thousand septillion
Septilliard
1048
Quindecillion
Octillion
Octillion
1051
Sexdecillion
Thousand octillion
Octilliard
1054
Septendecillion
Nonillion
Nonillion
1057
Octodecillion
Thousand nonillion
Nonilliard
1060
Novemdecillion
Decillion
Decillion
1063
Vigintillion
Thousand decillion
Decilliard
1066
Unvigintillion
Undecillion
Undecillion
1069
Duovigintillion
Thousand undecillion
Undecilliard
1072
Trevigintillion
Duodecillion
Duodecillion
1075
Quattuorvigintillion
Thousand duodecillion
Duodecilliard
1078
Quinvigintillion
Tredecillion
Tredecillion
1081
Sexvigintillion
Thousand tredecillion
Tredecilliard
1084
Septenvigintillion
Quattuordecillion
Quattuordecillion
1087
Octovigintillion
Thousand quattuordecillion
Quattuordecilliard
1090
Novemvigintillion
Quindecillion
Quindecillion
1093
Trigintillion
Thousand quindecillion
Quindecilliard
1096
Untrigintillion
Sexdecillion
Sexdecillion
1099
Duotrigintillion
Thousand sexdecillion
Sexdecilliard
10100
Googol
(Ten Duotrigintillion)
Ten Thousand sexdecillion
Ten sexdecilliard
10102
Tretrigintillion
Septendecillion
Septendecillion
10105
Quattuortrigintillion
Thousand septendecillion
Septendecilliard
10108
Quintrigintillion
Octodecillion
Octodecillion
10111
Sextrigintillion
Thousand octodecillion
Octodecilliard
10114
Septentrigintillion
Novemdecillion
Novemdecillion
10117
Octotrigintillion
Thousand novemdecillion
Novemdecilliard
10120
Novemtrigintillion
Vigintillion
Vigintillion
10123
Quadragintillion
Thousand vigintillion
Vigintilliard
10126
Unquadragintillion
Unvigintillion
Unvigintillion
10129
Duoquadragintillion
Thousand unvigintillion
Unvigintilliard
10132
Trequadragintillion
Duovigintillion
Duovigintillion
10135
Quattuorquadragintillion
Thousand duovigintillion
Duovigintilliard
10138
Quinquadragintillion
Trevigintillion
Trevigintillion
10141
Sexquadragintillion
Thousand trevigintillion
Trevigintilliard
10144
Septenquadragintillion
Quattuorvigintillion
Quattuorvigintillion
10147
Octoquadragintillion
Thousand quattuorvigintillion
Quattuorvigintilliard
10150
Novemquadragintillion
Quinvigintillion
Quinvigintillion
10153
Quinquagintillion
Thousand quinvigintillion
Quinvigintilliard
10156
Unquinquagintillion
Sexvigintillion
Sexvigintillion
10159
Duoquinquagintillion
Thousand sexvigintillion
Sexvigintilliard
10162
Trequinquagintillion
Septenvigintillion
Septenvigintillion
10165
Quattuorquinquagintillion
Thousand septenvigintillion
Septenvigintilliard
10168
Quinquinquagintillion
Octovigintillion
Octovigintillion
10171
Sexquinquagintillion
Thousand octovigintillion
Octovigintilliard
10174
Septenquinquagintillion
Novemvigintillion
Novemvigintillion
10177
Octoquinquagintillion
Thousand novemvigintillion
Novemvigintilliard
10180
Novemquinquagintillion
Trigintillion
Trigintillion
10183
Sexagintillion
Thousand trigintillion
Trigintilliard
10186
Unsexagintillion
...
...
10189
Duosexagintillion
...
...
10192
Tresexagintillion
...
...
10195
Quattuorsexagintillion
...
...
10198
Quinsexagintillion
...
...
10201
Sexsexagintillion
...
...
10204
Septsexagintillion
...
...
10207
Octosexagintillion
...
...
10210
Novemsexagintillion
...
...
10213
Septuagintillion
Thousand quintrigintillion
Quintrigintilliard
10216
Unseptuagintillion
...
...
10219
Duoseptuagintillion
...
...
10222
Treseptuagintillion
...
...
10225
Quattuorseptuagintillion
...
...
10228
Quinseptuagintillion
...
...
10231
Sexseptuagintillion
...
...
10234
Septseptuagintillion
...
...
10237
Octoseptuagintillion
...
...
10240
Novemseptuagintillion
Quadragintillion
Quadragintillion
10243
Octogintillion
Thousand quadragintillion
Quadragintilliard
10246
Unoctogintillion
...
...
10249
Duooctogintillion
...
...
10252
Treoctogintillion
...
...
10255
Quattuoroctogintillion
...
...
10258
Quinoctogintillion
...
...
10261
Sexoctogintillion
...
...
10264
Septoctogintillion
...
...
10267
Octooctogintillion
...
...
10270
Novemoctogintillion
...
...
10273
Nonagintillion
Quinquadragintillion
Quinquadragintilliard
10276
Unnonagintillion
...
...
10279
Duononagintillion
...
...
10282
Trenonagintillion
...
...
10285
Duattuornonagintillion
...
...
10288
Quinnonagintillion
...
...
10291
Sexnonagintillion
...
...
10294
Septnonagintillion
...
...
10297
Octononagintillion
...
...
10300
Novemnonagintillion
Quinquagintillion
Quinquagintillion
10303
Centillion
Thousand quinquagintillion
Quinquagintilliard
10306
Cenuntillion
...
...
10309
Duocentillion
...
...
10312
Centretillion
...
...
10360
Sexagintillion
Sexagintillion
10363
Thousand sexagintillion
Sexagintilliard
10420
Septuagintillion
Septuagintillion
10423
Thousand septuagintillion
Septuagintilliard
10480
Octogintillion
Octogintillion
10483
Thousand octogintillion
Octogintilliard
10540
...
Nonagintillion
Nonagintillion
10543
...
Thousand nonagintillion
Nonagintilliard
10600
...
Centillion
Centillion
10603
Ducentillion
Thousand centillion
Centilliard
10903
Trecentillion
...
...
101203
Quadringentillion
...
..
101503
Quingentillion
...
...
101803
Sescentillion
...
...
102103
Septingentillion
...
...
102403
Octingentillion
...
...
102703
Nongentillion
...
...
103003
Millillion
...
...
10googol
Googolplex
...
...
Conway numbers
This system will itself become ambiguous for numbers much larger than this, with exponents of a size which the Romans rarely counted to, like 10 6,000,258. John Horton Conway and Richard Guy have proposed
a consistent set of conventions which permit the system to provide "English names", in principle, for any integer whatever.Other large numbers used in mathematics
See also
External links
References
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