Names of large numbers

From Academic Kids

Contents

The "standard dictionary numbers"

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".

Name American and modern
British value
("short scale")
Traditional British value
("long scale")
Authorities Google hits
AHD4 COD Dcom OED2 OEDnew RHD2 SOED3 W3
million 106 106 54,900,000 Pages
6,880,000 Articles
milliard   109 24,300 English pages
466,700 other pages
41,000 Articles
billion 109 1012 12,700,000 English pages (mostly using 109)
3,000,000 other pages
2,200,000 Articles
trillion 1012 1018 1,460,000 Pages
303,000 Articles
quadrillion 1015 1024 63,200 Pages
52,800 Articles
quintillion 1018 1030 15,300 Pages
42,900 Articles
sextillion 1021 1036 6,260 Pages
38,500 Articles
septillion 1024 1042 3,460 Pages
37,800 Articles
octillion 1027 1048 3,440 Pages
39,700 Articles
nonillion 1030 1054 1,980 Pages
715 Articles
decillion 1033 1060 4,940 Pages
783 Articles
undecillion 1036 1066 854 Pages
294 Articles
duodecillion 1039 1072 1250 Pages
249 Articles
tredecillion 1042 1078 885 Pages
182 Articles
quattuordecillion 1045 1084 830 Pages
154 Articles
quindecillion (quinquadecillion) 1048 1090 686 Pages
151 Articles
sexdecillion (sedecillion) 1051 1096 759 Pages
145 Articles
septendecillion 1054 10102 699 Pages
138 Articles
octodecillion 1057 10108 863 Pages
157 Articles
novemdecillion (novendecillion) 1060 10114 941 Pages
182 Articles
vigintillion 1063 10120 1550 Pages
410 Articles
googol 10100 10100 82,500 Pages
10,200 Articles
centillion 10303 10600 8560 Pages
25 Articles
googolplex 10googol 10googol 28,100 Pages
14,400 Articles

Note the oddity that some dictionaries list "decillion" and "vigintillion", but not of the numbers in between (undecillion, duodecillion, etc.)

Vigintillion 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 Kasners' nephew. None include any higher names in the googol family (googolplexplex, etc.) The Shorter Oxford English Dictionary comments that googol and googolplex are "not in formal mathematical use".

Dictionaries cited

√ means the word is included in the dictionary

"Google hits" means the number of hits reported by Google when searching the entire web and the Google USENET archives as of June 4 2004.

Usage of names of large numbers

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 $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.

It can be argued that a number such as a "trigintillion" has no existence in the physical world. There simply is not a trigintillion of anything, and no practical need for such a name.

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" 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 to American and British audiences—assuming that there are any listeners who understand this number without consulting a dictionary). 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. Glancing at the table for the Googol family, it is clear what the value of a "Googolquattuordeciplex" would be if there were such a word, but that does not mean that such a word exists, nor that there is any value in coining it.

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.

Chuquet and the origins of the "standard dictionary numbers"

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:

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).

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 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".
  • The words "bymillions" and "trimillions" were first recorded in 1475 in a work by Jehan Adam.
  • 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, did extend the naming scheme and invent the names for the higher powers.
  • Notice that 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, a bimillion (Adam) 1012, and a trimillion was 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:

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 = <math>\,\!10^{10^{100}}<math> Googolplex Kasner and Newman, dictionaries (see above)
10googolplex = <math>\,\!10^{10^{10^{100}}}<math> Googolplexplex or Googolduplex
10googolduplex = <math>\,\!10^{10^{10^{10^{100}}}}<math>Googolplexplexplex or Googoltriplex
10googoltriplex = <math>\,\!10^{10^{10^{10^{10^{100}}}}}<math> Googolplexplexplexplex or Googolquadriplex

Extensions of the standard dictionary numbers

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.

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, 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, respectively, 6630, 102, 312, 7, and 127 hits, respectively.

Names of reciprocals of large numbers do not need to be listed here, because they regularly formed by adding -th, e.g. quattuordecillionth, centillionth, etc. Googolminex has been proposed as a name for 1/googol but is not in any real use.

For additional details, see: billion and long scale.

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 ... ...
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 Novenonagintacentillion Centillion Centillion
10603 Ducentillion Thousand centillion Centilliard

For non-standard conventions that assign regular names to even larger numbers, see Other names of large numbers.

See also

External links

References

pl:Nazwy dużych liczb ru:Системы наименования чисел sl:imena velikih števil

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