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Names of large numbers

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Introduction

Some large numbers have real referents in human experience. Their names are real words, encountered in many contexts. For example, today, Google News shows 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:" "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."

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 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.

Since then, many others have engaged in the pursuit of conceptualizing and naming numbers that really have no existence outside of the imagination. Such names, 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 or 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.

The "standard dictionary numbers"

These names of numbers are found in many English dictionaries and as such have a special claim to being "real words," even though those above quadrillion are very rarely used.

Name American value British value Authorities
million 106 106 √ AHD4, √ OED2, √ RHD2-1987, √ ShorterOED3-1993, √ W3-1993
billion 109 1012 √ AHD4, √ OED2, √ RHD2-1987, √ ShorterOED3-1993, √ W3-1993
trillion 1012 1018 √ AHD4, √ OED2, √ RHD2-1987, √ ShorterOED3-1993, √ W3-1993
quadrillion 1015 1024 √ AHD4, √ OED2, √ RHD2-1987, √ ShorterOED3-1993, √ W3-1993
quintillion 1018 1030 √ AHD4, √ OED2, √ RHD2-1987, √ ShorterOED3-1993, √ W3-1993
sextillion 1021 1036 √ AHD4, √ OED2, √ RHD2-1987, √ ShorterOED3-1993, √ W3-1993
septillion 1024 1042 √ AHD4, √ OED2, √ RHD2-1987, √ ShorterOED3-1993, √ W3-1993
octillion 1027 1048 √ AHD4, √ OED2, √ OEDnew, √ RHD2-1987, √ ShorterOED3-1993, √ W3-1993
nonillion 1030 1054 √ AHD4, √ OED2, √ RHD2-1987, √ ShorterOED3-1993, √ W3-1993
decillion 1033 1060 √ AHD4, √ OED2, √ RHD2-1987, √ ShorterOED3-1993, √ W3-1993
undecillion 1036 1066 × AHD4, √ RHD2-1987, × ShorterOED3-1993, √ W3-1993
duodecillion 1039 1072 × AHD4, √ RHD2-1987, × ShorterOED3-1993, √ W3-1993
tredecillion 1042 1078 × AHD4, √ RHD2-1987, × ShorterOED3-1993, √ W3-1993
quattuordecillion 1045 1084 × AHD4, √ RHD2-1987, × ShorterOED3-1993, √ W3-1993
quindecillion 1048 1090 × AHD4, √ RHD2-1987, × ShorterOED3-1993, √ W3-1993
sexdecillion 1051 1096 × AHD4, √ RHD2-1987, × ShorterOED3-1993, √ W3-1993
septendecillion 1054 10102 × AHD4, √ RHD2-1987, × ShorterOED3-1993, √ W3-1993
octodecillion 1057 10108 × AHD4, √ RHD2-1987, × ShorterOED3-1993, √ W3-1993
novemdecillion 1060 10114 × AHD4, √ RHD2-1987, × ShorterOED3-1993, √ W3-1993
vigintillion 1063 10120 √ AHD4, √ OED2, √ RHD2-1987, √ ShorterOED3-1993, √ W3-1993
googol 10100 10100 √ AHD4, √ OED2, √ RHD2-1987, √ ShorterOED3-1993, √ W3-1993
centillion 10303 10600 √ AHD4, √ OED2
googolplex 10googol 10googol √ AHD4, √ OED2, √ RHD2-1987, √ ShorterOED3-1993, √ W3-1993

Note the oddity that two of the dictionaries consulted list "decillion" and "vigintillion," but none 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:
AHD4: American Heritage Dictionary, 4th edition, at http://www.bartleby.com/61
ShorterOED3-1993: Shorter Oxford English Dictionary, 3rd edition, 1993, Oxford: Clarendon Press
RHD2-1987: The Random House Dictionary, 2nd Unabridged Edition, 1987, Random House
W3-1993: Webster's Third New International Dictionary, Unabridged, 1993, Merriam-Webster
OED2: Oxford English Dictionary, 2nd edition, Oxford: Oxford University Press ; (and addendums since publication in 1989)
OEDnew: Oxford English Dictionary, New Edition, Oxford: Oxford University Press ; at http://www.oed.com (subscription required)
Dcom: Dictionary.com

√ means the word is included in the dictionary
× means the word is not included in the dictionary

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).

Because of this passage, 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.

Both Adam and Chuquet applied the names to powers of a million; that is, a bimillion (Adam) or byllion (Chuquet) was 1012, and a trimillion (Adam) or tryllion (Chuquet) was 1018.

The Googol family

The googol and googolplex were introduced in Kasner and Newman's 1940 book, Mathematics and the Imagination, thus:

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 as 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)
1010100 Googolplex Kasner and Newman, dictionaries (see above)
101010100 Googolplexplex or Googolduplex
10101010100 Googolplexplexplex or Googoltriplex
1010101010100 Googolplexplexplexplex

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 used new names for each power of one million. 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) uses new names for each power of one thousand, with a "billion" being 109, a "trillion" being 1012, and so forth.

As early as 1929 Fowler's Modern English Usage recommended the use of the American nomenclature, and in 1974 the British Government adopted it for official British reports and statistics.

Traditional French usage has varied; in 1948, France, which had been using the American-style terminology, reverted to the Chuquet-style nomenclature.

The term milliard is unambiguous and always means 109. It is almost never seen in American usage, sometimes in British usage, and frequently in European usage. The term is sometimes attributed to a French mathematician named Pelletier circa 1550, but the American Heritage Dictionary states a derivation from Old French milliart.

For additional details, see: billion

Value USA Traditional British Traditional European (Pelletier)
103 Thousand Thousand Thousand
106 Million Million Million
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
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
... ... ... ...
10213 Septuagintillion Thousand quintrigintillion Quintrigintilliard
... ... ... ...
10240 Novemseptuagintillion Quadragintillion Quadragintillion
10243 Octogintillion Thousand quadragintillion Quadragintilliard
... ... ... ...
10273 Nonagintillion Thousand quinquadragintillion Quinquadragintilliard
... ... ... ...
10300 Novemnonagintillion Quinquagintillion Quinquagintillion
10303 Centillion Thousand quinquagintillion Quinquagintilliard
... ... ... ...
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   Thousand centillion Centilliard
... ... ... ...

Chuquet-ized Numbering

This table contains a list of numbers that extend to very large sums. Note that there is no standard way of naming after centilliard or centillion.

This is a list of obscenely large numbers
Value Extended American
(Chuquet)
Extended European
(Pelletier)
Extended Myriadic
(Knuth)
Extended Myriadic
(Knuth-Pelletier)
0 Zero Zero Zero Zero
101 One One One One
103 Thousand Thousand Thousand Thousand
104 Ten thousand Ten thousand Myriad Myriad
106 Million Million Hundred myriad Hundred myriad
108 Hundred million Hundred million Myllion Myllion
109 Billion Milliard Ten myllion Ten myllion
1012 Trillion Billion Myriad myllion Myriad myllion
1015 Quadrillion Billiard Thousand myriad myllion Thousand myriad myllion
1016 Ten quadrillion Ten billiard Byllion Mylliard
1024 Septillion Quadrillion Myllion byllion Myllion mylliard
1032 Hundred nonillion Hundred quintillion Tryllion Byllion
1033 Decillion Quintilliard Ten tryllion Ten byllion
1060 Novemdecillion Decillion Myriad myllion byllion tryllion Myriad myllion mylliard byllion
1063 Vigintillion Decilliard Thousand myriad myllion byllion tryllion Thousand myriad myllion mylliard byllion
1064 Ten vigintillion Ten decilliard Quadryllion Bylliard
10303 Centillion Quinquagintilliard Thousand myriad myllion tryllion sextyllion Thousand myriad myllion byllion trylliard
10600 Novenonacentillion? Centillion Myllion byllion quadryllion septyllion Myllion mylliard trylliard quadryllion
10603 Duecentillion Centilliard Thousand myllion byllion quadryllion septyllion Thousand myllion mylliard trylliard quadryllion
103003 Millillion Pentacentilliard? ? ?
106000 Novenonaennecentumillillion? Millillion ? ?
106003 Duemillillion? Millilliard ? ?
1030,003 Myrillion Pentamillilliard? ? ?
1060,000 Novenonaennecentupentamillimyrillion? Myrillion ? ?
1060,003 Duemyrillion? Myrilliard ? ?
10300,003 ? ? ? ?
10600,000 ? ? ? ?
10600,003 ? ? ? ?
103,000,003 Micrillion ? ? ?
103,000,000,003 Nanillion ? ? ?
103*1012+3 Picillion ? ? ?
103*1015+3 Femtillion ? ? ?
103*1018+3 Attillion ? ? ?
103*1021+3 Zeptillion ? ? ?
103*1024+3 Yoctillion ? ? ?
103*1027+3 Xonillion ? ? ?
103*1030+3 Dekillion
Vecillion
Contillion
? ? ?
103*1033+3 Mecillion ? ? ?
103*1036+3 Duecillion ? ? ?
103*1039+3 Trecillion ? ? ?
103*1042+3 Tetrecillion ? ? ?
103*1045+3 Pentecillion ? ? ?
103*1048+3 Hexecillion ? ? ?
103*1051+3 Heptecillion ? ? ?
103*1054+3 Octecillion ? ? ?
103*1057+3 Ennecillion ? ? ?
103*1060+3 Icocillion
Ducontillion
? ? ?
103*1090+3 Triacontillion ? ? ?
103*10120+3 Tetracontillion ? ? ?
103*10150+3 Pentacontillion ? ? ?
103*10180+3 Hexacontillion ? ? ?
103*10210+3 Heptacontillion ? ? ?
103*10240+3 Octacontillion ? ? ?
103*10270+3 Ennacontillion ? ? ?
103*10300+3 Hectillion
Icocontillion
? ? ?
103*103000+3 Killillion
Onillion
? ? ?
103*103,000,000+3 Megillion ? ? ?
103*103,000,000,000+3 Gigillion ? ? ?
103*103*1012+3 Terillion ? ? ?
103*103*1015+3 Petillion ? ? ?
103*103*1018+3 Exillion ? ? ?
103*103*1021+3 Zettillion ? ? ?
103*103*1024+3 Yottillion ? ? ?
103*103*1027+3 Xennillion ? ? ?
103*103*1030+3 Vekillion
Teenillion
? ? ?
103*103*1033+3 Mekillion ? ? ?
103*103*1036+3 Duekillion ? ? ?
103*103*1039+3 Trekillion ? ? ?
103*103*1042+3 Tetrekillion ? ? ?
103*103*1045+3 Pentekillion ? ? ?
103*103*1048+3 Hexekillion ? ? ?
103*103*1051+3 Heptekillion ? ? ?
103*103*1054+3 Octekillion ? ? ?
103*103*1057+3 Ennekillion ? ? ?
103*103*1060+3 Icokillion
Twentillion
? ? ?
103*103*1090+3 Thirtillion ? ? ?
103*103*10120+3 Fortillion ? ? ?
103*103*10150+3 Fiftillion ? ? ?
103*103*10180+3 Sixtillion ? ? ?
103*103*10210+3 Seventillion ? ? ?
103*103*10240+3 Eightillion ? ? ?
103*103*10270+3 Nintillion ? ? ?
103*103*10300+3 Hundrillion ? ? ?
103*103*103000+3 Thousillion ? ? ?
103*103*1030,000+3 Myriaillion
Manillion
? ? ?
103*103*10300,000+3 Lakhillion ? ? ?
103*103*103,000,000+3 ? ? ? ?
103*103*1030,00,000+3 Crorillion ? ? ?
103*103*10300,000,000+3 Awkillion ? ? ?
106*103*10300,000,000 ? Awkillion ? ?
?106*103*10300,000,000+3? ? ?Awkilliard? ? ?
103*103*103,000,000,000+3 ? ? ? ?
103*103*103*1012+3 ? ? ? ?
103*103*103*1015+3 ? ? ? ?
103*103*103*1018+3 ? ? ? ?
103*103*103*1021+3 ? ? ? ?
103*103*103*1024+3 ? ? ? ?
103*103*103*1027+3 ? ? ? ?
103*103*103*1030+3 ? ? ? ?
103*103*103*1033+3 ? ? ? ?
... ... ... ... ...

[1] [2] [3] [4] [5] Commercial Product using Yoctillion

See Also

References

  • Kasner, Edward and James Newman, Mathematics and the Imagination, 1940, Simon and Schuster, New York.
  • Weisstein, Math World, Wolfram Research, http://mathworld.wolfram.com