Names of large numbers

From Wikipedia, the free encyclopedia
  (Redirected from Googolplexplex)
Jump to: navigation, search

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[edit]

Name Short scale
(U.S., Canada and
modern British)
Long scale
(continental Europe,
older British)
Authorities
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 1048 1090        
Sexdecillion (Sedecillion) 1051 1096        
Septendecillion 1054 10102        
Octodecillion 1057 10108        
Novemdecillion (Novendecillion) 1060 10114        
Vigintillion 1063 10120  
Centillion 10303 10600      

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.).

Name Value Authorities
AHD4 CED COD OED2 OEDnew RHD2 SOED3 W3 UM
Googol 10100  
Googolplex 10Googol

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[edit]

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"[edit]

Chuquet.gif

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 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[edit]

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[edit]

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)
10googol = \,\!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 10N. This gives rise to the name googolplexplex for 10googolplex. 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[edit]

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 French mathematician 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 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 K. 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 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.[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

Binary prefixes[edit]

The International System of Quantities (ISQ) defines a series of prefixes denoting integer powers of 1024 between 10241 and 10248.[16]

Power Value ISQ
Symbol
ISQ
Prefix
1 10241 Ki Kibi-
2 10242 Mi Mebi-
3 10243 Gi Gibi-
4 10244 Ti Tebi-
5 10245 Pi Pebi-
6 10246 Ei Exbi-
7 10247 Zi Zebi-
8 10248 Yi Yobi-

Proposals for new naming system[edit]

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.[17]

Value Name
103 Thousand
106 Million
109 Gillion
1012 Tetrillion
1015 Pentillion
1018 Hexillion
1021 Heptillion
1024 Oktillion
1027 Ennillion
1030 Dekillion
Value Name
1033 Hendekillion
1036 Dodekillion
1039 Trisdekillion
1042 Tetradekillion
1045 Pentadekillion
1048 Hexadekillion
1051 Heptadekillion
1054 Oktadekillion
1057 Enneadekillion
1060 Icosillion
Value Name
1063 Icosihenillion
1066 Icosidillion
1069 Icositrillion
1072 Icositetrillion
1075 Icosipentillion
1078 Icosihexillion
1081 Icosiheptillion
1084 Icosioktillion
1087 Icosiennillion
1090 Triacontillion

Other large numbers used in mathematics and physics[edit]

See also[edit]

References[edit]

  1. ^ American Heritage Dictionary, 4th edition, ISBN 0-395-82517-2. [1]
  2. ^ Collins English Dictionary, 11th Edition, HarperCollins Publishers.
  3. ^ Cambridge Dictionaries Online, Cambridge, UK: Cambridge University Press.
  4. ^ Oxford English Dictionary, 2nd edition, Oxford, UK: Oxford University Press. ISBN 0-19-861186-2 (and addendums since publication in 1989.)
  5. ^ Oxford English Dictionary, New Edition, Oxford, UK: Oxford University Press. [2] (subscription required), checked April 2007
  6. ^ The Random House Dictionary, 2nd Unabridged Edition, 1987, Random House.
  7. ^ Shorter Oxford English Dictionary, 3rd edition, 1993, Oxford: Clarendon Press.
  8. ^ Webster's Third New International Dictionary, Unabridged, 1993, Merriam-Webster.
  9. ^ "How Many? A Dictionary of Units of Measures". Russ Rowlett and the University of North Carolina at Chapel Hill. Retrieved 2009-08-15. 
  10. ^ p. 316, The History of the English Language, Oliver Farrar Emerson, New York, London: Macmillan and Co., 1894.
  11. ^ Entry for centillion in the American Heritage Dictionary
  12. ^ "Zimbabwe rolls out Z$100tr note". BBC News. January 16, 2009. Retrieved 2009-01-16. 
  13. ^ Kasner, Edward and James Newman, Mathematics and the Imagination, 1940, Simon and Schuster, New York.
  14. ^ a b c d The Book of Numbers, J. H. Conway and R. K. Guy, New York: Springer-Verlag, 1996, pp. 15–16. ISBN 0-387-97993-X.
  15. ^ Bowers, Jonathan. "Infinity Scrapers". Polytope, 2010.
  16. ^ "IEC 80000-13:2008". International Organization for Standardization. Retrieved 21 July 2013. 
  17. ^ Rowlett, Russ (2001-11-01). "Names for large numbers". University of North Carolina. Retrieved 2008-01-31. 

External links[edit]