 Names of large numbers

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 nonEnglishspeaking areas, including continental Europe and Spanishspeaking 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
Name Short scale
(U.S. and
modern British)Long scale
(continental Europe,
older British)Authorities AHD4^{[1]} COD^{[2]} OED2^{[3]} OEDnew^{[4]} RHD2^{[5]} SOED3^{[6]} W3^{[7]} UM^{[8]} Million 10^{6} 10^{6} ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Milliard 10^{9} ✓ ✓ ✓ ✓ Billion 10^{9} 10^{12} ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Billiard 10^{15} ✓ ✓ ✓ ✓ Trillion 10^{12} 10^{18} ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Quadrillion 10^{15} 10^{24} ✓ ✓ ✓ ✓ ✓ ✓ ✓ Quintillion 10^{18} 10^{30} ✓ ✓ ✓ ✓ ✓ ✓ ✓ Sextillion 10^{21} 10^{36} ✓ ✓ ✓ ✓ ✓ ✓ ✓ Septillion 10^{24} 10^{42} ✓ ✓ ✓ ✓ ✓ ✓ ✓ Octillion 10^{27} 10^{48} ✓ ✓ ✓ ✓ ✓ ✓ ✓ Nonillion 10^{30} 10^{54} ✓ ✓ ✓ ✓ ✓ ✓ ✓ Decillion 10^{33} 10^{60} ✓ ✓ ✓ ✓ ✓ ✓ ✓ Undecillion 10^{36} 10^{66} ✓ ✓ ✓ ✓ Duodecillion 10^{39} 10^{72} ✓ ✓ ✓ ✓ Tredecillion 10^{42} 10^{78} ✓ ✓ ✓ ✓ Quattuordecillion 10^{45} 10^{84} ✓ ✓ ✓ ✓ Quindecillion (Quinquadecillion) 10^{48} 10^{90} ✓ ✓ ✓ ✓ Sexdecillion (Sedecillion) 10^{51} 10^{96} ✓ ✓ ✓ ✓ Septendecillion 10^{54} 10^{102} ✓ ✓ ✓ ✓ Octodecillion 10^{57} 10^{108} ✓ ✓ ✓ ✓ Novemdecillion (Novendecillion) 10^{60} 10^{114} ✓ ✓ ✓ ✓ Vigintillion 10^{63} 10^{120} ✓ ✓ ✓ ✓ ✓ ✓ ✓ Centillion 10^{303} 10^{600} ✓ ✓ ✓ ✓ Name Value Authorities AHD4 COD OED2 OEDnew RHD2 SOED3 W3 UM Googol 10^{100} ✓ ✓ ✓ ✓ ✓ ✓ ✓ Googolplex 10^{Googol} ✓ ✓ ✓ ✓ ✓ ✓ ✓ Apart from million, the words in this list ending with illion are all derived by adding prefixes (bi, tri, etc.) to the stem illion.^{[9]} Centillion^{[10]} 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".
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 (10^{21} or 1 milliard bilpengő as printed) printed in Hungary in 1946. In 2009, Zimbabwe printed a 100 trillion (10^{14}) Zimbabwean dollar note, which at the time of printing was only worth about US$30.^{[11]}
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 wellestablished 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 Xray emission of the radio galaxy is 1.3×10^{45} ergs." When a number such as 10^{45} needs to be referred to in words, it is simply read out: "ten to the fortyfifth". 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 (10^{8}) "first numbers" and called 10^{8} 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, 10^{8}·10^{8}=10^{16}. 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 10^{8}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" (10^{63}).
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"
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 cinq^{e} quyllion Le six^{e} sixlion Le sept.^{e} septyllion Le huyt^{e} ottyllion Le neuf^{e} 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 wellknown, 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 10^{12}, and Adam's trimillion (Chuquet's tryllion) denoted 10^{18}.
An aidememoire
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 shortscale trillion in scientific notation: one trillion = 10^{12}.
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 longscale billion in scientific notation: one billion = 10^{12}. 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 10^{45}. 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
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,^{[12]} in the following passage:
The name "googol" was invented by a child (Dr. Kasner's nineyearold 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 10^{100} Googol Kasner and Newman, dictionaries (see above) 10^{googol} = Googolplex Kasner and Newman, dictionaries (see above) Conway and Guy ^{[13]} have suggested that Nplex be used as a name for 10^{N}. This gives rise to the name googolplexplex for 10^{googolplex}. This number (ten to the power of a googolplex) is also known as a googolduplex. ^{[14]} Conway and Guy ^{[13]} have proposed that Nminex 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
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; 1,000,000^{2} = 1 billion; 1,000,000^{3} = 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 × 1000^{2} = 10^{9}; a trillion is 1000 × 1000^{3} = 10^{12}; 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 10^{9}. 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 ChuquetPeletier system), but the Oxford English Dictionary states that the term derives from postClassical Latin term milliartum, which became milliare and then milliart and finally our modern term.
With regard to names ending in illiard for numbers 10^{6n+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 10^{3n+3} (short scale) or 10^{6n} (long scale) and concatenating Latin roots for its units, tens, and hundreds place, together with the suffix illion. In this way, numbers up to 10^{3·999+3} = 10^{3000} (short scale) or 10^{6·999} = 10^{5994} (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 10^{6,000,258}, 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.^{[13]}
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. and modern British
(short scale)Traditional British
(long scale)Traditional European (Peletier)
(long scale)SI
SymbolSI
Prefix1 10^{6} Million Million Million M mega 2 10^{9} Billion Thousand million Milliard G giga 3 10^{12} Trillion Billion Billion T tera 4 10^{15} Quadrillion Thousand billion Billiard P peta 5 10^{18} Quintillion Trillion Trillion E exa 6 10^{21} Sextillion Thousand trillion Trilliard Z zetta 7 10^{24} Septillion Quadrillion Quadrillion Y yotta 8 10^{27} Octillion Thousand quadrillion Quadrilliard 9 10^{30} Nonillion Quintillion Quintillion 10 10^{33} Decillion Thousand quintillion Quintilliard 11 10^{36} Undecillion Sextillion Sextillion 12 10^{39} Duodecillion Thousand sextillion Sextilliard 13 10^{42} Tredecillion Septillion Septillion 14 10^{45} Quattuordecillion Thousand septillion Septilliard 15 10^{48} Quindecillion Octillion Octillion 16 10^{51} Sexdecillion Thousand octillion Octilliard 17 10^{54} Septendecillion Nonillion Nonillion 18 10^{57} Octodecillion Thousand nonillion Nonilliard 19 10^{60} Novemdecillion Decillion Decillion 20 10^{63} Vigintillion Thousand decillion Decilliard 21 10^{66} Unvigintillion Undecillion Undecillion 22 10^{69} Duovigintillion Thousand undecillion Undecilliard 23 10^{72} Tresvigintillion Duodecillion Duodecillion 24 10^{75} Quattuorvigintillion Thousand duodecillion Duodecilliard 25 10^{78} Quinquavigintillion Tredecillion Tredecillion 26 10^{81} Sesvigintillion Thousand tredecillion Tredecilliard 27 10^{84} Septemvigintillion Quattuordecillion Quattuordecillion 28 10^{87} Octovigintillion Thousand quattuordecillion Quattuordecilliard 29 10^{90} Novemvigintillion Quindecillion Quindecillion 30 10^{93} Trigintillion Thousand quindecillion Quindecilliard 31 10^{96} Untrigintillion Sexdecillion Sexdecillion 32 10^{99} Duotrigintillion Thousand sexdecillion Sexdecilliard 33 10^{102} Trestrigintillion Septendecillion Septendecillion 34 10^{105} Quattuortrigintillion Thousand septendecillion Septendecilliard 35 10^{108} Quinquatrigintillion Octodecillion Octodecillion 36 10^{111} Sestrigintillion Thousand octodecillion Octodecilliard 37 10^{114} Septentrigintillion Novemdecillion Novemdecillion 38 10^{117} Octotrigintillion Thousand novemdecillion Novemdecilliard 39 10^{120} Novemtrigintillion Vigintillion Vigintillion 40 10^{123} Quadragintillion Thousand vigintillion Vigintilliard 50 10^{153} Quinquagintillion Thousand quinquavigintillion Quinquavigintilliard 60 10^{183} Sexagintillion Thousand trigintillion Trigintilliard 70 10^{213} Septuagintillion Thousand quinquatrigintillion Quinquatrigintilliard 80 10^{243} Octogintillion Thousand quadragintillion Quadragintilliard 90 10^{273} Nonagintillion Thousand quinquaquadragintillion Quinquaquadragintilliard 100 10^{303} Centillion Thousand quinquagintillion Quinquagintilliard 101 10^{306} Uncentillion Unquinquagintillion Unquinquagintillion 102 10^{309} Duocentillion Thousand unquinquagintillion Unquinquagintilliard 103 10^{312} Trescentillion Duoquinquagintillion Duoquinquagintillion 110 10^{333} Decicentillion Thousand quinquaquinquagintillion Quinquaquinquagintilliard 111 10^{336} Undecicentillion Sexaquinquagintillion Sexaquinquagintillion 120 10^{363} Viginticentillion Thousand sexagintillion Sexagintilliard 121 10^{366} Unviginticentillion Unsexagintillion Unsexagintillion 130 10^{393} Trigintacentillion Thousand quinquasexagintillion Quinquasexagintilliard 140 10^{423} Quadragintacentillion Thousand septuagintillion Septuagintilliard 150 10^{453} Quinquagintacentillion Thousand quinquaseptuagintillion Quinquaseptuagintilliard 160 10^{483} Sexagintacentillion Thousand octogintillion Octogintilliard 170 10^{513} Septuagintacentillion Thousand quinquaoctogintillion Quinquaoctogintilliard 180 10^{543} Octogintacentillion Thousand nonagintillion Nonagintilliard 190 10^{573} Nonagintacentillion Thousand quinquanonagintillion Quinquanonagintilliard 200 10^{603} Ducentillion Thousand centillion Centilliard 300 10^{903} Trecentillion Thousand quinquagintacentillion Quinquagintacentilliard 400 10^{1203} Quadringentillion Thousand ducentillion Ducentilliard 500 10^{1503} Quingentillion Thousand quinquagintaducentillion Quinquagintaducentilliard 600 10^{1803} Sescentillion Thousand trecentillion Trecentilliard 700 10^{2103} Septingentillion Thousand quinquagintatrecentillion Quinquagintatrecentilliard 800 10^{2403} Octingentillion Thousand quadringentillion Quadringentilliard 900 10^{2703} Nongentillion Thousand quinquagintaquadringentillion Quinquagintaquadringentilliard 1000 10^{3003} Millinillion Thousand quingentillion Quingentilliard Value U.S. and modern British
(short scale)Traditional British
(long scale)Traditional European (Peletier)
(long scale)10^{100} Googol (Ten duotrigintillion) Googol (Ten thousand sexdecillion) Googol (Ten sexdecilliard) Googolplex Googolplex Googolplex Proposals for new naming system
See also: yllionIn 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 Latinbased short scale and long scale systems should be replaced by an unambiguous Greekbased system for naming large numbers that would be based on powers of one thousand.^{[15]}
Value Name 10^{3} Thousand 10^{6} Million 10^{9} Gillion 10^{12} Tetrillion 10^{15} Pentillion 10^{18} Hexillion 10^{21} Heptillion 10^{24} Oktillion 10^{27} Ennillion 10^{30} Dekillion Value Name 10^{33} Hendekillion 10^{36} Dodekillion 10^{39} Trisdekillion 10^{42} Tetradekillion 10^{45} Pentadekillion 10^{48} Hexadekillion 10^{51} Heptadekillion 10^{54} Oktadekillion 10^{57} Enneadekillion 10^{60} Icosillion Value Name 10^{63} Icosihenillion 10^{66} Icosidillion 10^{69} Icositrillion 10^{72} Icositetrillion 10^{75} Icosipentillion 10^{78} Icosihexillion 10^{81} Icosiheptillion 10^{84} Icosioktillion 10^{87} Icosiennillion 10^{90} Triacontillion Other large numbers used in mathematics and physics
 Avogadro's number
 Graham's number
 Skewes' number
 Steinhaus–Moser notation
See also
 Names of small numbers
 Nicolas Chuquet
 Number names
 Number prefix
 Orders of magnitude
 Orders of magnitude (numbers)
References
 ^ American Heritage Dictionary, 4th edition, ISBN 0395825172. [1]
 ^ Cambridge Dictionaries Online, Cambridge, UK: Cambridge University Press.
 ^ Oxford English Dictionary, 2nd edition, Oxford, UK: Oxford University Press. ISBN 0198611862 (and addendums since publication in 1989.)
 ^ Oxford English Dictionary, New Edition, Oxford, UK: Oxford University Press. [2] (subscription required), checked April 2007
 ^ The Random House Dictionary, 2nd Unabridged Edition, 1987, Random House.
 ^ Shorter Oxford English Dictionary, 3rd edition, 1993, Oxford: Clarendon Press.
 ^ Webster's Third New International Dictionary, Unabridged, 1993, MerriamWebster.
 ^ "How Many? A Dictionary of Units of Measures". Russ Rowlett and the University of North Carolina at Chapel Hill. http://www.unc.edu/~rowlett/units/index.html. Retrieved 20090815.
 ^ p. 316, The History of the English Language, Oliver Farrar Emerson, New York, London: Macmillan and Co., 1894.
 ^ Entry for centillion in the American Heritage Dictionary
 ^ "Zimbabwe rolls out Z$100tr note". BBC News. January 16, 2009. http://news.bbc.co.uk/2/hi/africa/7832601.stm. Retrieved 20090116.
 ^ Kasner, Edward and James Newman, Mathematics and the Imagination, 1940, Simon and Schuster, New York.
 ^ ^{a} ^{b} ^{c} The Book of Numbers, J. H. Conway and R. K. Guy, New York: SpringerVerlag, 1996, pp. 15–16. ISBN 038797993X.
 ^ Bowers, Jonathan. "Infinity Scrapers". Polytope, 2010.
 ^ Rowlett, Russ (20011101). "Names for large numbers". University of North Carolina. http://www.unc.edu/~rowlett/units/large.html. Retrieved 20080131.
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 10^{n} and by word length
 Big numbers Educational site, which can name any numbers put into it (up to centillion)
 The English name of a number An online tool that prints names of numbers of any size
Subarticles Names of large numbers · History of large numbers
Examples in numerical order million · googol · googolplex · Skewes' number · Moser's number · Graham's number · Transfinite numbers · Infinity
Expression methods Notations Operators Hyper operators (Tetration) · Ackermann function
Related articles Categories: Large numbers
 Numeration
 Numerals
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