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Depending on context (e.g. language, culture, region), some have that allow for describing large quantities in a textual form; not mathematics. For very large values, the text is generally shorter than a decimal numeric representation although longer than scientific notation.
Two naming scales for large numbers have been used in English and other European languages since the early modern era: the long and short scales. Most English variants use the short scale today, but the long scale remains dominant in many non-English-speaking areas, including continental Europe and Spanish language-speaking countries in Latin America. These naming procedures are based on taking the number n occurring in 103 n+3 (short scale) or 106 n (long scale) and concatenating Latin roots for its units, tens, and hundreds place, together with the suffix -illion.
Names of numbers above a trillion are rarely used in practice; such large numbers have practical usage primarily in the scientific domain, where powers of ten are expressed as 10 with a numeric superscript. However, these somewhat rare names are considered acceptable for approximate statements. For example, the statement "There are approximately 7.1 octillion atoms in an adult human body" is understood to be in short scale of the table below (and is only accurate if referring to short scale rather than long scale).
The Indian numbering system uses the named numbers common between the long and short scales up to ten thousand. For larger values, it includes named numbers at each multiple of 100; including lakh (105) and crore (107).
English also has words, such as zillion, that are used informally to mean large but unspecified amounts.
Usage:
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. Centillion 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 (see below). 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".
Names of larger numbers, however, have a tenuous, artificial existence, rarely found outside definitions, lists, and discussions of how large numbers are named. Even well-established names like sextillion are rarely used, since in the context of science, including astronomy, 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 ." When a number such as 1045 needs to be referred to in words, it is simply read out as "ten to the forty-fifth" or "ten to the forty-five". This is easier to say 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, 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 way people try to conceptualize and understand them.
One of the earliest 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. 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" (1063).
Since then, many others have engaged in the pursuit of conceptualizing and naming numbers that have no existence outside 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.
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.
John Horton Conway and Richard K. Guy have suggested that N-plex be used as a name for 10N. This gives rise to the name googolplexplex for 10googolplex = 10. 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, which is written as 10. None of these names are in wide use.
The names googol and googolplex inspired the name of the Dot-com company Google and its corporate headquarters, the Googleplex, respectively.
Traditional British usage assigned new names for each power of one million (the long scale): ; ; ; and so on. It was adapted from French usage, and is similar to the system that was documented or invented by Nicolas Chuquet.
Traditional American usage (which was also adapted from French usage but at a later date), Canadian, and modern British usage assign 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 originally popularized the short scale worldwide, reverted to the long scale.
The term milliard is unambiguous and always means 109. It is seldom seen in American usage and rarely in British usage, but frequently in continental European usage. The term is sometimes attributed to French mathematician Jacques Peletier du Mans (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.
Concerning names ending in -illiard for numbers 106 n+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 "milliardo" in Italian, "Milliarde" in German, "miljard" in Dutch, "milyar" in Turkish, and "миллиард," milliard (transliterated) in Russian, are standard usage when discussing financial topics.
The naming procedure for large numbers is based on taking the number n occurring in 103 n+3 (short scale) or 106 n (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 9 or smaller. For larger n (between 10 and 999), prefixes can be constructed based on a system described by Conway and Guy. Today, sexdecillion and novemdecillion are standard dictionary numbers and, using the same reasoning as Conway and Guy did for the numbers up to nonillion, could probably be used to form acceptable prefixes. The Conway–Guy system for forming prefixes:
The Conway–Guy system disagrees with some standard dictionary names, like "quindecillion," "sexdecillion," and "novemdecillion." Oliver Miakinen argued that since "quindecillion" is a widely accepted term, and the Latin for 15 is actually quindecim and not quinquadecim, the prefix "quinqua-" should be replaced with "quin-". This new prefix is more commonly used nowadays.
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 co-devised with Allan Wechsler the following set of consistent conventions that permit, in principle, the extension of this system indefinitely to provide English short-scale names for any integer whatsoever. The name of a number 103 n+3, where n is greater than or equal to 1000, is formed by concatenating the names of the numbers of the form 103 m+3, where m represents each group of comma-separated digits of n, with each but the last "-illion" trimmed to "-illi-", or, in the case of m = 0, either "-nilli-" or "-nillion". For example, 103,000,012, the 1,000,003rd "-illion" number, equals one "millinillitrillion"; 1033,002,010,111, the 11,000,670,036th "-illion" number, equals one "undecillinilliseptuagintasescentillisestrigintillion"; and 1029,629,629,633, the 9,876,543,210th "-illion" number, equals one "nonilliseseptuagintaoctingentillitresquadragintaquingentillideciducentillion".
The following table shows number names generated by the system described by Conway and Guy for the short and long scales.
Standard dictionary numbers
Million 106 ✓ Milliard 109 ✓ Billion 1012 ✓ Trillion 1018 ✓ Quadrillion 1024 ✓ Quintillion 1030 ✓ Sextillion 1036 ✓ Septillion 1042 ✓ Octillion 1048 ✓ Nonillion 1054 ✓ Decillion 1060 ✓ Undecillion 1066 ✓ Duodecillion 1072 ✓ Tredecillion 1078 ✓ Quattuordecillion 1084 ✓ Quindecillion 1090 ✓ Sexdecillion 1096 ✓ Septendecillion 10102 ✓ Octodecillion 10108 ✓ Novemdecillion 10114 ✓ Vigintillion 10120 ✓ Centillion 10600 ✓
Googol 10100 ✓ 10googol (1010100) ✓
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, particularly in finance and economics. 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 worth about US$30. In global economics, the name of a significantly larger number was used in 2024, when the Russian news outlet RBK Group stated that the sum of legal claims against Google in Russia totalled 2 undecillion (2) Russian Ruble, or US $20 decillion (US $2); a value worth more than all financial assets in the world combined. A Kremlin spokesperson, Dmitry Peskov, stated that this value was symbolic.
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 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).
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:
Kasner and Newman, dictionaries (see above) Kasner and Newman, dictionaries (see above)
Extensions of the standard dictionary numbers
This section illustrates several systems for naming large numbers, and shows how they can be extended past vigintillion.
! Units
! Tens
! Hundreds 1 1 106 Million Million Million M Mega- 2 1 109 Billion Thousand million Milliard G Giga- 3 2 1012 Trillion Billion Billion T Tera- 4 2 1015 Quadrillion Thousand billion Billiard P Peta- 5 3 1018 Quintillion Trillion Trillion E Exa- 6 3 1021 Sextillion Thousand trillion Trilliard Z Zetta- 7 4 1024 Septillion Quadrillion Quadrillion Y Yotta- 8 4 1027 Octillion Thousand quadrillion Quadrilliard R Ronna- 9 5 1030 Nonillion Quintillion Quintillion Q Quetta- 10 5 1033 Decillion Thousand quintillion Quintilliard 11 6 1036 Undecillion Sextillion Sextillion 12 6 1039 Duodecillion Thousand sextillion Sextilliard 13 7 1042 Tredecillion Septillion Septillion 14 7 1045 Quattuordecillion Thousand septillion Septilliard 15 8 1048 Quindecillion Octillion Octillion 16 8 1051 Sedecillion Thousand octillion Octilliard 17 9 1054 Septendecillion Nonillion Nonillion 18 9 1057 Octodecillion Thousand nonillion Nonilliard 19 10 1060 Novendecillion Decillion Decillion 20 10 1063 Vigintillion Thousand decillion Decilliard 21 11 1066 Unvigintillion Undecillion Undecillion 22 11 1069 Duovigintillion Thousand undecillion Undecilliard 23 12 1072 Tresvigintillion Duodecillion Duodecillion 24 12 1075 Quattuorvigintillion Thousand duodecillion Duodecilliard 25 13 1078 Quinvigintillion Tredecillion Tredecillion 26 13 1081 Sesvigintillion Thousand tredecillion Tredecilliard 27 14 1084 Septemvigintillion Quattuordecillion Quattuordecillion 28 14 1087 Octovigintillion Thousand quattuordecillion Quattuordecilliard 29 15 1090 Novemvigintillion Quindecillion Quindecillion 30 15 1093 Trigintillion Thousand quindecillion Quindecilliard 31 16 1096 Untrigintillion Sedecillion Sedecillion 32 16 1099 Duotrigintillion Thousand sedecillion Sedecilliard 33 17 10102 Trestrigintillion Septendecillion Septendecillion 34 17 10105 Quattuortrigintillion Thousand septendecillion Septendecilliard 35 18 10108 Quintrigintillion Octodecillion Octodecillion 36 18 10111 Sestrigintillion Thousand octodecillion Octodecilliard 37 19 10114 Septentrigintillion Novendecillion Novendecillion 38 19 10117 Octotrigintillion Thousand novendecillion Novendecilliard 39 20 10120 Noventrigintillion Vigintillion Vigintillion 40 20 10123 Quadragintillion Thousand vigintillion Vigintilliard 50 25 10153 Quinquagintillion Thousand quinvigintillion Quinvigintilliard 60 30 10183 Sexagintillion Thousand trigintillion Trigintilliard 70 35 10213 Septuagintillion Thousand quintrigintillion Quintrigintilliard 80 40 10243 Octogintillion Thousand quadragintillion Quadragintilliard 90 45 10273 Nonagintillion Thousand quinquadragintillion Quinquadragintilliard 100 50 10303 Centillion Thousand quinquagintillion Quinquagintilliard 101 51 10306 Uncentillion Unquinquagintillion Unquinquagintillion 110 55 10333 Decicentillion Thousand quinquinquagintillion Quinquinquagintilliard 111 56 10336 Undecicentillion Sesquinquagintillion Sesquinquagintillion 120 60 10363 Viginticentillion Thousand sexagintillion Sexagintilliard 121 61 10366 Unviginticentillion Unsexagintillion Unsexagintillion 130 65 10393 Trigintacentillion Thousand quinsexagintillion Quinsexagintilliard 140 70 10423 Quadragintacentillion Thousand septuagintillion Septuagintilliard 150 75 10453 Quinquagintacentillion Thousand quinseptuagintillion Quinseptuagintilliard 160 80 10483 Sexagintacentillion Thousand octogintillion Octogintilliard 170 85 10513 Septuagintacentillion Thousand quinoctogintillion Quinoctogintilliard 180 90 10543 Octogintacentillion Thousand nonagintillion Nonagintilliard 190 95 10573 Nonagintacentillion Thousand quinnonagintillion Quinnonagintilliard 200 100 10603 Ducentillion Thousand centillion Centilliard 300 150 10903 Trecentillion Thousand quinquagintacentillion Quinquagintacentilliard 400 200 101203 Quadringentillion Thousand ducentillion Ducentilliard 500 250 101503 Quingentillion Thousand quinquagintaducentillion Quinquagintaducentilliard 600 300 101803 Sescentillion Thousand trecentillion Trecentilliard 700 350 102103 Septingentillion Thousand quinquagintatrecentillion Quinquagintatrecentilliard 800 400 102403 Octingentillion Thousand quadringentillion Quadringentilliard 900 450 102703 Nongentillion Thousand quinquagintaquadringentillion Quinquagintaquadringentilliard 1000 500 103003 Millinillion Thousand quingentillion Quingentilliard 10100 Googol Ten duotrigintillion Ten thousand sedecillion Ten sedecilliard 10 Googolplex Ten 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Binary prefixes
The International System of Quantities (ISQ) defines a series of prefixes denoting integer powers of 1024 between 10241 and 10248.
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-
Other named large numbers used in mathematics, physics and chemistry
See also
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