Names of large numbers
Encyclopedia
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.
Apart from million, the words in this list ending with -illion are all derived by adding prefixes (bi-, tri-, etc.) 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. 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".
(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.
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 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 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: "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 prefix
es 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.,
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 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.
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:
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 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.
's nephew, Milton Sirotta, and introduced in Kasner and Newman's 1940 book,
Mathematics and the Imagination,
in the following passage:
Conway and Guy
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. 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. None of these names are in wide use, nor are any currently found in dictionaries.
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 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 × 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 a French mathematician named 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 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 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.
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
.
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.
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
Continental Europe
Continental Europe, also referred to as mainland Europe or simply the Continent, is the continent of Europe, explicitly excluding European islands....
and Spanish
Spanish language
Spanish , also known as Castilian , is a Romance language in the Ibero-Romance group that evolved from several languages and dialects in central-northern Iberia around the 9th century and gradually spread with the expansion of the Kingdom of Castile into central and southern Iberia during the...
-speaking countries in Latin America
Latin America
Latin America is a region of the Americas where Romance languages – particularly Spanish and Portuguese, and variably French – are primarily spoken. Latin America has an area of approximately 21,069,500 km² , almost 3.9% of the Earth's surface or 14.1% of its land surface area...
; see Long and short scales
Long and short scales
The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale...
.
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 Long and short scales The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale... (U.S. and modern British) |
Long scale Long and short scales The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale... (continental Europe, older British) |
Authorities | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| AHD4 | COD | OED2 | OEDnew | RHD2 | SOED3 | W3 | UM | |||
| Million Million One million or one thousand thousand, is the natural number following 999,999 and preceding 1,000,001. The word is derived from the early Italian millione , from mille, "thousand", plus the augmentative suffix -one.In scientific notation, it is written as or just 106... |
106 | 106 | ||||||||
| Milliard | 109 | |||||||||
| Billion 1000000000 (number) 1,000,000,000 is the natural number following 999,999,999 and preceding 1,000,000,001.In scientific notation, it is written as 109.... |
109 | 1012 | ||||||||
| Billiard | 1015 | |||||||||
| 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 (Quinquadecillion) | 1048 | 1090 | ||||||||
| Sexdecillion (Sedecillion) | 1051 | 1096 | ||||||||
| Septendecillion | 1054 | 10102 | ||||||||
| Octodecillion | 1057 | 10108 | ||||||||
| Novemdecillion (Novendecillion) | 1060 | 10114 | ||||||||
| Vigintillion | 1063 | 10120 | ||||||||
| Centillion Centillion Centillion is a big number. It refers to different quantities based on locality of usage.- North American system :* In Canadian and U.S. usage, two centillion is 10303+10303.... |
10303 | 10600 | ||||||||
| Name | Value | Authorities | |||||||
|---|---|---|---|---|---|---|---|---|---|
| AHD4 | COD | OED2 | OEDnew | RHD2 | SOED3 | W3 | UM | ||
| Googol Googol A googol is the large number 10100, that is, the digit 1 followed by 100 zeros:The term was coined in 1938 by 9-year-old Milton Sirotta , nephew of American mathematician Edward Kasner... |
10100 | ||||||||
| Googolplex | 10Googol Googol A googol is the large number 10100, that is, the digit 1 followed by 100 zeros:The term was coined in 1938 by 9-year-old Milton Sirotta , nephew of American mathematician Edward Kasner... |
||||||||
Apart from million, the words in this list ending with -illion are all derived by adding prefixes (bi-, tri-, etc.) 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. 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 pengőPengo
Pengo may refer to:*Pengo , a 1982 video arcade game from Sega*Hungarian pengő, the name of an old Hungarian currency*Pengo language, a Dravidian language spoken in south central India...
(1021 or 1 milliard bilpengő as printed) printed in Hungary
Hungary
Hungary , officially the Republic of Hungary , is a landlocked country in Central Europe. It is situated in the Carpathian Basin and is bordered by Slovakia to the north, Ukraine and Romania to the east, Serbia and Croatia to the south, Slovenia to the southwest and Austria to the west. The...
in 1946. In 2009, Zimbabwe
Zimbabwe
Zimbabwe is a landlocked country located in the southern part of the African continent, between the Zambezi and Limpopo rivers. It is bordered by South Africa to the south, Botswana to the southwest, Zambia and a tip of Namibia to the northwest and Mozambique to the east. Zimbabwe has three...
printed a 100 trillion (1014) Zimbabwean dollar note, which at the time of printing was only worth about US$30.
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 large numbers often occur, numbers are usually written using scientific notation
Scientific notation
Scientific notation is a way of writing numbers that are too large or too small to be conveniently written in standard decimal notation. Scientific notation has a number of useful properties and is commonly used in calculators and by scientists, mathematicians, doctors, and engineers.In scientific...
. 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: "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 prefix
SI prefix
The International System of Units specifies a set of unit prefixes known as SI prefixes or metric prefixes. An SI prefix is a name that precedes a basic unit of measure to indicate a decadic multiple or fraction of the unit. Each prefix has a unique symbol that is prepended to the unit symbol...
es 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
Parsec
The parsec is a unit of length used in astronomy. It is about 3.26 light-years, or just under 31 trillion kilometres ....
and light year or the particle physicist's barn
Barn (unit)
A barn is a unit of area. Originally used in nuclear physics for expressing the cross sectional area of nuclei and nuclear reactions, today it is used in all fields of high energy physics to express the cross sections of any scattering process, and is best understood as a measure of the...
.
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
The Sand Reckoner
The Sand Reckoner is a work by Archimedes in which he set out to determine an upper bound for the number of grains of sand that fit into the universe. In order to do this, he had to estimate the size of the universe according to the then-current model, and invent a way to talk about extremely...
, in which Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
gave a system for naming large numbers. To do this, he called the numbers up to a myriad
Myriad
Myriad , "numberlesscountless, infinite", is a classical Greek word for the number 10,000. In modern English, the word refers to an unspecified large quantity.-History and usage:...
myriad
Myriad
Myriad , "numberlesscountless, infinite", is a classical Greek word for the number 10,000. In modern English, the word refers to an unspecified large quantity.-History and usage:...
(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 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"
Jehan Adam
Jehan Adam was a French mathematician who flourished in the 15th century. He was secretary to Nicholle Tilhart, who was notary, secretary and auditor of accounts to King Louis XI of France....
. Subsequently, Nicolas Chuquet
Nicolas Chuquet
Nicolas Chuquet was a French mathematician whose great work, Triparty en la science des nombres , was unpublished in his lifetime...
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
Estienne de La Roche
Estienne de La Roche was a French mathematician.Sometimes known as Estienne de Villefranche, La Roche was born in Lyon, but his family also owned property in Villefranche-sur-Saône, where he lived during his youth. He studied mathematics with Nicolas Chuquet. Having in his possession Chuquet's...
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
Long and short scales
The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short 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
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
Long and short scales
The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale...
.
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 KasnerEdward Kasner
Edward Kasner who studied under Cassius Jackson Keyser, was a prominent American mathematician who was appointed Tutor on Mathematics in the Columbia University Mathematics Department...
's nephew, Milton Sirotta, and introduced in Kasner and Newman's 1940 book,
Mathematics and the Imagination,
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 CarneraPrimo CarneraPrimo Carnera was an Italian boxer, nicknamed the Ambling Alp, who became the world heavyweight champion.-Biography:...
a better mathematician than Dr. EinsteinAlbert EinsteinAlbert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
, 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 Googol A googol is the large number 10100, that is, the digit 1 followed by 100 zeros:The term was coined in 1938 by 9-year-old Milton Sirotta , nephew of American mathematician Edward Kasner... |
Kasner and Newman, dictionaries (see above) |
10googol =  |
Googolplex | Kasner and Newman, dictionaries (see above) |
Conway and Guy
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. 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. 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
Long and short scales
The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale...
): ; ; ; and so on. It was adapted from French usage, and is similar to the system that was documented or invented by Chuquet
Nicolas Chuquet
Nicolas Chuquet was a French mathematician whose great work, Triparty en la science des nombres , was unpublished in his lifetime...
.
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
Long and short scales
The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use 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
United Nations
The United Nations is an international organization whose stated aims are facilitating cooperation in international law, international security, economic development, social progress, human rights, and achievement of world peace...
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 a French mathematician named Jacques Peletier du Mans
Jacques Peletier du Mans
Jacques Pelletier du Mans, also spelled Peletier, in Latin: Peletarius , was a humanist, poet and mathematician of the French Renaissance....
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
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
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
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
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 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.
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
Billion
Billion may refer to: In numbers:*Long and short scales**1,000,000,000 , one thousand million, 109, in the short scale**1,000,000,000,000 , one million million, 1012, in the long scaleIn other usage:...
and long and short scales
Long and short scales
The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale...
.
| Base -illion (short scale Long and short scales The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale... ) |
Value | U.S. and modern British (short scale Long and short scales The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale... ) |
Traditional British (long scale Long and short scales The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale... ) |
Traditional European (Peletier Jacques Peletier du Mans Jacques Pelletier du Mans, also spelled Peletier, in Latin: Peletarius , was a humanist, poet and mathematician of the French Renaissance.... ) (long scale Long and short scales The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short 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 | Quindecillion | Octillion | Octillion | ||
| 16 | 1051 | Sexdecillion | Thousand octillion | Octilliard | ||
| 17 | 1054 | Septendecillion | Nonillion | Nonillion | ||
| 18 | 1057 | Octodecillion | Thousand nonillion | Nonilliard | ||
| 19 | 1060 | Novemdecillion | 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 | Sexdecillion | Sexdecillion | ||
| 32 | 1099 | Duotrigintillion | Thousand sexdecillion | Sexdecilliard | ||
| 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 | Novemdecillion | Novemdecillion | ||
| 38 | 10117 | Octotrigintillion | Thousand novemdecillion | Novemdecilliard | ||
| 39 | 10120 | Novemtrigintillion | 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 | Sexaquinquagintillion | Sexaquinquagintillion | ||
| 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. and modern British (short scale Long and short scales The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale... ) |
Traditional British (long scale Long and short scales The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale... ) |
Traditional European (Peletier Jacques Peletier du Mans Jacques Pelletier du Mans, also spelled Peletier, in Latin: Peletarius , was a humanist, poet and mathematician of the French Renaissance.... ) (long scale Long and short scales The long and short scales are two of several different large-number naming systems used throughout the world for integer powers of ten. Many countries, including most in continental Europe, use the long scale whereas most English-speaking countries use the short scale... ) |
|---|---|---|---|
| 10100 | Googol Googol A googol is the large number 10100, that is, the digit 1 followed by 100 zeros:The term was coined in 1938 by 9-year-old Milton Sirotta , nephew of American mathematician Edward Kasner... (Ten duotrigintillion) |
Googol Googol A googol is the large number 10100, that is, the digit 1 followed by 100 zeros:The term was coined in 1938 by 9-year-old Milton Sirotta , nephew of American mathematician Edward Kasner... (Ten thousand sexdecillion) |
Googol Googol A googol is the large number 10100, that is, the digit 1 followed by 100 zeros:The term was coined in 1938 by 9-year-old Milton Sirotta , nephew of American mathematician Edward Kasner... (Ten sexdecilliard) |
 |
Googolplex | Googolplex | Googolplex |
Proposals for new naming system
In 2001, Russ Rowlett, Director of the Center for Mathematics and Science Education at the University of North Carolina at Chapel HillUniversity of North Carolina at Chapel Hill
The University of North Carolina at Chapel Hill is a public research university located in Chapel Hill, North Carolina, United States...
proposed that, to avoid confusion, the Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
-based short scale and long scale systems should be replaced by an unambiguous Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
-based system for naming large numbers that would be based on powers of one thousand.
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Other large numbers used in mathematics and physics
- Avogadro's numberAvogadro's numberIn chemistry and physics, the Avogadro constant is defined as the ratio of the number of constituent particles N in a sample to the amount of substance n through the relationship NA = N/n. Thus, it is the proportionality factor that relates the molar mass of an entity, i.e...
- Graham's numberGraham's numberGraham's number, named after Ronald Graham, is a large number that is an upper bound on the solution to a certain problem in Ramsey theory.The number gained a degree of popular attention when Martin Gardner described it in the "Mathematical Games" section of Scientific American in November 1977,...
- Skewes' number
- Steinhaus–Moser notationSteinhaus–Moser notationIn mathematics, Steinhaus–Moser notation is a means of expressing certain extremely large numbers. It is an extension of Steinhaus’s polygon notation.- Definitions :...
See also
- Chinese numbers
- Indefinite and fictitious numbers
- Indian numbering systemIndian numbering systemThe South Asian numbering system, used today in the Indian subcontinent , is based on grouping by two decimal places, rather than the three decimal places commonplace in most parts of the world. This system of measurement introduces separators into numbers in places appropriate to the two-digit...
- Knuth's up-arrow notationKnuth's up-arrow notationIn mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976. It is closely related to the Ackermann function and especially to the hyperoperation sequence. The idea is based on the fact that multiplication can be viewed as iterated...
- Law of large numbersLaw of large numbersIn probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times...
- List of numbers
- Names of small numbersNames of small numbersThis article lists and discusses the usage and derivation of names of small numbers.-Table of names:The following table lists English language names of small numbers used in the long and short scales, along with the power of ten, engineering notation, and International System of Units symbols and...
- Nicolas ChuquetNicolas ChuquetNicolas Chuquet was a French mathematician whose great work, Triparty en la science des nombres , was unpublished in his lifetime...
- Number namesNumber namesIn linguistics, number names are specific words in a natural language that represent numbers.In writing, numerals are symbols also representing numbers...
- Number prefix
- Orders of magnitude
- Orders of magnitude (numbers)Orders of magnitude (numbers)This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantity and probabilities. Each number is given a name in the short scale, which is used in English speaking countries, as well as a name in the long scale, which is used in some of the...
External links
- Robert Munafo's Large Numbers
- How high can you count? by Landon Curt NollLandon Curt NollLandon Curt Noll is an American computer scientist, co-discoverer of the 25th Mersenne prime and discoverer of the 26th, which he found while still enrolled in high school and concurrently at Cal State Hayward....
. - Full list of large number names list sorted by 10n 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