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= Names_of_large_numbers =
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Introduction
======================================================================
Depending on context (e.g. language, culture, region), some large
numbers have names that allow for describing large quantities in a
textual form; not mathematical. 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-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.
Standard dictionary numbers
======================================================================
rowspan="2" | x rowspan="2" valign="top" | Name (SS/LS, LS)
rowspan="2" valign="top" | SS (103x+3) rowspan="2" valign="top" | LS
(106x, 106x+3) colspan="9" | Authorities
AHD4 CED COD OED2 OEDweb RHD2 SOED3 W3 HM
1 Million 106 106 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
Milliard 109 ✓ ✓ ✓ ✓ ✓ ✓ ✓
2 Billion 109 1012 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
3 Trillion 1012 1018 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
4 Quadrillion 1015 1024 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
5 Quintillion 1018 1030 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
6 Sextillion 1021 1036 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
7 Septillion 1024 1042 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
8 Octillion 1027 1048 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
9 Nonillion 1030 1054 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
10 Decillion 1033 1060 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
11 Undecillion 1036 1066 ✓ ✓ ✓ ✓ ✓
12 Duodecillion 1039 1072 ✓ ✓ ✓ ✓ ✓ ✓
13 Tredecillion 1042 1078 ✓ ✓ ✓ ✓ ✓
14 Quattuordecillion 1045 1084 ✓ ✓ ✓ ✓ ✓
15 Quindecillion 1048 1090 ✓ ✓ ✓ ✓ ✓
16 Sexdecillion 1051 1096 ✓ ✓ ✓ ✓ ✓
17 Septendecillion 1054 10102 ✓ ✓ ✓ ✓ ✓
18 Octodecillion 1057 10108 ✓ ✓ ✓ ✓ ✓
19 Novemdecillion 1060 10114 ✓ ✓ ✓ ✓ ✓
20 Vigintillion 1063 10120 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
100 Centillion 10303 10600 ✓ ✓ ✓ ✓ ✓ ✓
Usage:
* Short scale: US, English Canada, modern British, Australia, and
Eastern Europe
* Long scale: French Canada, older British, Western & Central
Europe
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.).
rowspan="2" valign="top" | Name rowspan="2" valign="top" | Value
colspan="9" | Authorities
AHD4 CED COD OED2 OEDweb RHD2 SOED3 W3 HM
Googol 10100 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
Googolplex 10googol (1010100) ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
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".
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 stated that the sum of legal claims against
Google in Russia totalled 2 undecillion (2) rubles, 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.
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, 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 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.
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).
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.
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:
width=200 | Value Name Authority
10100 Googol Kasner and Newman, dictionaries (see above)
10googol = 10 Googolplex Kasner and Newman, dictionaries (see
above)
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{{sup|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 Internet
company Google and its corporate headquarters, the Googleplex,
respectively.
Extensions of the standard dictionary numbers
======================================================================
This section 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): ; ; ; 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 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:
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 Quin 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
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.
Base -illion (short scale) Base -illion (long scale) Value US, Canada
and modern British (short scale) Traditional British (long scale)
Traditional European (Peletier long scale) SI Symbol SI Prefix
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 colspan="3" | Quintilliard
11 6 1036 Undecillion Sextillion colspan="3" | Sextillion
12 6 1039 Duodecillion Thousand sextillion colspan="3" | Sextilliard
13 7 1042 Tredecillion Septillion colspan="3" | Septillion
14 7 1045 Quattuordecillion Thousand septillion colspan="3" |
Septilliard
15 8 1048 Quindecillion Octillion colspan="3" | Octillion
16 8 1051 Sedecillion Thousand octillion colspan="3" | Octilliard
17 9 1054 Septendecillion Nonillion colspan="3" | Nonillion
18 9 1057 Octodecillion Thousand nonillion colspan="3" | Nonilliard
19 10 1060 Novendecillion Decillion colspan="3" | Decillion
20 10 1063 Vigintillion Thousand decillion colspan="3" | Decilliard
21 11 1066 Unvigintillion Undecillion colspan="3" | Undecillion
22 11 1069 Duovigintillion Thousand undecillion colspan="3" |
Undecilliard
23 12 1072 Tresvigintillion Duodecillion colspan="3" | Duodecillion
24 12 1075 Quattuorvigintillion Thousand duodecillion colspan="3" |
Duodecilliard
25 13 1078 Quinvigintillion Tredecillion colspan="3" | Tredecillion
26 13 1081 Sesvigintillion Thousand tredecillion colspan="3" |
Tredecilliard
27 14 1084 Septemvigintillion Quattuordecillion colspan="3" |
Quattuordecillion
28 14 1087 Octovigintillion Thousand quattuordecillion colspan="3" |
Quattuordecilliard
29 15 1090 Novemvigintillion Quindecillion colspan="3" |
Quindecillion
30 15 1093 Trigintillion Thousand quindecillion colspan="3" |
Quindecilliard
31 16 1096 Untrigintillion Sedecillion colspan="3" | Sedecillion
32 16 1099 Duotrigintillion Thousand sedecillion colspan="3" |
Sedecilliard
33 17 10102 Trestrigintillion Septendecillion colspan="3" |
Septendecillion
34 17 10105 Quattuortrigintillion Thousand septendecillion
colspan="3" | Septendecilliard
35 18 10108 Quintrigintillion Octodecillion colspan="3" |
Octodecillion
36 18 10111 Sestrigintillion Thousand octodecillion colspan="3" |
Octodecilliard
37 19 10114 Septentrigintillion Novendecillion colspan="3" |
Novendecillion
38 19 10117 Octotrigintillion Thousand novendecillion colspan="3" |
Novendecilliard
39 20 10120 Noventrigintillion Vigintillion colspan="3" |
Vigintillion
40 20 10123 Quadragintillion Thousand vigintillion colspan="3" |
Vigintilliard
50 25 10153 Quinquagintillion Thousand quinvigintillion colspan="3" |
Quinvigintilliard
60 30 10183 Sexagintillion Thousand trigintillion colspan="3" |
Trigintilliard
70 35 10213 Septuagintillion Thousand quintrigintillion colspan="3" |
Quintrigintilliard
80 40 10243 Octogintillion Thousand quadragintillion colspan="3" |
Quadragintilliard
90 45 10273 Nonagintillion Thousand quinquadragintillion colspan="3"
| Quinquadragintilliard
100 50 10303 Centillion Thousand quinquagintillion colspan="3" |
Quinquagintilliard
101 51 10306 Uncentillion Unquinquagintillion colspan="3" |
Unquinquagintillion
110 55 10333 Decicentillion Thousand quinquinquagintillion
colspan="3" | Quinquinquagintilliard
111 56 10336 Undecicentillion Sesquinquagintillion colspan="3" |
Sesquinquagintillion
120 60 10363 Viginticentillion Thousand sexagintillion colspan="3" |
Sexagintilliard
121 61 10366 Unviginticentillion Unsexagintillion colspan="3" |
Unsexagintillion
130 65 10393 Trigintacentillion Thousand quinsexagintillion
colspan="3" | Quinsexagintilliard
140 70 10423 Quadragintacentillion Thousand septuagintillion
colspan="3" | Septuagintilliard
150 75 10453 Quinquagintacentillion Thousand quinseptuagintillion
colspan="3" | Quinseptuagintilliard
160 80 10483 Sexagintacentillion Thousand octogintillion colspan="3"
| Octogintilliard
170 85 10513 Septuagintacentillion Thousand quinoctogintillion
colspan="3" | Quinoctogintilliard
180 90 10543 Octogintacentillion Thousand nonagintillion colspan="3"
| Nonagintilliard
190 95 10573 Nonagintacentillion Thousand quinnonagintillion
colspan="3" | Quinnonagintilliard
200 100 10603 Ducentillion Thousand centillion colspan="3" |
Centilliard
300 150 10903 Trecentillion Thousand quinquagintacentillion
colspan="3" | Quinquagintacentilliard
400 200 101203 Quadringentillion Thousand ducentillion colspan="3" |
Ducentilliard
500 250 101503 Quingentillion Thousand quinquagintaducentillion
colspan="3" | Quinquagintaducentilliard
600 300 101803 Sescentillion Thousand trecentillion colspan="3" |
Trecentilliard
700 350 102103 Septingentillion Thousand quinquagintatrecentillion
colspan="3" | Quinquagintatrecentilliard
800 400 102403 Octingentillion Thousand quadringentillion colspan="3"
| Quadringentilliard
900 450 102703 Nongentillion Thousand quinquagintaquadringentillion
colspan="3" | Quinquagintaquadringentilliard
1000 500 103003 Millinillion Thousand quingentillion colspan="3" |
Quingentilliard
Value Name Equivalent
US, Canadian and modern British (short scale) Traditional British
(long scale) Traditional European (Peletier long scale)
10100 Googol Ten duotrigintillion Ten thousand sedecillion Ten
sedecilliard
10 Googolplex Ten
trillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentilliduotrigintatrecentillion
Ten thousand
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Ten
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{{notelist|refs=
{{efn|name=googleplex-shortscale-calculation|text=Googolplex's short
scale name is derived from it equal to ten of the
3,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,332nd
"-illion"s (This is the value of n when 10 × 10((3n + 3)) = 10)}}
{{efn|name=googleplex-longscale-calculation|text=Googolplex's long
scale name (both traditional British and traditional European) is
derived from it being equal to ten thousand of the
1,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666th
"-illion"s (This is the value of n when 10,000 × 10(6n) = 10).}}}}
Binary prefixes
======================================================================
The International System of Quantities (ISQ) defines a series of
prefixes denoting integer powers of 1024 between 10241 and 10248.
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-
Other named large numbers used in mathematics, physics and chemistry
======================================================================
* Avogadro number
* Graham's number
* Skewes's number
* Steinhaus-Moser notation
* TREE(3)
* Rayo's number
* SSCG(3)
References
======================================================================
{{reflist|30em|refs=
}}
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=========
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Original Article:
http://en.wikipedia.org/wiki/Names_of_large_numbers
