Names of large numbers (wikipedia)

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This article lists and discusses the usage
and derivation of names of large
numbers , together with their possible
extensions.
The following table lists those names of
large numbers which are found in many
English dictionaries and thus have a
special claim to being "real words". The
"Traditional British" values shown are
unused in American English and are
becoming rare in British English, but their
other-language variants are dominant in
many non-English-speaking areas,
including continental Europe and Spanish -
speaking countries in Latin America ; see
Long and short scales.
English also has many words, such as
"zillion", used informally to mean large
but unspecified amounts; see indefinite
and fictitious numbers.
Standard dictionary numbers
Name
Short scale
(U.S., Canada and
modern British)
Long scale
(continental Europe,
older British)
Authorities
AHD4 [1]
CED [2]
COD [3]
OED2 [4]
OEDnew[5]
RHD2 [6]
SOED3 [7]
W3 [8]
UM[9]
Million
106
106
✓
✓
✓
✓
✓
✓
✓
✓
✓
Milliard
109
✓
✓
✓
✓
✓
✓
Billion
109
10 12
✓
✓
✓
✓
✓
✓
✓
✓
✓
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
10 48
10 90
✓
✓
✓
✓
✓
Sexdecillion (Sedecillion)
10 51
10 96
✓
✓
✓
✓
✓
Septendecillion
10 54
10102
✓
✓
✓
✓
✓
Octodecillion
10 57
10108
✓
✓
✓
✓
✓
Novemdecillion (Novendecillion)
10 60
10114
✓
✓
✓
✓
✓
Vigintillion
10 63
10120
✓
✓
✓
✓
✓
✓
✓
✓
Centillion
10303
10600
✓
✓
✓
✓
✓
✓
Apart from million, the words in this list
ending with -illion are all derived by
adding prefixes (bi -, tri-, etc., derived
from Latin) to the stem -illion .[10]
Centillion [11] appears to be the highest
name ending in -"illion" that is included in
these dictionaries. Trigintillion, often
cited as a word in discussions of names of
large numbers, is not included in any of
them, nor are any of the names that can
easily be created by extending the
naming pattern ( unvigintillion ,
duovigintillion , duoquinquagintillion ,
etc.).
Name
Value
Authorities
AHD4
CED
COD
OED2
OEDnew
RHD2
SOED3
W3
UM
Googol
10100
✓
✓
✓
✓
✓
✓
✓
✓
✓
Googolplex
10Googol (10 10100 )
✓
✓
✓
✓
✓
✓
✓
✓
✓
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
hyperinflation . 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 (1014 ) Zimbabwean dollar note,
which at the time of printing was worth
about US$30. [12]
Names of larger numbers, however, have
a tenuous, artificial existence, rarely
found outside definitions, lists, and
discussions of the ways in which large
numbers are named. Even well-
established names like sextillion are
rarely used, since in the contexts of
science, astronomy, and engineering,
where such large numbers often occur,
they are nearly always written using
scientific notation . In this notation,
powers of ten are expressed as 10 with a
numeric superscript, e.g., "The X-ray
emission of the radio galaxy is 1.3×10 45
ergs." When a number such as 1045
needs to be referred to in words, it is
simply read out: "ten to the forty-fifth".
This is just as easy to say, easier to
understand, and less ambiguous than
"quattuordecillion", which means
something different in the long scale and
the short scale.
When a number represents a quantity
rather than a count, SI prefixes can be
used—thus "femtosecond", not "one
quadrillionth of a second"—although
often powers of ten are used instead of
some of the very high and very low
prefixes. In some cases, specialized units
are used, such as the astronomer's parsec
and light year or the particle physicist's
barn.
Nevertheless, large numbers have an
intellectual fascination and are of
mathematical interest, and giving them
names is one of the ways in which people
try to conceptualize and understand
them.
One of the first examples of this is The
Sand Reckoner , in which Archimedes
gave a system for naming large numbers.
To do this, he called the numbers up to a
myriad myriad (108 ) "first numbers" and
called 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
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 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 neufe nonyllion et
ainsi des ault' s se plus oultre on
vouloit preceder
”
“
(Or if you prefer the first mark
can signify million, the second
mark byllion, the third mark
tryllion, the fourth quadrillion,
the fifth quyillion, the sixth
sixlion, the seventh septyllion,
the eighth ottyllion, the ninth
nonyllion and so on with others
as far as you wish to go).
”
Chuquet is sometimes credited with
inventing the names million, billion,
trillion, quadrillion, and so forth. This is
an oversimplification.
Million was certainly not invented by
Adam or Chuquet. Milion is an Old
French word thought to derive from
Italian milione, an intensification of
mille, a thousand. That is, a million is a
big thousand .
From the way in which Adam and
Chuquet use the words, it can be inferred
that they were recording usage rather
than inventing it. One obvious possibility
is that words similar to billion and
trillion were already in use and well-
known, but that Chuquet, an expert in
exponentiation, extended the naming
scheme and invented the names for the
higher powers.
Chuquet's names are only similar to, not
identical to, the modern ones.
Adam and Chuquet used the long scale of
powers of a million; that is, Adam's
bymillion (Chuquet's byllion) denoted
1012 , and Adam's trimillion (Chuquet's
tryllion ) denoted 1018.
An aide-memoire
It can be a problem to find the values,
either in scientific notation or in sheer
digits, for names of large numbers. Every
number name larger than a million listed
in this article 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
quadri llion, 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 = 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 1045.
Doubling the prefix and adding one then
multiplying the result by three would give
the same result.
These mechanisms are illustrated in the
table in the article on long and short
scales.
Note that when writing out large numbers
using this system, one should place a
comma or space after every three digits,
starting from the right and moving left.
The googol family
The names googol and googolplex were
invented by Edward Kasner 's nephew,
Milton Sirotta, and introduced in Kasner
and Newman's 1940 book, Mathematics
and the Imagination ,[13] in the
following passage:
“
The name "googol" was invented
by a child (Dr. Kasner's nine-
year-old nephew) who was asked
to think up a name for a very big
number, namely 1 with one
hundred zeroes after it. He was
very certain that this number
was not infinite, and therefore
equally certain that it had to
have a name. At the same time
that he suggested "googol" he
gave a name for a still larger
number: "Googolplex". A
googolplex is much larger than a
googol, but is still finite, as the
inventor of the name was quick
to point out. It was first
suggested that a googolplex
should be 1, followed by writing
zeros until you got tired. This is a
description of what would
actually happen if one actually
tried to write a googolplex, but
different people get tired at
different times and it would
never do to have Carnera a
better mathematician than Dr.
Einstein, simply because he had
more endurance. The googolplex
is, then, a specific finite number,
equal to 1 with a googol zeros
after it.
”
Value
Name
Authority
10100
Googol
Kasner and Newman, dictionaries (see
above)
10googol = 1010100
Googolplex
Kasner and Newman, dictionaries (see
above)
Conway and Guy[14] have suggested that
N-plex be used as a name for 10N. This
gives rise to the name googolplexplex
for 10 googolplex. This number (ten to the
power of a googolplex) is also known as a
googolduplex and googolplexian. [15]
Conway and Guy[14] have proposed that
N-minex be used as a name for 10 −N ,
giving rise to the name googolminex for
the reciprocal of a googolplex. None of
these names are in wide use, nor are any
currently found in dictionaries.
The names googol and googolplex have
inspired the name of the Internet
company Google and its corporate
headquarters, the Googleplex ,
respectively.
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), Canadian and
modern British usage, assigns new names
for each power of one thousand (the
short scale.) Thus, a billion is 1000 ×
1000 2 = 109 ; a trillion is 1000 × 1000 3 =
1012 ; and so forth. Due to its dominance
in the financial world (and by the US
dollar ), this was adopted for official
United Nations documents.
Traditional French usage has varied; in
1948, France, which had been using the
short scale, reverted to the long scale.
The term milliard is unambiguous and
always means 109 . It is almost never
seen in American usage, rarely in British
usage, and frequently in European usage.
The term is sometimes attributed to
French mathematician Jacques Peletier du
Mans circa 1550 (for this reason, the long
scale is also known as the Chuquet-
Peletier system), but the Oxford English
Dictionary states that the term derives
from post-Classical Latin term
milliartum , which became milliare and
then milliart and finally our modern
term.
With regard to names ending in -illiard for
numbers 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 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 10 3·999+3 = 103000
(short scale) or 106·999 = 105994 (long
scale) may be named. The choice of roots
and the concatenation procedure is that
of the standard dictionary numbers if n is
20 or smaller, and, for larger n (between
21 and 999), is due to John Horton
Conway and Richard K. Guy: [14]
Units
Tens
Hundreds
1
Un
N Deci
NX Centi
2
Duo
MS Viginti
N Ducenti
3
Tre (*)
NS Triginta
NS Trecenti
4
Quattuor
NS Quadraginta
NS Quadringenti
5
Quinqua
NS Quinquaginta
NS Quingenti
6
Se (*)
N Sexaginta
N Sescenti
7
Septe (*)
N Septuaginta
N Septingenti
8
Octo
MX Octoginta
MX Octingenti
9
Nove (*)
Nonaginta
Nongenti
(*) ^ When preceding a component
marked S or X , “tre” changes to “tres”
and “se” to “ses” or “sex”; similarly,
when preceding a component marked
M or N, “septe” and “nove” change to
“septem” and “novem” or “septen” and
“noven”.
Since the system of using Latin prefixes
will become ambiguous for numbers with
exponents of a size which the Romans
rarely counted to, like 106,000,258 ,
Conway and Guy have also proposed a
consistent set of conventions which
permit, in principle, the extension of this
system to provide English names for any
integer whatsoever. [14]
Names of reciprocals of large numbers
do not need to be listed here, because
they are regularly formed by adding -th,
e.g. quattuordecillionth, centillionth,
etc.
For additional details, see billion and long
and short scales.
Base -illion
( short scale)
Value
U.S., Canada and modern British
( short scale)
Traditional British
( long scale)
Traditional European ( Peletier)
( long scale)
SI
Symbol
SI
Prefix
1
106
Million
Million
Million
M
Mega-
2
109
Billion
Thousand million
Milliard
G
Giga-
3
1012
Trillion
Billion
Billion
T
Tera-
4
1015
Quadrillion
Thousand billion
Billiard
P
Peta-
5
1018
Quintillion
Trillion
Trillion
E
Exa-
6
1021
Sextillion
Thousand trillion
Trilliard
Z
Zetta-
7
1024
Septillion
Quadrillion
Quadrillion
Y
Yotta-
8
1027
Octillion
Thousand quadrillion
Quadrilliard
9
1030
Nonillion
Quintillion
Quintillion
10
1033
Decillion
Thousand quintillion
Quintilliard
11
1036
Undecillion
Sextillion
Sextillion
12
1039
Duodecillion
Thousand sextillion
Sextilliard
13
1042
Tredecillion
Septillion
Septillion
14
1045
Quattuordecillion
Thousand septillion
Septilliard
15
1048
Quinquadecillion
Octillion
Octillion
16
1051
Sedecillion
Thousand octillion
Octilliard
17
1054
Septendecillion
Nonillion
Nonillion
18
1057
Octodecillion
Thousand nonillion
Nonilliard
19
1060
Novendecillion
Decillion
Decillion
20
1063
Vigintillion
Thousand decillion
Decilliard
21
1066
Unvigintillion
Undecillion
Undecillion
22
1069
Duovigintillion
Thousand undecillion
Undecilliard
23
1072
Tresvigintillion
Duodecillion
Duodecillion
24
1075
Quattuorvigintillion
Thousand duodecillion
Duodecilliard
25
1078
Quinquavigintillion
Tredecillion
Tredecillion
26
1081
Sesvigintillion
Thousand tredecillion
Tredecilliard
27
1084
Septemvigintillion
Quattuordecillion
Quattuordecillion
28
1087
Octovigintillion
Thousand quattuordecillion
Quattuordecilliard
29
1090
Novemvigintillion
Quindecillion
Quindecillion
30
1093
Trigintillion
Thousand quindecillion
Quindecilliard
31
1096
Untrigintillion
Sedecillion
Sedecillion
32
1099
Duotrigintillion
Thousand sedecillion
Sedecilliard
33
10102
Trestrigintillion
Septendecillion
Septendecillion
34
10105
Quattuortrigintillion
Thousand septendecillion
Septendecilliard
35
10108
Quinquatrigintillion
Octodecillion
Octodecillion
36
10111
Sestrigintillion
Thousand octodecillion
Octodecilliard
37
10114
Septentrigintillion
Novendecillion
Novendecillion
38
10117
Octotrigintillion
Thousand novendecillion
Novendecilliard
39
10120
Noventrigintillion
Vigintillion
Vigintillion
40
10123
Quadragintillion
Thousand vigintillion
Vigintilliard
50
10153
Quinquagintillion
Thousand quinquavigintillion
Quinquavigintilliard
60
10183
Sexagintillion
Thousand trigintillion
Trigintilliard
70
10213
Septuagintillion
Thousand quinquatrigintillion
Quinquatrigintilliard
80
10243
Octogintillion
Thousand quadragintillion
Quadragintilliard
90
10273
Nonagintillion
Thousand quinquaquadragintillion
Quinquaquadragintilliard
100
10303
Centillion
Thousand quinquagintillion
Quinquagintilliard
101
10306
Uncentillion
Unquinquagintillion
Unquinquagintillion
102
10309
Duocentillion
Thousand unquinquagintillion
Unquinquagintilliard
103
10312
Trescentillion
Duoquinquagintillion
Duoquinquagintillion
110
10333
Decicentillion
Thousand quinquaquinquagintillion
Quinquaquinquagintilliard
111
10336
Undecicentillion
Sesquinquagintillion
Sesquinquagintillion
120
10363
Viginticentillion
Thousand sexagintillion
Sexagintilliard
121
10366
Unviginticentillion
Unsexagintillion
Unsexagintillion
130
10393
Trigintacentillion
Thousand quinquasexagintillion
Quinquasexagintilliard
140
10423
Quadragintacentillion
Thousand septuagintillion
Septuagintilliard
150
10453
Quinquagintacentillion
Thousand quinquaseptuagintillion
Quinquaseptuagintilliard
160
10483
Sexagintacentillion
Thousand octogintillion
Octogintilliard
170
10513
Septuagintacentillion
Thousand quinquaoctogintillion
Quinquaoctogintilliard
180
10543
Octogintacentillion
Thousand nonagintillion
Nonagintilliard
190
10573
Nonagintacentillion
Thousand quinquanonagintillion
Quinquanonagintilliard
200
10603
Ducentillion
Thousand centillion
Centilliard
300
10903
Trecentillion
Thousand quinquagintacentillion
Quinquagintacentilliard
400
101203
Quadringentillion
Thousand ducentillion
Ducentilliard
500
101503
Quingentillion
Thousand quinquagintaducentillion
Quinquagintaducentilliard
600
101803
Sescentillion
Thousand trecentillion
Trecentilliard
700
102103
Septingentillion
Thousand quinquagintatrecentillion
Quinquagintatrecentilliard
800
102403
Octingentillion
Thousand quadringentillion
Quadringentilliard
900
102703
Nongentillion
Thousand quinquagintaquadringentillion
Quinquagintaquadringentilliard
1000
103003
Millinillion
Thousand quingentillion
Quingentilliard
Value
U.S., Canada and modern British
( short scale)
Traditional British
( long scale)
Traditional European ( Peletier)
( long scale)
10100
Googol (Ten duotrigintillion)
Googol (Ten thousand sedecillion)
Googol (Ten sedecilliard)
1010100
Googolplex
Googolplex
Googolplex
Binary prefixes
The International System of Quantities
(ISQ) defines a series of prefixes denoting
integer powers of 1024 between 10241
and 10248 . [16]
Power
Value
ISQ
Symbol
ISQ
Prefix
1
10241
Ki
Kibi-
2
10242
Mi
Mebi-
3
10243
Gi
Gibi-
4
10244
Ti
Tebi-
5
10245
Pi
Pebi-
6
10246
Ei
Exbi-
7
10247
Zi
Zebi-
8
10248
Yi
Yobi-