# Names of large numbers (wikipedia)

## Names of large numbers (wikipedia)

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-

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-

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