Как пишется число триллиард

Названия больших чисел

Существует десять цифр: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Числа состоят из цифр. Число 52 состоит из двух цифр: 5 и 2. Числа с 1 впереди и последующими нулями имеют названия. Всем известны: 10 — десять, 100 — сто, 1000 — тысяча, 1 000 000 — миллион. Так как большие числа с большим числом нулей записывать неудобно, используют сокращения в виде степеней: запись 10¹¹ означает число с 11-ю нулями, запись 10⁵² означает число с 52-мя нулями и т.д. Приведем названия чисел с десятками и сотнями нулей.

Самого большого числа в мире не существует, так как любое большое число всегда можно увеличить, умножить, возвести в степень, и получится другое большее число. Бесконечность не является числом.

Из известных самых больших чисел, имеющих название (математическое доказательство) можно выделить: число TREE(3), число SCG(13), число Лоудера, число Мозера, число Скьюза, число Райо, число Грэма, инфитеиплеон.

Таблица больших чисел с указанием количества нулей и названиями на русском и английском.

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 10³ⁿ⁺³ (short scale) or 10⁶ⁿ (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.

Indian English does not use millions, but has its own system of large numbers including lakhs and crores.^[1] English also has many words, such as «zillion», used informally to mean large but unspecified amounts; see indefinite and fictitious numbers.

Standard dictionary numbers

x	Name (SS/LS, LS)	SS (103x+3)	LS (106x, 106x+3)	Authorities
AHD4[2]	CED[3]	COD[4]	OED2[5]	OEDweb[6]	RHD2[7]	SOED3[8]	W3[9]	HM[10]
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.^[11] Centillion^[12] 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 (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. 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.^[13]

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 1.3×10⁴⁵ joules.» When a number such as 10⁴⁵ needs to be referred to in words, it is simply read out as «ten to the forty-fifth». 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.^{[citation needed]}

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.^{[citation needed]}

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 neuf^e 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^[14] 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 happen if one 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)

John Horton Conway and Richard K. Guy^[15] have suggested that N-plex be used as a name for 10^N. This gives rise to the name googolplexplex for 10^googolplex = 10^1010100. Conway and Guy^[15] 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^-(10100). 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): 1,000,000 = 1 million; 1,000,000² = 1 billion; 1,000,000³ = 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 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 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.

Concerning 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,» milliard (transliterated) in Russian, are standard usage when discussing financial topics.

For additional details, see billion and long and short scale.

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 10^3·999+3 = 10³⁰⁰⁰ (short scale) or 10^6·999 = 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.^[15] 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	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 10^6,000,258, 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.^[15] The name of a number 10³ⁿ⁺³, where n is greater than or equal to 1000, is formed by concatenating the names of the numbers of the form 10^3m+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».^[15] For example, 10^3,000,012, the 1,000,003rd «-illion» number, equals one «millinillitrillion»; 10^{33,002,010,111}, the 11,000,670,036th «-illion» number, equals one «undecillinilli­septua­ginta­ses­centilli­sestrigint­illion»; and 10^{29,629,629,633}, the 9,876,543,210th «-illion» number, equals one «nonillise­septua­ginta­octingentillitres­quadra­ginta­quingentillideciducent­illion».^[15]

The following table shows number names generated by the system described by Conway and Guy for the short and long scales.^[16]

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	Quintilliard
11	6	1036	Undecillion	Sextillion	Sextillion
12	6	1039	Duodecillion	Thousand sextillion	Sextilliard
13	7	1042	Tredecillion	Septillion	Septillion
14	7	1045	Quattuordecillion	Thousand septillion	Septilliard
15	8	1048	Quindecillion	Octillion	Octillion
16	8	1051	Sedecillion	Thousand octillion	Octilliard
17	9	1054	Septendecillion	Nonillion	Nonillion
18	9	1057	Octodecillion	Thousand nonillion	Nonilliard
19	10	1060	Novendecillion	Decillion	Decillion
20	10	1063	Vigintillion	Thousand decillion	Decilliard
21	11	1066	Unvigintillion	Undecillion	Undecillion
22	11	1069	Duovigintillion	Thousand undecillion	Undecilliard
23	12	1072	Tresvigintillion	Duodecillion	Duodecillion
24	12	1075	Quattuor­vigint­illion	Thousand duodecillion	Duodecilliard
25	13	1078	Quinvigintillion	Tredecillion	Tredecillion
26	13	1081	Sesvigintillion	Thousand tredecillion	Tredecilliard
27	14	1084	Septemvigintillion	Quattuordecillion	Quattuordecillion
28	14	1087	Octovigintillion	Thousand quattuordecillion	Quattuordecilliard
29	15	1090	Novemvigintillion	Quindecillion	Quindecillion
30	15	1093	Trigintillion	Thousand quindecillion	Quindecilliard
31	16	1096	Untrigintillion	Sedecillion	Sedecillion
32	16	1099	Duotrigintillion	Thousand sedecillion	Sedecilliard
33	17	10102	Trestrigintillion	Septendecillion	Septendecillion
34	17	10105	Quattuor­trigint­illion	Thousand septendecillion	Septendecilliard
35	18	10108	Quintrigintillion	Octodecillion	Octodecillion
36	18	10111	Sestrigintillion	Thousand octodecillion	Octodecilliard
37	19	10114	Septentrigintillion	Novendecillion	Novendecillion
38	19	10117	Octotrigintillion	Thousand novendecillion	Novendecilliard
39	20	10120	Noventrigintillion	Vigintillion	Vigintillion
40	20	10123	Quadragintillion	Thousand vigintillion	Vigintilliard
50	25	10153	Quinquagintillion	Thousand quinvigintillion	Quinvigintilliard
60	30	10183	Sexagintillion	Thousand trigintillion	Trigintilliard
70	35	10213	Septuagintillion	Thousand quintrigintillion	Quintrigintilliard
80	40	10243	Octogintillion	Thousand quadragintillion	Quadragintilliard
90	45	10273	Nonagintillion	Thousand quin­quadra­gint­illion	Quin­quadra­gint­illiard
100	50	10303	Centillion	Thousand quinquagintillion	Quinquagintilliard
101	51	10306	Uncentillion	Unquinquagintillion	Unquinquagintillion
110	55	10333	Decicentillion	Thousand quin­quinqua­gint­illion	Quin­quinqua­gint­illiard
111	56	10336	Undecicentillion	Ses­quinqua­gint­illion	Ses­quinqua­gint­illion
120	60	10363	Viginticentillion	Thousand sexagintillion	Sexagintilliard
121	61	10366	Unviginticentillion	Unsexagintillion	Unsexagintillion
130	65	10393	Trigintacentillion	Thousand quinsexagintillion	Quinsexagintilliard
140	70	10423	Quadra­gintacent­illion	Thousand septuagintillion	Septuagintilliard
150	75	10453	Quinqua­gintacent­illion	Thousand quin­septua­gint­illion	Quin­septua­gint­illiard
160	80	10483	Sexagintacentillion	Thousand octogintillion	Octogintilliard
170	85	10513	Septuagintacentillion	Thousand quinoctogintillion	Quinoctogintilliard
180	90	10543	Octogintacentillion	Thousand nonagintillion	Nonagintilliard
190	95	10573	Nonagintacentillion	Thousand quinnonagintillion	Quinnonagintilliard
200	100	10603	Ducentillion	Thousand centillion	Centilliard
300	150	10903	Trecentillion	Thousand quinqua­gintacent­illion	Quinqua­gintacent­illiard
400	200	101203	Quadringentillion	Thousand ducentillion	Ducentilliard
500	250	101503	Quingentillion	Thousand quinqua­gintaducent­illion	Quinqua­gintaducent­illiard
600	300	101803	Sescentillion	Thousand trecentillion	Trecentilliard
700	350	102103	Septingentillion	Thousand quinqua­gintatrecent­illion	Quinqua­gintatrecent­illiard
800	400	102403	Octingentillion	Thousand quadringentillion	Quadringentilliard
900	450	102703	Nongentillion	Thousand quinqua­ginta­quadringent­illion	Quinqua­ginta­quadringent­illiard
1000	500	103003	Millillion (alt. millinillion)[17]	Thousand quingentillion	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
1010100	Googolplex	[1] Ten trillitrestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centilliduotrigintatre­centillion	[2] Ten thousand milli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentillion	[2] Ten milli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexaginta­sescentilliard

^[1] 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 X 10^{(3n + 3)} = 10¹⁰¹⁰⁰)

^[2] 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 X 10⁶ⁿ = 10¹⁰¹⁰⁰).

Binary prefixes

The International System of Quantities (ISQ) defines a series of prefixes denoting integer powers of 1024 between 1024¹ and 1024⁸.^[18]

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 large numbers used in mathematics and physics

See also

References

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 10³ⁿ⁺³ (short scale) or 10⁶ⁿ (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.

Indian English does not use millions, but has its own system of large numbers including lakhs and crores.^[1] English also has many words, such as «zillion», used informally to mean large but unspecified amounts; see indefinite and fictitious numbers.

Standard dictionary numbers

x	Name (SS/LS, LS)	SS (103x+3)	LS (106x, 106x+3)	Authorities
AHD4[2]	CED[3]	COD[4]	OED2[5]	OEDweb[6]	RHD2[7]	SOED3[8]	W3[9]	HM[10]
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.^[11] Centillion^[12] 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 (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. 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.^[13]

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 1.3×10⁴⁵ joules.» When a number such as 10⁴⁵ needs to be referred to in words, it is simply read out as «ten to the forty-fifth». 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.^{[citation needed]}

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.^{[citation needed]}

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 neuf^e 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^[14] 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 happen if one 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)

John Horton Conway and Richard K. Guy^[15] have suggested that N-plex be used as a name for 10^N. This gives rise to the name googolplexplex for 10^googolplex = 10^1010100. Conway and Guy^[15] 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^-(10100). 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): 1,000,000 = 1 million; 1,000,000² = 1 billion; 1,000,000³ = 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 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 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.

Concerning 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,» milliard (transliterated) in Russian, are standard usage when discussing financial topics.

For additional details, see billion and long and short scale.

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 10^3·999+3 = 10³⁰⁰⁰ (short scale) or 10^6·999 = 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.^[15] 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	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 10^6,000,258, 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.^[15] The name of a number 10³ⁿ⁺³, where n is greater than or equal to 1000, is formed by concatenating the names of the numbers of the form 10^3m+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».^[15] For example, 10^3,000,012, the 1,000,003rd «-illion» number, equals one «millinillitrillion»; 10^{33,002,010,111}, the 11,000,670,036th «-illion» number, equals one «undecillinilli­septua­ginta­ses­centilli­sestrigint­illion»; and 10^{29,629,629,633}, the 9,876,543,210th «-illion» number, equals one «nonillise­septua­ginta­octingentillitres­quadra­ginta­quingentillideciducent­illion».^[15]

The following table shows number names generated by the system described by Conway and Guy for the short and long scales.^[16]

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	Quintilliard
11	6	1036	Undecillion	Sextillion	Sextillion
12	6	1039	Duodecillion	Thousand sextillion	Sextilliard
13	7	1042	Tredecillion	Septillion	Septillion
14	7	1045	Quattuordecillion	Thousand septillion	Septilliard
15	8	1048	Quindecillion	Octillion	Octillion
16	8	1051	Sedecillion	Thousand octillion	Octilliard
17	9	1054	Septendecillion	Nonillion	Nonillion
18	9	1057	Octodecillion	Thousand nonillion	Nonilliard
19	10	1060	Novendecillion	Decillion	Decillion
20	10	1063	Vigintillion	Thousand decillion	Decilliard
21	11	1066	Unvigintillion	Undecillion	Undecillion
22	11	1069	Duovigintillion	Thousand undecillion	Undecilliard
23	12	1072	Tresvigintillion	Duodecillion	Duodecillion
24	12	1075	Quattuor­vigint­illion	Thousand duodecillion	Duodecilliard
25	13	1078	Quinvigintillion	Tredecillion	Tredecillion
26	13	1081	Sesvigintillion	Thousand tredecillion	Tredecilliard
27	14	1084	Septemvigintillion	Quattuordecillion	Quattuordecillion
28	14	1087	Octovigintillion	Thousand quattuordecillion	Quattuordecilliard
29	15	1090	Novemvigintillion	Quindecillion	Quindecillion
30	15	1093	Trigintillion	Thousand quindecillion	Quindecilliard
31	16	1096	Untrigintillion	Sedecillion	Sedecillion
32	16	1099	Duotrigintillion	Thousand sedecillion	Sedecilliard
33	17	10102	Trestrigintillion	Septendecillion	Septendecillion
34	17	10105	Quattuor­trigint­illion	Thousand septendecillion	Septendecilliard
35	18	10108	Quintrigintillion	Octodecillion	Octodecillion
36	18	10111	Sestrigintillion	Thousand octodecillion	Octodecilliard
37	19	10114	Septentrigintillion	Novendecillion	Novendecillion
38	19	10117	Octotrigintillion	Thousand novendecillion	Novendecilliard
39	20	10120	Noventrigintillion	Vigintillion	Vigintillion
40	20	10123	Quadragintillion	Thousand vigintillion	Vigintilliard
50	25	10153	Quinquagintillion	Thousand quinvigintillion	Quinvigintilliard
60	30	10183	Sexagintillion	Thousand trigintillion	Trigintilliard
70	35	10213	Septuagintillion	Thousand quintrigintillion	Quintrigintilliard
80	40	10243	Octogintillion	Thousand quadragintillion	Quadragintilliard
90	45	10273	Nonagintillion	Thousand quin­quadra­gint­illion	Quin­quadra­gint­illiard
100	50	10303	Centillion	Thousand quinquagintillion	Quinquagintilliard
101	51	10306	Uncentillion	Unquinquagintillion	Unquinquagintillion
110	55	10333	Decicentillion	Thousand quin­quinqua­gint­illion	Quin­quinqua­gint­illiard
111	56	10336	Undecicentillion	Ses­quinqua­gint­illion	Ses­quinqua­gint­illion
120	60	10363	Viginticentillion	Thousand sexagintillion	Sexagintilliard
121	61	10366	Unviginticentillion	Unsexagintillion	Unsexagintillion
130	65	10393	Trigintacentillion	Thousand quinsexagintillion	Quinsexagintilliard
140	70	10423	Quadra­gintacent­illion	Thousand septuagintillion	Septuagintilliard
150	75	10453	Quinqua­gintacent­illion	Thousand quin­septua­gint­illion	Quin­septua­gint­illiard
160	80	10483	Sexagintacentillion	Thousand octogintillion	Octogintilliard
170	85	10513	Septuagintacentillion	Thousand quinoctogintillion	Quinoctogintilliard
180	90	10543	Octogintacentillion	Thousand nonagintillion	Nonagintilliard
190	95	10573	Nonagintacentillion	Thousand quinnonagintillion	Quinnonagintilliard
200	100	10603	Ducentillion	Thousand centillion	Centilliard
300	150	10903	Trecentillion	Thousand quinqua­gintacent­illion	Quinqua­gintacent­illiard
400	200	101203	Quadringentillion	Thousand ducentillion	Ducentilliard
500	250	101503	Quingentillion	Thousand quinqua­gintaducent­illion	Quinqua­gintaducent­illiard
600	300	101803	Sescentillion	Thousand trecentillion	Trecentilliard
700	350	102103	Septingentillion	Thousand quinqua­gintatrecent­illion	Quinqua­gintatrecent­illiard
800	400	102403	Octingentillion	Thousand quadringentillion	Quadringentilliard
900	450	102703	Nongentillion	Thousand quinqua­ginta­quadringent­illion	Quinqua­ginta­quadringent­illiard
1000	500	103003	Millillion (alt. millinillion)[17]	Thousand quingentillion	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
1010100	Googolplex	[1] Ten trillitrestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centilliduotrigintatre­centillion	[2] Ten thousand milli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentillion	[2] Ten milli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexaginta­sescentilliard

^[1] 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 X 10^{(3n + 3)} = 10¹⁰¹⁰⁰)

^[2] 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 X 10⁶ⁿ = 10¹⁰¹⁰⁰).

Binary prefixes

The International System of Quantities (ISQ) defines a series of prefixes denoting integer powers of 1024 between 1024¹ and 1024⁸.^[18]

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 large numbers used in mathematics and physics

See also

References

Названия больших чисел

• Таблица больших чисел с указанием количества нулей и названиями на русском и английском
• Как называется самое большое простое число
• Принципы построения названий и список больших чисел

Существует десять цифр: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Числа состоят из цифр. Число 52 состоит из двух цифр: 5 и 2. Числа с 1 впереди и последующими нулями имеют названия.

Всем известны:

• 10 — десять,
• 100 — сто,
• 1000 — тысяча,
• 1 000 000 — миллион.

Так как большие числа с большим числом нулей записывать неудобно, используют сокращения в виде степеней: запись 10¹¹ означает число с 11-ю нулями, запись 10⁵² означает число с 52-мя нулями и т.д. Приведем названия чисел с десятками и сотнями нулей.

Таблица больших чисел с указанием количества нулей и названиями на русском и английском

Названия «круглых» чисел, которые можно встретить в школьной программе:

• 1 000 000 — миллион (6 нулей)
• 1 000 000 000 — миллиард или биллион (9 нулей)
• 1 000 000 000 000 — триллион (12 нулей)
• 1 000 000 000 000 000 — квадриллион (15 нулей)
• 1 000 000 000 000 000 000 — квинтиллион (18 нулей)
• 1 000 000 000 000 000 000 000 — секстиллион (21 нуль)
• 1 000 000 000 000 000 000 000 000 — септиллион (24 нуля)
• 1 000 000 000 000 000 000 000 000 000 — октиллион (27 нулей)
• 1 000 000 000 000 000 000 000 000 000 000 — нониллион (30 нулей)
• 1 000 000 000 000 000 000 000 000 000 000 000 — дециллион (33 нуля)

Еще некоторые примеры интересных названий:

• 10¹⁰⁰ — гугол, googol (100 нулей)
• 10¹⁰¹⁰⁰ — гуголплекс, googolplex (десять в степени гугол)
• 10¹⁴⁰ — асанкхейя, asankhyeya или сто квинквадрагинтиллионов
• 10³⁰³ — центиллион, centillion
• 10³⁰⁰³ — миллиллион, millillion
• 10^3000003 — милли-миллиллион, milli-millillion

Самого большого числа в мире не существует, так как любое большое число всегда можно увеличить, умножить, возвести в степень, и получится другое большее число. Бесконечность не является числом.

Из известных самых больших чисел, имеющих название (математическое доказательство) можно выделить: число TREE(3), число SCG(13), число Лоудера, число Мозера, число Скьюза, число Райо, число Грэма, инфитеиплеон.

Таблица больших чисел с указанием количества нулей и названиями на русском и английском.

Число нулей	Краткая запись	Название	Название на английском
3	103	тысяча	thousand
6	106	миллион	million
9	109	миллиард (биллион)	billion
12	1012	триллион	trillion
15	1015	квадриллион	quadrillion
18	1018	квинтиллион	quintillion
21	1021	секстиллион	sextillion
24	1024	септиллион	septillion
27	1027	октиллион	octillion
30	1030	нониллион	nonillion
33	1033	дециллион	decillion
36	1036	ундециллион	undecillion
39	1039	дуодециллион	duodecillion
42	1042	тредециллион	tredecillion
45	1045	кватуордециллион	quattuordecillion
48	1048	квиндециллион	quindecillion
51	1051	сексдециллион	sexdecillion
54	1054	септендециллион	septendecillion
57	1057	октодециллион	octodecillion
60	1060	новемдециллион	novemdecillion
63	1063	вигинтиллион	vigintillion
66	1066	унвигинтиллион	unvigintillion
69	1069	дуовигинтиллион	duovigintillion
72	1072	тревигинтиллион	trevigintillion
75	1075	кватуорвигинтиллион	quattuorvigintillion
78	1078	квинвигинтиллион	quinvigintillion
81	1081	сексвигинтиллион	sexvigintillion
84	1084	септенвигинтиллион	septenvigintillion
87	1087	октовигинтиллион	octovigintillion
90	1090	новемвигинтиллион	novemvigintillion
93	1093	тригинтиллион	trigintillion
96	1096	унтригинтиллион	untrigintillion
99	1099	дуотригинтиллион	duotrigintillion
102	10102	третригинтиллион	trestrigintillion
105	10105	кватортригинтиллион	quattuortrigintillion
108	10108	квинтригинтиллион	quintrigintillion
111	10111	секстригинтиллион	sextrigintillion
114	10114	септентригинтиллион	septentrigintillion
117	10117	октотригинтиллион	octotrigintillion
120	10120	новемтригинтиллион	novemtrigintillion
123	10123	квадрагинтиллион	quadragintillion
126	10126	унквадрагинтиллион	unquadragintillion
129	10129	дуоквадрагинтиллион	duoquadragintillion
132	10132	треквадрагинтиллион	trequadragintillion
135	10135	кваторквадрагинтиллион	quattuorquadragintillion
138	10138	квинквадрагинтиллион	quinquadragintillion
141	10141	сексквадрагинтиллион	sexquadragintillion
144	10144	септенквадрагинтиллион	septenquadragintillion
147	10147	октоквадрагинтиллион	octoquadragintillion
150	10150	новемквадрагинтиллион	novemquadragintillion
153	10153	квинквагинтиллион	quinquagintillion
156	10156	унквинкагинтиллион	unquinquagintillion
159	10159	дуоквинкагинтиллион	duoquinquagintillion
162	10162	треквинкагинтиллион	trequinquagintillion
165	10165	кваторквинкагинтиллион	quattuorquinquagintillion
168	10168	квинквинкагинтиллион	quinquinquagintillion
171	10171	сексквинкагинтиллион	sexquinquagintillion
174	10174	септенквинкагинтиллион	septenquinquagintillion
177	10177	октоквинкагинтиллион	octoquinquagintillion
180	10180	новемквинкагинтиллион	novemquinquagintillion
183	10183	сексагинтиллион	sexagintillion
186	10186	унсексагинтиллион	unsexagintillion
189	10189	дуосексагинтиллион	duosexagintillion
192	10192	тресексагинтиллион	tresexagintillion
195	10195	кваторсексагинтиллион	quattuorsexagintillion
198	10198	квинсексагинтиллион	quinsexagintillion
201	10201	секссексагинтиллион	sexsexagintillion
204	10204	септенсексагинтиллион	septensexagintillion
207	10207	октосексагинтиллион	octosexagintillion
210	10210	новемсексагинтиллион	novemsexagintillion
213	10213	септагинтиллион	septuagintillion
216	10216	унсептагинтиллион	unseptuagintillion
219	10219	дуосептагинтиллион	duoseptuagintillion
222	10222	тресептагинтиллион	treseptuagintillion
225	10225	кваторсептагинтиллион	quattuorseptuagintillion
228	10228	квинсептагинтиллион	quinseptuagintillion
231	10231	секссептагинтиллион	sexseptuagintillion
234	10234	септенсептагинтиллион	septenseptuagintillion
237	10237	октосептагинтиллион	octoseptuagintillion
240	10240	новемсептагинтиллион	novemseptuagintillion
243	10243	октогинтиллион	octogintillion
246	10246	уноктогинтиллион	unoctogintillion
249	10249	дуооктогинтиллион	duooctogintillion
252	10252	треоктогинтиллион	treoctogintillion
255	10255	кватороктогинтиллион	quattuoroctogintillion
258	10258	квиноктогинтиллион	quinoctogintillion
261	10261	сексоктогинтиллион	sexoctogintillion
264	10264	септоктогинтиллион	septoctogintillion
267	10267	октооктогинтиллион	octooctogintillion
270	10270	новемоктогинтиллион	novemoctogintillion
273	10273	нонагинтиллион	nonagintillion
276	10276	уннонагинтиллион	unnonagintillion
279	10279	дуононагинтиллион	duononagintillion
282	10282	тренонагинтиллион	trenonagintillion
285	10285	кваторнонагинтиллион	quattuornonagintillion
288	10288	квиннонагинтиллион	quinnonagintillion
291	10291	секснонагинтиллион	sexnonagintillion
294	10294	септеннонагинтиллион	septennonagintillion
297	10297	октононагинтиллион	octononagintillion
300	10300	новемнонагинтиллион	novemnonagintillion
303	10303	центиллион	centillion

Как называется самое большое простое число

Простое число — то, которое делится только на себя и на единицу. В конце 2018 года американец Патрик Лярош представил научному миру самое большое простое число.

• Длина его — 24 862 048 символов. Для сравнения: в эпохальном произведении Л.Н. Толстого «Война и мир» около 6-7 миллионов символов, если учитывать знаки препинания и пробелы.
• Это число можно записать следующим образом: 2^82589933-1
• А читается оно так: два в степени 82589933 минус один.
• Существует целый онлайн-проект GIMPS, нацеленный как раз на поиск самых больших простых чисел.

Принципы построения названий и список больших чисел

Все названия больших чисел построены довольно простым образом: в начале идет латинское порядковое числительное, а в конце к нему добавляется суффикс —иллион. Исключение составляет название «миллион» которое является названием числа тысяча (mille) и увеличительного суффикса —иллион. В мире существует два основных типа названий больших чисел:

• система 3х+3 (где х — латинское порядковое числительное) — эта система используется в России, Франции, США, Канаде, Италии, Турции, Бразилии, Греции
• система 6х (где х — латинское порядковое числительное) — эта система наиболее распространена в мире (например: Испания, Германия, Венгрия, Португалия, Польша, Чехия, Швеция, Дания, Финляндия). В ней отсутствующие промежуточные 6х+3 заканчиваются суффиксом -иллиард (из нее мы заимствовали миллиард, который еще называется биллион).

Общий список чисел используемых в России:

Число	Название	Латинское числительное	Увеличивающая приставка СИ	Уменьшаяющая приставка СИ	Практическое значение
101	десять		дека-	деци-	Число пальцев на 2 руках
102	сто		гекто-	санти-	Примерно половина числа всех государств на Земле
103	тысяча		кило-	милли-	Примерное число дней в 3 годах
106	миллион	unus (I)	мега-	микро-	В 5 раз больше числа капель в 10-литровом ведере воды
109	миллиард (биллион)	duo (II)	гига-	нано-	Примерная численность населения Индии
1012	триллион	tres (III)	тера-	пико-	1/13 внутреннего валового продукта России в рублях за 2003 год
1015	квадриллион	quattor (IV)	пета-	фемто-	1/30 длины парсека в метрах
1018	квинтиллион	quinque (V)	экса-	атто-	1/18 числа зерен из легендарной награды изобретателю шахмат
1021	секстиллион	sex (VI)	зетта-	цепто-	1/6 массы планеты Земля в тоннах
1024	септиллион	septem (VII)	йотта-	йокто-	Число молекул в 37,2 л воздуха
1027	октиллион	octo (VIII)	неа-	сито-	Половина массы Юпитера в килограммах
1030	нониллион	novem (IX)	деа-	тредо-	1/5 числа всех микроорганизмов на планете
1033	дециллион	decem (X)	уна-	рево-	Половина массы Солнца в граммах

Произношение чисел, идущих далее, часто различается.

Число	Название	Латинское числительное	Практическое значение
1036	андециллион	undecim (XI)
1039	дуодециллион	duodecim (XII)
1042	тредециллион	tredecim (XIII)	1/100 от количества молекул воздуха на Земле
1045	кваттордециллион	quattuordecim (XIV)
1048	квиндециллион	quindecim (XV)
1051	сексдециллион	sedecim (XVI)
1054	септемдециллион	septendecim (XVII)
1057	октодециллион		Столько элементарных частиц на Солнце
1060	новемдециллион
1063	вигинтиллион	viginti (XX)
1066	анвигинтиллион	unus et viginti (XXI)
1069	дуовигинтиллион	duo et viginti (XXII)
1072	тревигинтиллион	tres et viginti (XXIII)
1075	кватторвигинтиллион
1078	квинвигинтиллион
1081	сексвигинтиллион		Столько элементарных частиц во вселенной
1084	септемвигинтиллион
1087	октовигинтиллион
1090	новемвигинтиллион
1093	тригинтиллион	triginta (XXX)
1096	антригинтиллион

• 10¹⁰⁰ — гугол (число придумал 9-летний племянник американского математика Эдварда Каснера)
• 10¹²³ — квадрагинтиллион (quadraginta, XL)
• 10¹⁵³ — квинквагинтиллион (quinquaginta, L)
• 10¹⁸³ — сексагинтиллион (sexaginta, LX)
• 10²¹³ — септуагинтиллион (septuaginta, LXX)
• 10²⁴³ — октогинтиллион (octoginta, LXXX)
• 10²⁷³ — нонагинтиллион (nonaginta, XC)
• 10³⁰³ — центиллион (Centum, C)

Дальнейшие названия могут быть получены либо прямым, либо обратным порядком латинских числительных (как правильно, не известно):

• 10³⁰⁶ — анцентиллион или центуниллион
• 10³⁰⁹ — дуоцентиллион или центдуоллион
• 10³¹² — трецентиллион или центтриллион
• 10³¹⁵ — кватторцентиллион или центквадриллион
• 10⁴⁰² — третригинтацентиллион или центтретригинтиллион

Оказывается, существуют две системы наименования чисел. Но не арабская, египетская, или любых других древних цивилизаций, а — американская и английская.

В американской системе числа называются так: берется латинское числительное + — иллион (суффикс). Таким образом, получаются числа:

• Триллион — 1 000 000 000 000 (12 нулей)
• Квадриллион — 1 000 000 000 000 000 (15 нулей)
• Квинтиллион — 1 и 18 нулей
• Секстиллион — 1 и 21 нуль
• Септиллион — 1 и 24 нуля
• октиллион — 1 и 27 нулей
• Нониллион — 1 и 30 нулей
• Дециллион — 1 и 33 нуля

Формула проста: 3·x+3 (х — латинское числительное)

По идее должны быть еще числа анилион (unus в латинском языке — один) и дуолион (duo — два), но, по-моему, такие названия вообще не используются.

Английская система наименования чисел распространена в большей степени.

Здесь тоже берется латинское числительное и к нему добавляется суффикс -иллион. Однако название следующего числа, которое больше предыдущего в 1 000 раз, образуется с помощью того же латинского числа и суффикса — иллиард. То бишь:

• Триллион — 1 и 21 нуль (в американской системе — секстиллион!)
• Триллиард — 1 и 24 нуля (в американской системе — септиллион)
• Квадриллион — 1 и 27 нулей
• Квадриллиард — 1 и 30 нулей
• Квинтиллион — 1 и 33 нуля
• Квиниллиард — 1 и 36 нулей
• Секстиллион — 1 и 39 нулей
• Секстиллиард — 1 и 42 нуля

Формулы для подсчета количества нулей, таковы:

• Для чисел, оканчивающихся на — иллион — 6·x+3
• Для чисел, оканчивающихся на — иллиард — 6·x+6

В России принята американская система наименования чисел.

Из английской системы мы позаимствовали название числа «миллиард» — 1 000 000 000 = 10⁹

А где же «заветный» биллион? — Да ведь биллион — это и есть миллиард! По-американски. А мы, хоть и пользуемся американской системой, а «миллиард» взяли из английской.

Пользуясь латинскими наименованиями чисел и американской системой назовем числа:

— вигинтиллион — 1 и 63 нуля

— центиллион — 1 и 303 нуля

— миллеиллион — единица и 3003 нуля! О-го-го…

Но и это, оказывается, не все. Есть еще числа внесистемные.

И первое из них, наверное, мириада — сотня сотен = 10⁰⁰⁰

Гугол (именно в честь него названа известная поисковая система) — единица и сто нулей

В одном из буддийских трактатов названо число асанкхейя — единица и сто сорок нулей!

Какие же самые большие числа имеют практическое применение?

Наибольшее известное простое число — 2^82 589 933 − 1. Оно было найдено в рамках масштабного проекта по поиску простых чисел GIMPS  в 2018 году и содержит 24 862 048 десятичных цифр.

10⁸⁰ – примерное число элементарных частиц в известной Вселенной.

10¹⁰⁰ – гугол. В честь этого числа назвали известную поисковую систему – мол, она знает 10¹⁰⁰ разных фактов. Скажем так, это преувеличение. Для чего такое число может понадобиться? Например, наша Вселенная, по разным прогнозам, просуществует от 10¹⁰⁰ до 1,5*10¹⁰⁰ лет.

8,5*10¹⁸⁵ – объем известной Вселенной, вычисленный в кубиках, со стороной, равной длине Планка. Длина Планка — это 1,616199 *10^-35 метра. Это примерно в 10²⁰ раз меньше диаметра протона. Предполагается, что на этом уровне формируется «пена» пространства-времени.

А какое же число все-таки рекордное? Книга Рекодов Гиннесса утверждает, что самое большое число в математике — это число Грэма. Это гигантское число, которое, будучи записано десятичными цифрами, просто не поместилось бы во Вселенной. Поэтому его записывают так: G=g64. Это специальная запись для невообразимо больших чисел. Что она означает?

Число Грэма, названное в честь математика Рональда Грэма, было придумано им для решения теоремы Рамсея в многомерном пространстве. Оно обозначает верхний предел пространственных измерений, за которым решение теоремы Рамсея уже наверняка невозможно. Для справки: число пространственных измерений нашего мира – три, но математики проводят вычисления в абстрактных моделях, где измерений может быть множество.

Число Грэма — это 3 в очень большой степени. Чтобы прикинуть, насколько это огромная степень, можно попробовать представить начало вычислений. Итак, по методу стрелочной нотации Кнута (особый способ представления больших чисел) 3³ можно записать как 3^3, а 3^{3^3} как 3^3^3. Еще более сокращенная форма записи будет выглядеть как 3^². Соответственно 3^3^3^3^3 = 3^⁴. Обозначи 3^⁴ как g₁. Это был первый шаг вычислений.

Так вот, запись G=g64 означает 64 шага. Они разворачиваются вверх, как огромная башня. На первом шаге мы вычисляем g₁. На втором шаге – g₂ = 3(^g₁). На третьем шаге g₃ = 3(^g₂). И так далее, вплоть до g₆₄ = 3(^g₆₃). Это и есть число Грэма. На сегодняшний день это самое большое число, когда-либо использованное в математическом доказательстве.

Уровень8 класс ПредметМатематика СложностьСложная

Самые длинные десятичные дроби

Первое место здесь удерживает число Пи, вычисленное с точностью до 31415926535897 знаков после запятой, или, для большей наглядности, 3,1415926535897*10¹³ знаков. Число π  = 3,141592653589793… — это соотношение длины окружности к ее диаметру. Набрать его на клавиатуре можно так же, как и ∞, только вместо 8734 нужно напечатать 960.

На втором месте, сильно уступая π, находится число е=2,718281828459045 – основание натурального логарифма. Оно вычислено с точностью до 2 миллионов знаков.

Оба достижения являются чисто спортивными. Отсюда и совпадение числа вычисленных знаков после запятой с π.  На практике, даже ведущие научные центры, такие, как NASA, расчеты которых требуют высочайшей точности, не используют более 15 знаков после запятой для этих чисел.

Впрочем, игры с числами — отдельная тема. Есть очень много интересных цифровых последовательностей с глубоким смыслом, заложенным в них. К примеру, числа Фибоначчи.

Уровень8 класс ПредметМатематика СложностьСложная

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• Числа 1, 2, 3, 4, 5, …, использующиеся для счета предметов или для указания порядкового номера того или иного предмета среди однородных предметов, называют натуральными.

• Рациональное число — число, которое можно представить в виде обыкновенной дроби, где числитель является целым числом, а знаменатель — натуральным.

• К целым числам относятся натуральные числа, ноль, а также числа, противоположные натуральным. Натуральные числа — это положительные целые числа.

• Я́года — маленький сочный или мясистый плод, обычно кустарниковых или травянистых растений, который при употреблении в пищу не требуется откусывать или разрезать.

• Система древнерусских мер длины включала в себя следующие основные меры: версту, сажень, аршин, локоть, пядь и вершок.

• Суммой двух векторов a и b называется третий вектор c, проведенный из начала a к концу b, если начало вектора b совпадает с концом вектора a. Разностью двух векторов a и b называется вектор c при условии: c = a − b, если c + b =a.

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 10³ⁿ⁺³ (short scale) or 10⁶ⁿ (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.

Indian English does not use millions, but has its own system of large numbers including lakhs and crores.^[1] English also has many words, such as «zillion», used informally to mean large but unspecified amounts; see indefinite and fictitious numbers.

Standard dictionary numbers

x	Name (SS/LS, LS)	SS (103x+3)	LS (106x, 106x+3)	Authorities
AHD4[2]	CED[3]	COD[4]	OED2[5]	OEDweb[6]	RHD2[7]	SOED3[8]	W3[9]	HM[10]
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.^[11] Centillion^[12] 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 (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. 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.^[13]

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 1.3×10⁴⁵ joules.» When a number such as 10⁴⁵ needs to be referred to in words, it is simply read out as «ten to the forty-fifth». 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.^{[citation needed]}

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.^{[citation needed]}

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 neuf^e 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^[14] 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 happen if one 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)

John Horton Conway and Richard K. Guy^[15] have suggested that N-plex be used as a name for 10^N. This gives rise to the name googolplexplex for 10^googolplex = 10^1010100. Conway and Guy^[15] 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^-(10100). 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): 1,000,000 = 1 million; 1,000,000² = 1 billion; 1,000,000³ = 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 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 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.

Concerning 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,» milliard (transliterated) in Russian, are standard usage when discussing financial topics.

For additional details, see billion and long and short scale.

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 10^3·999+3 = 10³⁰⁰⁰ (short scale) or 10^6·999 = 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.^[15] 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	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 10^6,000,258, 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.^[15] The name of a number 10³ⁿ⁺³, where n is greater than or equal to 1000, is formed by concatenating the names of the numbers of the form 10^3m+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».^[15] For example, 10^3,000,012, the 1,000,003rd «-illion» number, equals one «millinillitrillion»; 10^{33,002,010,111}, the 11,000,670,036th «-illion» number, equals one «undecillinilli­septua­ginta­ses­centilli­sestrigint­illion»; and 10^{29,629,629,633}, the 9,876,543,210th «-illion» number, equals one «nonillise­septua­ginta­octingentillitres­quadra­ginta­quingentillideciducent­illion».^[15]

The following table shows number names generated by the system described by Conway and Guy for the short and long scales.^[16]

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	Quintilliard
11	6	1036	Undecillion	Sextillion	Sextillion
12	6	1039	Duodecillion	Thousand sextillion	Sextilliard
13	7	1042	Tredecillion	Septillion	Septillion
14	7	1045	Quattuordecillion	Thousand septillion	Septilliard
15	8	1048	Quindecillion	Octillion	Octillion
16	8	1051	Sedecillion	Thousand octillion	Octilliard
17	9	1054	Septendecillion	Nonillion	Nonillion
18	9	1057	Octodecillion	Thousand nonillion	Nonilliard
19	10	1060	Novendecillion	Decillion	Decillion
20	10	1063	Vigintillion	Thousand decillion	Decilliard
21	11	1066	Unvigintillion	Undecillion	Undecillion
22	11	1069	Duovigintillion	Thousand undecillion	Undecilliard
23	12	1072	Tresvigintillion	Duodecillion	Duodecillion
24	12	1075	Quattuor­vigint­illion	Thousand duodecillion	Duodecilliard
25	13	1078	Quinvigintillion	Tredecillion	Tredecillion
26	13	1081	Sesvigintillion	Thousand tredecillion	Tredecilliard
27	14	1084	Septemvigintillion	Quattuordecillion	Quattuordecillion
28	14	1087	Octovigintillion	Thousand quattuordecillion	Quattuordecilliard
29	15	1090	Novemvigintillion	Quindecillion	Quindecillion
30	15	1093	Trigintillion	Thousand quindecillion	Quindecilliard
31	16	1096	Untrigintillion	Sedecillion	Sedecillion
32	16	1099	Duotrigintillion	Thousand sedecillion	Sedecilliard
33	17	10102	Trestrigintillion	Septendecillion	Septendecillion
34	17	10105	Quattuor­trigint­illion	Thousand septendecillion	Septendecilliard
35	18	10108	Quintrigintillion	Octodecillion	Octodecillion
36	18	10111	Sestrigintillion	Thousand octodecillion	Octodecilliard
37	19	10114	Septentrigintillion	Novendecillion	Novendecillion
38	19	10117	Octotrigintillion	Thousand novendecillion	Novendecilliard
39	20	10120	Noventrigintillion	Vigintillion	Vigintillion
40	20	10123	Quadragintillion	Thousand vigintillion	Vigintilliard
50	25	10153	Quinquagintillion	Thousand quinvigintillion	Quinvigintilliard
60	30	10183	Sexagintillion	Thousand trigintillion	Trigintilliard
70	35	10213	Septuagintillion	Thousand quintrigintillion	Quintrigintilliard
80	40	10243	Octogintillion	Thousand quadragintillion	Quadragintilliard
90	45	10273	Nonagintillion	Thousand quin­quadra­gint­illion	Quin­quadra­gint­illiard
100	50	10303	Centillion	Thousand quinquagintillion	Quinquagintilliard
101	51	10306	Uncentillion	Unquinquagintillion	Unquinquagintillion
110	55	10333	Decicentillion	Thousand quin­quinqua­gint­illion	Quin­quinqua­gint­illiard
111	56	10336	Undecicentillion	Ses­quinqua­gint­illion	Ses­quinqua­gint­illion
120	60	10363	Viginticentillion	Thousand sexagintillion	Sexagintilliard
121	61	10366	Unviginticentillion	Unsexagintillion	Unsexagintillion
130	65	10393	Trigintacentillion	Thousand quinsexagintillion	Quinsexagintilliard
140	70	10423	Quadra­gintacent­illion	Thousand septuagintillion	Septuagintilliard
150	75	10453	Quinqua­gintacent­illion	Thousand quin­septua­gint­illion	Quin­septua­gint­illiard
160	80	10483	Sexagintacentillion	Thousand octogintillion	Octogintilliard
170	85	10513	Septuagintacentillion	Thousand quinoctogintillion	Quinoctogintilliard
180	90	10543	Octogintacentillion	Thousand nonagintillion	Nonagintilliard
190	95	10573	Nonagintacentillion	Thousand quinnonagintillion	Quinnonagintilliard
200	100	10603	Ducentillion	Thousand centillion	Centilliard
300	150	10903	Trecentillion	Thousand quinqua­gintacent­illion	Quinqua­gintacent­illiard
400	200	101203	Quadringentillion	Thousand ducentillion	Ducentilliard
500	250	101503	Quingentillion	Thousand quinqua­gintaducent­illion	Quinqua­gintaducent­illiard
600	300	101803	Sescentillion	Thousand trecentillion	Trecentilliard
700	350	102103	Septingentillion	Thousand quinqua­gintatrecent­illion	Quinqua­gintatrecent­illiard
800	400	102403	Octingentillion	Thousand quadringentillion	Quadringentilliard
900	450	102703	Nongentillion	Thousand quinqua­ginta­quadringent­illion	Quinqua­ginta­quadringent­illiard
1000	500	103003	Millillion (alt. millinillion)[17]	Thousand quingentillion	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
1010100	Googolplex	[1] Ten trillitrestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centilliduotrigintatre­centillion	[2] Ten thousand milli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentillion	[2] Ten milli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexaginta­sescentilliard

^[1] 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 X 10^{(3n + 3)} = 10¹⁰¹⁰⁰)

^[2] 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 X 10⁶ⁿ = 10¹⁰¹⁰⁰).

Binary prefixes

The International System of Quantities (ISQ) defines a series of prefixes denoting integer powers of 1024 between 1024¹ and 1024⁸.^[18]

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 large numbers used in mathematics and physics

See also

References

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 10³ⁿ⁺³ (short scale) or 10⁶ⁿ (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.

Indian English does not use millions, but has its own system of large numbers including lakhs and crores.^[1] English also has many words, such as «zillion», used informally to mean large but unspecified amounts; see indefinite and fictitious numbers.

Standard dictionary numbers

x	Name (SS/LS, LS)	SS (103x+3)	LS (106x, 106x+3)	Authorities
AHD4[2]	CED[3]	COD[4]	OED2[5]	OEDweb[6]	RHD2[7]	SOED3[8]	W3[9]	HM[10]
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.^[11] Centillion^[12] 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 (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. 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.^[13]

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 1.3×10⁴⁵ joules.» When a number such as 10⁴⁵ needs to be referred to in words, it is simply read out as «ten to the forty-fifth». 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.^{[citation needed]}

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.^{[citation needed]}

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 neuf^e 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^[14] 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 happen if one 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)

John Horton Conway and Richard K. Guy^[15] have suggested that N-plex be used as a name for 10^N. This gives rise to the name googolplexplex for 10^googolplex = 10^1010100. Conway and Guy^[15] 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^-(10100). 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): 1,000,000 = 1 million; 1,000,000² = 1 billion; 1,000,000³ = 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 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 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.

Concerning 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,» milliard (transliterated) in Russian, are standard usage when discussing financial topics.

For additional details, see billion and long and short scale.

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 10^3·999+3 = 10³⁰⁰⁰ (short scale) or 10^6·999 = 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.^[15] 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	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 10^6,000,258, 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.^[15] The name of a number 10³ⁿ⁺³, where n is greater than or equal to 1000, is formed by concatenating the names of the numbers of the form 10^3m+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».^[15] For example, 10^3,000,012, the 1,000,003rd «-illion» number, equals one «millinillitrillion»; 10^{33,002,010,111}, the 11,000,670,036th «-illion» number, equals one «undecillinilli­septua­ginta­ses­centilli­sestrigint­illion»; and 10^{29,629,629,633}, the 9,876,543,210th «-illion» number, equals one «nonillise­septua­ginta­octingentillitres­quadra­ginta­quingentillideciducent­illion».^[15]

The following table shows number names generated by the system described by Conway and Guy for the short and long scales.^[16]

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	Quintilliard
11	6	1036	Undecillion	Sextillion	Sextillion
12	6	1039	Duodecillion	Thousand sextillion	Sextilliard
13	7	1042	Tredecillion	Septillion	Septillion
14	7	1045	Quattuordecillion	Thousand septillion	Septilliard
15	8	1048	Quindecillion	Octillion	Octillion
16	8	1051	Sedecillion	Thousand octillion	Octilliard
17	9	1054	Septendecillion	Nonillion	Nonillion
18	9	1057	Octodecillion	Thousand nonillion	Nonilliard
19	10	1060	Novendecillion	Decillion	Decillion
20	10	1063	Vigintillion	Thousand decillion	Decilliard
21	11	1066	Unvigintillion	Undecillion	Undecillion
22	11	1069	Duovigintillion	Thousand undecillion	Undecilliard
23	12	1072	Tresvigintillion	Duodecillion	Duodecillion
24	12	1075	Quattuor­vigint­illion	Thousand duodecillion	Duodecilliard
25	13	1078	Quinvigintillion	Tredecillion	Tredecillion
26	13	1081	Sesvigintillion	Thousand tredecillion	Tredecilliard
27	14	1084	Septemvigintillion	Quattuordecillion	Quattuordecillion
28	14	1087	Octovigintillion	Thousand quattuordecillion	Quattuordecilliard
29	15	1090	Novemvigintillion	Quindecillion	Quindecillion
30	15	1093	Trigintillion	Thousand quindecillion	Quindecilliard
31	16	1096	Untrigintillion	Sedecillion	Sedecillion
32	16	1099	Duotrigintillion	Thousand sedecillion	Sedecilliard
33	17	10102	Trestrigintillion	Septendecillion	Septendecillion
34	17	10105	Quattuor­trigint­illion	Thousand septendecillion	Septendecilliard
35	18	10108	Quintrigintillion	Octodecillion	Octodecillion
36	18	10111	Sestrigintillion	Thousand octodecillion	Octodecilliard
37	19	10114	Septentrigintillion	Novendecillion	Novendecillion
38	19	10117	Octotrigintillion	Thousand novendecillion	Novendecilliard
39	20	10120	Noventrigintillion	Vigintillion	Vigintillion
40	20	10123	Quadragintillion	Thousand vigintillion	Vigintilliard
50	25	10153	Quinquagintillion	Thousand quinvigintillion	Quinvigintilliard
60	30	10183	Sexagintillion	Thousand trigintillion	Trigintilliard
70	35	10213	Septuagintillion	Thousand quintrigintillion	Quintrigintilliard
80	40	10243	Octogintillion	Thousand quadragintillion	Quadragintilliard
90	45	10273	Nonagintillion	Thousand quin­quadra­gint­illion	Quin­quadra­gint­illiard
100	50	10303	Centillion	Thousand quinquagintillion	Quinquagintilliard
101	51	10306	Uncentillion	Unquinquagintillion	Unquinquagintillion
110	55	10333	Decicentillion	Thousand quin­quinqua­gint­illion	Quin­quinqua­gint­illiard
111	56	10336	Undecicentillion	Ses­quinqua­gint­illion	Ses­quinqua­gint­illion
120	60	10363	Viginticentillion	Thousand sexagintillion	Sexagintilliard
121	61	10366	Unviginticentillion	Unsexagintillion	Unsexagintillion
130	65	10393	Trigintacentillion	Thousand quinsexagintillion	Quinsexagintilliard
140	70	10423	Quadra­gintacent­illion	Thousand septuagintillion	Septuagintilliard
150	75	10453	Quinqua­gintacent­illion	Thousand quin­septua­gint­illion	Quin­septua­gint­illiard
160	80	10483	Sexagintacentillion	Thousand octogintillion	Octogintilliard
170	85	10513	Septuagintacentillion	Thousand quinoctogintillion	Quinoctogintilliard
180	90	10543	Octogintacentillion	Thousand nonagintillion	Nonagintilliard
190	95	10573	Nonagintacentillion	Thousand quinnonagintillion	Quinnonagintilliard
200	100	10603	Ducentillion	Thousand centillion	Centilliard
300	150	10903	Trecentillion	Thousand quinqua­gintacent­illion	Quinqua­gintacent­illiard
400	200	101203	Quadringentillion	Thousand ducentillion	Ducentilliard
500	250	101503	Quingentillion	Thousand quinqua­gintaducent­illion	Quinqua­gintaducent­illiard
600	300	101803	Sescentillion	Thousand trecentillion	Trecentilliard
700	350	102103	Septingentillion	Thousand quinqua­gintatrecent­illion	Quinqua­gintatrecent­illiard
800	400	102403	Octingentillion	Thousand quadringentillion	Quadringentilliard
900	450	102703	Nongentillion	Thousand quinqua­ginta­quadringent­illion	Quinqua­ginta­quadringent­illiard
1000	500	103003	Millillion (alt. millinillion)[17]	Thousand quingentillion	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
1010100	Googolplex	[1] Ten trillitrestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centilliduotrigintatre­centillion	[2] Ten thousand milli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentillion	[2] Ten milli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexaginta­sescentilliard

^[1] 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 X 10^{(3n + 3)} = 10¹⁰¹⁰⁰)

^[2] 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 X 10⁶ⁿ = 10¹⁰¹⁰⁰).

Binary prefixes

The International System of Quantities (ISQ) defines a series of prefixes denoting integer powers of 1024 between 1024¹ and 1024⁸.^[18]

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 large numbers used in mathematics and physics

See also

References