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Names of large numbers

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Depending on context (e.g. language, culture, region), somelarge numbers havenames that allow for describing large quantities in a textual form; notmathematical. For very large values, the text is generally shorter than adecimal numeric representation although longer thanscientific notation.

Two naming scales forlarge numbers have been used in English and other European languages since the early modern era: thelong and short scales. Most English variants use the short scale today, but the long scale remains dominant in many non-English-speaking areas, includingcontinental Europe andSpanish-speaking countries inLatin America. These naming procedures are based on taking the numbern 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 as10 with a numeric superscript. However, these somewhat rare names are considered acceptable for approximate statements. For example, the statement "There are approximately 7.1 octillion atoms in an adult human body" is understood to be in short scale of the table below (and is only accurate if referring to short scale rather than long scale).

TheIndian numbering system uses the named numbers common between the long and short scales up to ten thousand. For larger values, it includes named numbers at each multiple of 100; includinglakh (105) andcrore (107).[1]

English also has words, such aszillion, that are used informally to meanlarge but unspecified amounts.

Standard dictionary numbers

xName
(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]
1Million106106
Milliard109
2Billion1091012
3Trillion10121018
4Quadrillion10151024
5Quintillion10181030
6Sextillion10211036
7Septillion10241042
8Octillion10271048
9Nonillion10301054
10Decillion10331060
11Undecillion10361066
12Duodecillion10391072
13Tredecillion10421078
14Quattuordecillion10451084
15Quindecillion10481090
16Sexdecillion10511096
17Septendecillion105410102
18Octodecillion105710108
19Novemdecillion106010114
20Vigintillion106310120
100Centillion1030310600

Usage:

Apart frommillion, 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.).

NameValueAuthorities
AHD4[2]CED[3]COD[4]OED2[5]OEDweb[6]RHD2[7]SOED3[8]W3[9]HM[10]
Googol10100
Googolplex10googol (1010100)

All of the dictionaries includedgoogol andgoogolplex, 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.). TheOxford English Dictionary comments thatgoogol andgoogolplex are "not in formal mathematical use".

Usage of names of large numbers

Some names of large numbers, such asmillion,billion, andtrillion, have real referents in human experience, and are encountered in many contexts, particularly in finance and economics. At times, the names of large numbers have been forced into common usage as a result ofhyperinflation. The highest numerical value banknote ever printed was a note for 1 sextillionpengő (1021 or 1 milliard bilpengő as printed) printed inHungary 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] In global economics, the name of a significantly larger number was used in 2024, when the Russian news outletRBK stated that the sum of legal claims againstGoogle inRussia totalled 2 undecillion (2×1036)rubles, or US $20 decillion (US $2×1034); a value worth more than all financial assets in the world combined.[14] AKremlin spokesperson,Dmitry Peskov, stated that this value was symbolic.[15]

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 likesextillion are rarely used, since in the context of science, including astronomy, where such large numbers often occur, they are nearly always written usingscientific notation. In this notation, powers of ten are expressed as10 with a numeric superscript, e.g. "The X-ray emission of the radio galaxy is1.3×1045 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" or "ten to the forty-five". This is easier to say and less ambiguous than "quattuordecillion", which means something different in the long scale and the short scale.

When a number represents a quantity rather than a count,SI prefixes can be used—thus "femtosecond", not "one quadrillionth of a second"—although often powers of ten are used instead of some of the very high and very low prefixes. In some cases, specialized units are used, such as the astronomer'sparsec andlight year or the particle physicist'sbarn.

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 isThe Sand Reckoner, in whichArchimedes gave a system for naming large numbers. To do this, he called the numbers up to amyriad 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.(108)(108)=108108,{\displaystyle (10^{8})^{(10^{8})}=10^{8\cdot 10^{8}},} and embedded this construction within another copy of itself to produce names for numbers up to((108)(108))(108)=1081016.{\displaystyle ((10^{8})^{(10^{8})})^{(10^{8})}=10^{8\cdot 10^{16}}.} 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 wordgoogol, 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 wordsbymillion andtrimillion were first recorded in 1475 in a manuscript ofJehan Adam. Subsequently,Nicolas Chuquet wrote a bookTriparty en la science des nombres which was not published during Chuquet's lifetime. However, most of it was copied byEstienne de La Roche for a portion of his 1520 book,L'arismetique. Chuquet's book contains a passage in which he shows a large number marked off into groups of six digits, with the comment:

Ou qui veult le premier point peult signiffier million Le second point byllion Le tiers point tryllion Le quart quadrillion Le cinqe quyllion Le sixe sixlion Le sept.e septyllion Le huyte ottyllion Le neufe nonyllion et ainsi des ault's se plus oultre on vouloit preceder

(Or if you prefer the first mark can signify million, the second mark byllion, the third mark tryllion, the fourth quadrillion, the fifth quyillion, the sixth sixlion, the seventh septyllion, the eighth ottyllion, the ninth nonyllion and so on with others as far as you wish to go).

Adam and Chuquet used thelong scale of powers of a million; that is, Adam'sbymillion (Chuquet'sbyllion) denoted 1012, and Adam'strimillion (Chuquet'stryllion) denoted 1018.

The googol family

The namesgoogol andgoogolplex were invented byEdward Kasner's nephew Milton Sirotta and introduced in Kasner and Newman's 1940 bookMathematics and the Imagination[16] 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 haveCarnera a better mathematician thanDr. Einstein, simply because he had more endurance. The googolplex is, then, a specific finite number, equal to 1 with a googol zeros after it.

ValueNameAuthority
10100GoogolKasner and Newman, dictionaries (see above)
10googol = 1010100GoogolplexKasner and Newman, dictionaries (see above)

John Horton Conway andRichard K. Guy[17] have suggested thatN-plex be used as a name for 10N. This gives rise to the namegoogolplexplex for 10googolplex = 101010100. Conway and Guy[17] have proposed thatN-minex be used as a name for 10−N, giving rise to the namegoogolminex for thereciprocal of a googolplex, which is written as 10-(10100). None of these names are in wide use.

The namesgoogol andgoogolplex inspired the name of theInternet companyGoogle and itscorporate headquarters, theGoogleplex, respectively.[citation needed]

Extensions of the standard dictionary numbers

Further information:billion andLong and short scales

This section illustrates several systems for naming large numbers, and shows how they can be extended pastvigintillion.

Traditional British usage assigned new names for each power of one million (thelong 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 byChuquet.

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 (theshort scale). Thus, abillion is 1000 × 10002 = 109; atrillion is 1000 × 10003 = 1012; and so forth. Due to its dominance in the financial world (and by theUS dollar), this was adopted for officialUnited Nations documents.

Traditional French usage has varied; in 1948, France, which had originally popularized the short scale worldwide, reverted to the long scale.

The termmilliard 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 mathematicianJacques Peletier du Mansc. 1550 (for this reason, the long scale is also known as theChuquet-Peletier system), but theOxford English Dictionary states that the term derives from post-ClassicalLatin termmilliartum, which becamemilliare and thenmilliart 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 "milliardo" in Italian, "Milliarde" in German, "miljard" in Dutch, "milyar" in Turkish, and "миллиард," milliard (transliterated) in Russian, are standard usage when discussing financial topics.

The naming procedure for large numbers is based on taking the numbern occurring in 103n+3 (short scale) or 106n (long scale) and concatenating Latin roots for its units, tens, and hundreds place, together with the suffix-illion. In this way, numbers up to 103·999+3 = 103000 (short scale) or 106·999 = 105994 (long scale) may be named. The choice of roots and the concatenation procedure is that of the standard dictionary numbers ifn is 9 or smaller. For largern (between 10 and 999), prefixes can be constructed based on a system described by Conway and Guy.[17] 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:[17]: 15 

UnitsTensHundreds
1UnN DeciNX Centi
2DuoMS VigintiN Ducenti
3Tre[a]NS TrigintaNS Trecenti
4QuattuorNS QuadragintaNS Quadringenti
5QuinquaNS QuinquagintaNS Quingenti
6Se[a]N SexagintaN Sescenti
7Septe[a]N SeptuagintaN Septingenti
8OctoMX OctogintaMX Octingenti
9Nove[a]NonagintaNongenti
  1. ^abcdWhen preceding a component markedS orX, "tre" changes to "tres" and "se" to "ses" or "sex"; similarly, when preceding a component markedM orN, "septe" and "nove" change to "septem" and "novem" or "septen" and "noven".

Since the system of using Latin prefixes will become ambiguous for numbers with exponents of a size which the Romans rarely counted to, like 106,000,258, Conway and Guy 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.[17] The name of a number 103n+3, wheren is greater than or equal to 1000, is formed by concatenating the names of the numbers of the form 103m+3, wherem represents each group of comma-separated digits ofn, with each but the last "-illion" trimmed to "-illi-", or, in the case ofm = 0, either "-nilli-" or "-nillion".[17] For example, 103,000,012, the 1,000,003rd "-illion" number, equals one "millinillitrillion"; 1033,002,010,111, the 11,000,670,036th "-illion" number, equals one "undecillinilli­septua­ginta­ses­centilli­sestrigint­illion"; and 1029,629,629,633, the 9,876,543,210th "-illion" number, equals one "nonillise­septua­ginta­octingentillitres­quadra­ginta­quingentillideciducent­illion".[17]

The following table shows number names generated by the system described by Conway and Guy for the short and long scales.[18]

Base -illion
(short scale)
Base -illion
(long scale)
ValueUS, Canada and modern British
(short scale)
Traditional British
(long scale)
Traditional European (Peletier long scale)SI
Symbol
SI
Prefix
11106MillionMillionMillionMMega-
21109BillionThousand millionMilliardGGiga-
321012TrillionBillionBillionTTera-
421015QuadrillionThousand billionBilliardPPeta-
531018QuintillionTrillionTrillionEExa-
631021SextillionThousand trillionTrilliardZZetta-
741024SeptillionQuadrillionQuadrillionYYotta-
841027OctillionThousand quadrillionQuadrilliardRRonna-
951030NonillionQuintillionQuintillionQQuetta-
1051033DecillionThousand quintillionQuintilliard
1161036UndecillionSextillionSextillion
1261039DuodecillionThousand sextillionSextilliard
1371042TredecillionSeptillionSeptillion
1471045QuattuordecillionThousand septillionSeptilliard
1581048QuindecillionOctillionOctillion
1681051Sedecillion[a]Thousand octillionOctilliard
1791054SeptendecillionNonillionNonillion
1891057OctodecillionThousand nonillionNonilliard
19101060Novendecillion[a]DecillionDecillion
20101063VigintillionThousand decillionDecilliard
21111066UnvigintillionUndecillionUndecillion
22111069DuovigintillionThousand undecillionUndecilliard
23121072TresvigintillionDuodecillionDuodecillion
24121075Quattuor­vigint­illionThousand duodecillionDuodecilliard
25131078QuinvigintillionTredecillionTredecillion
26131081SesvigintillionThousand tredecillionTredecilliard
27141084SeptemvigintillionQuattuordecillionQuattuordecillion
28141087OctovigintillionThousand quattuordecillionQuattuordecilliard
29151090NovemvigintillionQuindecillionQuindecillion
30151093TrigintillionThousand quindecillionQuindecilliard
31161096UntrigintillionSedecillion[a]Sedecillion[a]
32161099DuotrigintillionThousand sedecillion[a]Sedecilliard[a]
331710102TrestrigintillionSeptendecillionSeptendecillion
341710105Quattuor­trigint­illionThousand septendecillionSeptendecilliard
351810108QuintrigintillionOctodecillionOctodecillion
361810111SestrigintillionThousand octodecillionOctodecilliard
371910114SeptentrigintillionNovendecillion[a]Novendecillion[a]
381910117OctotrigintillionThousand novendecillion[a]Novendecilliard[a]
392010120NoventrigintillionVigintillionVigintillion
402010123QuadragintillionThousand vigintillionVigintilliard
502510153QuinquagintillionThousand quinvigintillionQuinvigintilliard
603010183SexagintillionThousand trigintillionTrigintilliard
703510213SeptuagintillionThousand quintrigintillionQuintrigintilliard
804010243OctogintillionThousand quadragintillionQuadragintilliard
904510273NonagintillionThousand quin­quadra­gint­illionQuin­quadra­gint­illiard
1005010303CentillionThousand quinquagintillionQuinquagintilliard
1015110306UncentillionUnquinquagintillionUnquinquagintillion
1105510333DecicentillionThousand quin­quinqua­gint­illionQuin­quinqua­gint­illiard
1115610336UndecicentillionSes­quinqua­gint­illionSes­quinqua­gint­illion
1206010363ViginticentillionThousand sexagintillionSexagintilliard
1216110366UnviginticentillionUnsexagintillionUnsexagintillion
1306510393TrigintacentillionThousand quinsexagintillionQuinsexagintilliard
1407010423Quadra­gintacent­illionThousand septuagintillionSeptuagintilliard
1507510453Quinqua­gintacent­illionThousand quin­septua­gint­illionQuin­septua­gint­illiard
1608010483SexagintacentillionThousand octogintillionOctogintilliard
1708510513SeptuagintacentillionThousand quinoctogintillionQuinoctogintilliard
1809010543OctogintacentillionThousand nonagintillionNonagintilliard
1909510573NonagintacentillionThousand quinnonagintillionQuinnonagintilliard
20010010603DucentillionThousand centillionCentilliard
30015010903TrecentillionThousand quinqua­gintacent­illionQuinqua­gintacent­illiard
400200101203QuadringentillionThousand ducentillionDucentilliard
500250101503QuingentillionThousand quinqua­gintaducent­illionQuinqua­gintaducent­illiard
600300101803SescentillionThousand trecentillionTrecentilliard
700350102103SeptingentillionThousand quinqua­gintatrecent­illionQuinqua­gintatrecent­illiard
800400102403OctingentillionThousand quadringentillionQuadringentilliard
900450102703NongentillionThousand quinqua­ginta­quadringent­illionQuinqua­ginta­quadringent­illiard
1000500103003Millinillion[19]Thousand quingentillionQuingentilliard
ValueNameEquivalent
US, Canadian and modern British
(short scale)
Traditional British
(long scale)
Traditional European (Peletier long scale)
10100GoogolTen duotrigintillionTen thousand sedecillion[a]Ten sedecilliard[a]
1010100GoogolplexTen trilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­trestrigintatrecentilli­duotrigintatrecentillion[b]Ten thousand milli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentillion[c]Ten milli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilliard[c]
  1. ^abcdefghijklWhile, today, sexdecillion and novemdecillion are standard dictionary numbers, these numbers are called "sedecillion" and "novendecillion" respectively in the Conway and Guy system. The same applies to the long scale forms "sedecilliard" and "novendecilliard".
  2. ^Googolplex's short scale name is derived from it equal to ten of the 3,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​332nd "-illion"s (This is the value of n when 10 × 10(3n + 3) = 1010100)
  3. ^abGoogolplex'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 × 106n = 1010100).

Binary prefixes

TheInternational System of Quantities (ISQ) defines a series of prefixes denoting integer powers of 1024 between 10241 and 10248.[20]

PowerValueISQ
symbol
ISQ
prefix
110241KiKibi-
210242MiMebi-
310243GiGibi-
410244TiTebi-
510245PiPebi-
610246EiExbi-
710247ZiZebi-
810248YiYobi-

Other named large numbers used in mathematics, physics and chemistry

See also

References

  1. ^Bellos, Alex (2011).Alex's Adventures in Numberland. A&C Black. p. 114.ISBN 978-1-4088-0959-4.
  2. ^abThe American Heritage Dictionary of the English Language (4th ed.). Houghton Mifflin. 2000.ISBN 0-395-82517-2.
  3. ^ab"Collins English Dictionary". HarperCollins.
  4. ^ab"Cambridge Dictionaries Online". Cambridge University Press.
  5. ^abThe Oxford English Dictionary (2nd ed.). Clarendon Press. 1991.ISBN 0-19-861186-2.
  6. ^ab"Oxford English Dictionary". Oxford University Press.
  7. ^abThe Random House Dictionary of the English Language (2nd ed.). Random House. 1987.
  8. ^abBrown, Lesley; Little, William (1993).The New Shorter Oxford English Dictionary. Oxford University Press.ISBN 0198612710.
  9. ^abWebster, Noah (1981).Webster's Third New International Dictionary of the English Language, Unabridged. Merriam-Webster.ISBN 0877792011.
  10. ^abRowlett, Russ."How Many? A Dictionary of Units of Measures". Russ Rowlett and the University of North Carolina at Chapel Hill. Archived fromthe original on 1 March 2000. Retrieved25 September 2022.
  11. ^Emerson, Oliver Farrar (1894).The History of the English Language. Macmillan and Co. p. 316.
  12. ^"Entry forcentillion in dictionary.com". dictionary.com. Retrieved25 September 2022.
  13. ^"Zimbabwe rolls out Z$100tr note". BBC News. 16 January 2009. Retrieved25 September 2022.
  14. ^Cunningham, Doug (31 October 2024)."Russian court levies huge $20 decillion fine against Google".United Press International. Retrieved1 November 2024.
  15. ^"Russia says $20 decillion fine against Google is 'symbolic'".The Guardian. Agence France-Presse. 31 October 2024.ISSN 0261-3077. Retrieved1 November 2024.
  16. ^Kasner, Edward; Newman, James (1940).Mathematics and the Imagination. Simon and Schuster.ISBN 0-486-41703-4.
  17. ^abcdefgConway, J. H.; Guy, R. K. (1998).The Book of Numbers. Springer Science & Business Media. pp. 15-16.ISBN 0-387-97993-X.
  18. ^Fish."Conway's illion converter". Retrieved1 March 2023.
  19. ^Stewart, Ian (2017).Infinity: A Very Short Introduction. Oxford University Press. p. 20.ISBN 978-0-19-875523-4.
  20. ^"IEC 80000-13:2008".International Organization for Standardization. 15 April 2008. Retrieved25 September 2022.
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