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The -illion system is a suffix used to represent the powers of one thousand,[1] the only large number system that is widely in use except for systems in eastern Asia which are based on a myriad.

Short and long scale

The -illion system has two distinct forms, the long scale and the short scale. Accoring to long and short scale#Current usage, Most English-speaking and Arabic-speaking countries and regions use the short scale, while most continental European countries such as French and German use the long scale, and some countries use both. The long scale was also used in UK English until 1974, when the British usage and the American usage became identical.

The short scale goes as follows:

The long scale goes as follows:

Sbiis Saibian replaces the long-scale -illion with -illiad.[2]

Naming systems of zillion numbers

Conway-Wechsler system

Conway and Guy developed a system for extending the zillion numbers according to the rule in this table, which determines n-illion numbers for n<1000.[3][4] For n<10, standard English name is accepted.

N units tens hundreds
1 un N deci NX centi
2 duo MS viginti N ducenti
3 tre (*) NS triginta NS trecenti
4 quattuor NS quadraginta NS quadringenti
5 quin (**) NS quinquaginta NS quingenti
6 se (*) N sexaginta N sescenti
7 septe (*) N septuaginta N septingenti
8 octo MX octoginta MX octingenti
9 nove (*) nonaginta nongenti
  • (*) Note: when it is immediately before a component marked with S or N, "tre" increases to "tres" and "se" to "ses" or "sex" as appropriate. Similarly "septe" and "nove" increase to "septem" and "novem" or "septen" and "noven" immediately before components marked with M or N.
  • (**) quin is changed from original quinqua, as explained below.

Any zillion from the 10th to the 999th is made by combining parts from the appropriate columns of this table, and then replacing the final vowel by "illion". For example,

  • 234-th zillion number is named as "quattuor" + "triginta" + "ducenti" + "llion" = quattuortrigintaducentillion.
  • 34-th zillion number is named as "quattuor" + "triginta" = quattuortriginta, replace the final "a" with "illion" = quattuortrigintillion

Miakinen proposed[5] that as quindecillion is a common name and Latin name of 15 is quindecim, not quinquadecim, quinqua should be replaced to quin, and a similar change to all the Conway-Wechsler names involving the quinqua- prefix should be updated. Robert adopted the suggestion,[6] and call this modified version as the above table as the Conway-Wechsler system.

Conway and Guy wrote that this complete system of -illion words first appears in their book,[3] and this system has been widely accepted, with some modifications described below.

Extending the system

Conway and Guy[3] extended this system with Allan Wechsler for N≥1000 as follows.

With Allan Wechsler we propose to extend this system indefinitely by combining these according to the convention that "XilliYilliZillion" (say) denotes the (1000000X + 1000Y + Z)th zillion, using "nillion" for the zeroth "zillion" when this is needed as a placeholder. So for example the million-and-third zillion is a "millinillitrillion."

Modified Conway-Wechsler system

Even after changing quinqua to quin, Conway-Wechsler names have some difference from the common names, for example, sedecillion and novendecillion are better known as sexdecillion and novemdecillion respectively. Many people simplified the Conway-Wechsler system, where Cookiefonster summarized as follows.[7] Cookiefonster says that this modified system is faithful to the dictionary -illions and it is accepted so much. In googology wiki, this modified Conway-Wechsler system is accepted for the title of -illion numbers up to 999-illion.

N units tens hundreds
1 un deci centi
2 duo viginti ducenti
3 tre triginta trecenti
4 quattuor quadraginta quadringenti
5 quin quinquaginta quingenti
6 sex sexaginta sescenti
7 septen septuaginta septingenti
8 octo octoginta octingenti
9 novem nonaginta nongenti

Fish made a list of the -illion numbers of the Conway-Wechsler system and the modified system and a program which created the table.[8]

In googological notations

For the short scale:

Notation Expression
Fast-growing hierarchy between \(f_2(f_1^3(n))\) and \(f_2(f_1^4(n))\)
Hardy hierarchy between \(H_{\omega^2+\omega 3}(n)\) and \(H_{\omega^2+\omega 4}(n)\)
Slow-growing hierarchy \(g_{\omega^{n+1}}(1,000)\); just below \(g_{\omega^\omega}(n)\)

For the long scale:

Notation Expression
Fast-growing hierarchy between \(f_2(f_1^4(n))\) and \(f_2(f_1^5(n))\)
Hardy hierarchy between \(H_{\omega^2+\omega 4}(n)\) and \(H_{\omega^2+\omega 5}(n)\)
Slow-growing hierarchy \(g_{\omega^{2n}}(1,000)\); just below \(g_{\omega^\omega}(n)\)

For -illiard:

Notation Expression
Fast-growing hierarchy between \(f_2(f_1^4(n))\) and \(f_2(f_1^5(n))\)
Hardy hierarchy between \(H_{\omega^2+\omega 4}(n)\) and \(H_{\omega^2+\omega 5}(n)\)
Slow-growing hierarchy \(g_{\omega^{2n+1}}(1,000)\); just below \(g_{\omega^\omega}(n)\)

Etymology

All "-illion" terms derive from "million", the word for which came from the Italian millione; the augmentative suffix -one applied to the word for "thousand", mille. This gives "million" the literal definition of roughly "super thousand".[9] The pattern of extracting "illion" from it was established in 1484 by Nicolas Chuquet in his unpublished book Triparty en la science des nombres,[10][11] which defined the long-scale senses of Byllion up to Nonyllion.

Quote

  • Although we value [your] donations, we were somewhat surprised to note that none of them ended in "-illion."[12]


References

  1. Billion from Wolfram Mathworld
  2. https://sites.google.com/site/largenumbers/home/appendix/a/numbers
  3. 3.0 3.1 3.2 Conway and Guy (1995) "The book of Numbers" Copernicus
  4. Robert Munafo The Conway-Wechsler System
  5. Olivier Miakinen. Les zillions selon Conway, Wechsler... et Miakinen 21 May 2003
  6. Robert Munafo, More Conway-Wechsler Number Names
  7. Cookiefonster, 1.7. Extensions to the -illions I: Henkle, Conway, and Rowlett
  8. Fish Conway's zillion numbers with a program, July 25, 2021
  9. "Million" at Wiktionary
  10. La reception du "Triparty en la science des nombres" (1484) de Nicolas Chuquet aux XVIe et XVIIe siècles, in [Analisi linguistica e letteraria. Anno XII - N. 1/2, 2004][Milano : [S.l.] : Vita e Pensiero ; DSU Università Cattolica del Sacro Cuore, 2004.] - Permalink
  11. The entirety of Nicolas Chuquet's text concerning the systematic of zillions, worked out in the very first chapter of his Triparty (1484) florencetime.net (archived on 28 Sep 2007)
  12. Stephen Colbert Edit of Wikipedia

See also

Standard -illion numbers 103n+3 (106n)
Main article: -illion
1–9: million · billion · trillion · quadr · quint · sext · sept · oct · non
10–19: decillion · undec · duodec · tredec · quattuordec · quindec · sexdec · septendec · octodec · novemdec
20–29: vigintillion · unvigint · duovigint · trevigint · quattuorvigint · quinvigint · sexvigint · septenvigint · octovigint · novemvigint
30–39: trigintillion (un- · duo- · tre- · quattuor- · quin- · sex- · septen- · octo- · novem-)
40–49: quadragintillion (un- · duo- · tre- · quattuor- · quin- · sex- · septen- · octo- · novem-)
50–59: quinquagintillion (un- · duo- · tre- · quattuor- · quin- · sex- · septen- · octo- · novem-)
60–69: sexagintillion (un- · duo- · tre- · quattuor- · quin- · sex- · septen- · octo- · novem-)
70–79: septuagintillion (un- · duo- · tre- · quattuor- · quin- · sex- · septen- · octo- · novem-)
80–89: octogintillion (un- · duo- · tre- · quattuor- · quin- · sex- · septen- · octo- · novem-)
90–99: nonagintillion (un- · duo- · tre- · quattuor- · quin- · sex- · septen- · octo- · novem-)
100–900: centillion · unviginticent · unsexagintacent · ducent · trecent · duotrigintatrecent · quadringent · quingent · sescent · septuagintisescent · septingent · octingent · nongent
1,000–1024: millillion · dumill · dumillinonagintanongent · trimill · trimilliduotrigintatrecent · trimillisexoctogintaoctingent · quadrimill · quadrimilliquattuordecicent · quinmill · sexmill · septimill · octimill · nonimill · myr · decimilliquinsexagintasescent · dumyr · unquadragintamilliunquinquagintacent · centimill · micr · nan · pic · femt · att · zept · yoct
Googological affixes

Suffixes: -teen · -ty · -plex · -illion · -yllion · -exian · -chime · -toll · -gong · -bong · -throng · -illiob
Prefixes: gar- · fz- · fuga- · megafuga- · booga- · trooga- · googo- · googolple-
SI prefixes: deca- · hecto- · kilo- · mega- · giga- · tera- · peta- · exa- · zetta- · yotta-
Obsolete SI prefixes: myria-
Extended SI prefixes: 1.001 · 1.01 · 1.1 · 1.5 · 2 · 3 · 666 · 105 · 107 · 1027 · 1030 · 1033 · 1036 · 1039 · 1042 · 1045 · 1048 · 1051 · 1054 · 1057 · 1060 · 1063 · 1066 · 1069 · 1072 · 1075 · 1078 · 1081 · 1084 · 1087 · 1090 · 10100 · 10120 · 10150 · 10180 · 10210 · 10240 · 10270 · 10300 · 10600 · 10900 · 101200 · 101500 · 101800 · 102100 · 102400 · 102700 · 103000 and higher

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