The -illion system is a suffix used to represent the powers of ten,[1] the only large number system that is widely in use except for systems in eastern Asia which are based on a myriad.
Contents
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:
- million = \(10^6\)
- billion = \(10^9\)
- trillion = \(10^{12}\)
- quadrillion = \(10^{15}\)
- quintillion = \(10^{18}\)
- sextillion = \(10^{21}\)
- n-illion = \(10^{(3n + 3)}\) using Latin prefix n
The long scale goes as follows:
- million = \(10^6\)
- milliard = \(10^9\)
- billion = \(10^{12}\)
- billiard = \(10^{15}\)
- trillion = \(10^{18}\)
- trilliard = \(10^{21}\)
- quadrillion = \(10^{24}\)
- quadrilliard = \(10^{27}\)
- n-illion = \(10^{6n}\)
- n-illiard = \(10^{(6n + 3)}\)
Sbiis Saibian replaces the long-scale -illion with -illiad.[2]
Naming systems of -illion numbers
Conway-Wechsler system
Conway and Guy developed a system for extending the -illion 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.
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 \(g_{\omega^\omega}(n)\) |
Quote
- Although we value [your] donations, we were somewhat surprised to note that none of them ended in "-illion."[9]
References
- ↑ Billion from Wolfram Mathworld
- ↑ https://sites.google.com/site/largenumbers/home/appendix/a/numbers
- ↑ 3.0 3.1 3.2 Conway and Guy (1995) "The book of Numbers" Copernicus
- ↑ Robert Munafo The Conway-Wechsler System
- ↑ Olivier Miakinen. Les zillions selon Conway, Wechsler... et Miakinen 21 May 2003
- ↑ Robert Munafo, More Conway-Wechsler Number Names
- ↑ Cookiefonster, 1.7. Extensions to the -illions I: Henkle, Conway, and Rowlett
- ↑ Fish Conway's zillion numbers with a program, July 25, 2021
- ↑ Stephen Colbert Edit of Wikipedia
See also
- millillion
- googolillion
- zillion
- Long and short scales on Wikipedia
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: trigintallion (un- · duo- · tre- · quattuor- · quin- · sex- · septen- · octo- · novem-)
40–49: quadragintallion (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
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
Concepts: googology · large number · eventual domination · ordinal · cardinal
Important numbers and functions: googol · googolplex · -illions · hyper-operators · Graham's number · Extensible-E system · TREE sequence · Busy beaver function · fast-growing hierarchy
People: Chris Bird · Jonathan Bowers · Harvey Friedman · Lawrence Hollom · André Joyce · Leo Moser · Robert Munafo · Sbiis Saibian · Aarex Tiaokhiao
