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. It has two distinct forms, the long scale and the short scale. English uses the short scale, while other languages such as French and German use the long scale. 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]
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 \(g_{\omega^\omega}(n)\) |
References
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: 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
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
