Graph of various numbers prefixed with the fuga- family of prefixes. Fz- is the blue line. Garfz- is the purple line, Fzgar- is the green line, and Fuga- is the orange line.
Graph of their logarithms
Fz- is a prefix used on a number n to indicate \(n^n\) or equivalently \({^2}n\) (i.e n tetrated to 2).[1][2][3] It was invented by Allstair Cockburn for the sole purpose for continuing the gar- prefix invented by his son as mentioned in his A Fuga Really Big Numbers blog, noting that the expression "n ns" can be found out as the square of n by children as early as before junior high.
It is equivalent to putting a number inside a triangle in Steinhaus-Moser_Notation, and \(10^{n \log n}\) as an exponential expression. It is therefore faster-growing than the most common definition of the -plex suffix, which means to take this number and output ten to the power of this number, but still within singly exponential growth.
The first few values of fz-n are 1, 4, 27, 256, 3125, 46,656, 823,543, 16,777,216, 387,420,489, and 10,000,000,000.
The fz-n sequence is used by the game "Cookie Clicker" for the price of various upgrades in the Santa minigame.
Properties[]
- Since Fz-n is repeated multiplication of n, Fz-n is divisible by n.
In googological notations[]
| Notation | Expression |
|---|---|
| Fast-growing hierarchy | In between \(f_2(n)\) and \(f_3(n)\) |
| Hardy hierarchy | > \(H_{\omega^2}(n)\) |
| Slow-growing hierarchy | \(g_{\omega^\omega}(n)\) |
| Steinhaus-Moser Notation | Exactly \(n[3]\) (triangle(n)) |
Sources[]
- ↑ Cockburn, Alistair. A fuga really big numbers (archived 2016.
- ↑ Saibian, Sbiis. The Fz, The Fuga & The Megafuga. Retrieved 2021-07-22.
- ↑ Cookiefonster. 2.5A. Googological prefixes and suffixes - Pointless Large Number Stuff. Retrieved 2021-07-22.
See also[]
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- · ronna- · quetta-
Non-SI prefixes: 1.001 · 1.01 · 1.1 · 1.5 · 2 · 3 · 666 · 104 · 105 · 107 · 108 · 1010 · 1011 · 1013 · 1014 · 1016 · 1017 · 1019 · 1020 · 1022 · 1023 · 1025 · 1027 · 1028 · 1029 · 1030 · 1031 · 1032 · 1033 · 1034 · 1035 · 1036 · 1039 · 1042 · 1045 · 1048 · 1051 · 1054 · 1057 · 1060 · 1063 · 1066 · 1069 · 1072 · 1075 · 1090 · 10100 · 10120 · 10150 · 10180 · 10210 · 10240 · 10270 · 10300 · 10600 · 10900 · 101200 · 101500 · 101800 · 102100 · 102400 · 102700 · 103000 and higher
Gar-: garone · gartwo · garthree · garfour · garfive · garsix · garseven · gareight · garnine · garten · garhundred · garmillion · gargoogol · gareceton · gartrialogue · gargoogolplex · gargiggol
Fz-: fzone · fztwo · fzthree · fzfour · fzfive · fzsix · fzseven · fzeight · fznine · fzten · fztwenty · fzthirty · fzhundred · fzthousand · fzmillion · fzgoogol · fzmilliplexion · fzgoogolplex · fzgargoogolplex · fzgiggol
Fuga-: fugaone · fugatwo · fugathree · fugafour · fugafive · fugasix · fugaseven · fugaeight · fuganine · fugaten · fugahundred · fugagoogol · fugagoogolplex · fugagargoogolplex · fugagiggol
Megafuga-: megafugaone · megafugatwo · megafugathree · megafugafour · megafugafive · megafugasix · megafugaseven · megafugaeight · megafuganine · megafugaten · megafugahundred · megafugagoogol · megafugagoogolplex · megafugagargoogolplex · megafugagargantugoogolplex · megafugagrangol
Extensions:
Booga-:: boogaone · boogatwo · boogathree · boogafour · boogafive · boogasix · boogaten · boogahundred · boogagoogol · boogagoogolplex
Gag-:: gagone · gagtwo · gagthree · gagfour · gagfive · gagsix · gagseven · gageight · gagnine · gagten · gaggoogol · gaggoogolplex