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Difference between revisions of "Jonathan Bowers"
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|{10,{10,100,2},2} = 10 {2} 10 {2} 100 | |{10,{10,100,2},2} = 10 {2} 10 {2} 100 | ||
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− | | | + | |Googolplexian |
|<math>10^{10^{10^{100}}}</math> | |<math>10^{10^{10^{100}}}</math> | ||
|{10,{10,{10,100,2},2},2} = 10 {2} 10 {2} 10 {2} 100 | |{10,{10,{10,100,2},2},2} = 10 {2} 10 {2} 10 {2} 100 | ||
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|- | |- | ||
| [[Platillion]] || <math>{10}^{6000}</math> | | [[Platillion]] || <math>{10}^{6000}</math> | ||
+ | |- | ||
+ | | [[Myrillion]] || <math>{10}^{6000}</math> | ||
|- | |- | ||
| Micrillion || 10^ 3000003 | | Micrillion || 10^ 3000003 |
Revision as of 02:14, 28 March 2006
Jonathan Bowers, mathematician (November 27, 1969–)
Contents
Polychora
One of the participants in the Uniform Polychora Project, an attempt to name higher-dimensional polychora. He independently began his search for polychora in 1990. Circa 1993, he invented his short names for the uniform polychora, which have come to be known as the Bowers style acronyms (or pet name). In 1997, he contacted others who are also interested in the subject, such as Magnus Wenninger, Vincent Matsko, and George Olshevsky.
Very large numbers
Bowers has proposed a series of names (including giggol, gaggol, geegol, goggol, tridecal, tetratri, dutritri, xappol, dimendecal, gongulus, trimentri, goppatoth, golapulus, golapulusplex, golapulusplux, big boowa and guapamonga) for extremely Large numbers, which he terms infinity scrapers (a pun on skyscraper), many of which are so unspeakably and unprecedentedly large as to require special newly-devised extended mathematical notations in order to be expressed.infinity scrapers
He is also the inventor of the Array Notation as a means to represent very large numbers. This notion is very similar to the Conway chained arrow notation and relies on the tetration operator <math>{\ ^{b}a = \atop {\ }} {{\underbrace{a^{a^{\cdot^{\cdot^{a}}}}}} \atop b}</math>, and its higher order analogues: pentation, sexation, heptation. some examples of which are
- <math>\{a,b,1\} = a \{1\} b = a+b\;</math>
- <math>\{a,b,2\} = a \{2\} b = a*b\;</math>
- <math>\{a,b,3\} = a \{3\} b = a^b\;</math>
- <math>\{a,b,4\} = a \{4\} b = \ ^{b}a</math> a tetrated to b.
- <math>\{a,b,5\} = a \{5\} b = \ ^{\ ^{\ ^{\ ^a\cdot}\cdot}a}a</math> - a pentated to b - a tetrated to itself b times.
- <math>\{a,b,c,2\} = a \{\{c\}\}b\;</math>
- <math>\{a,2,1,2\} = a \{\{1\}\}2 = a \{a\}a\;</math>
- <math>\{a,3,1,2\} = a \{\{1\}\}3 = a \{a \{a\}a\}a\;</math>
- <math>\{a,b,1,2\} = a \{\{1\}\}b = a \{a\ldots\{a\}\ldots a\}a\;</math> - a expanded to b.
- <math>\{a,b,2,2\} = a \{\{2\}\}b\;</math> - a expanded to itself b times.
Group | Name | Value | Array notation |
---|---|---|---|
Googol group | Googol | <math>10^{100}=10^{10^2}</math> | {10,100,2} = 10 {2} 100 |
Googolplex | <math>10^{10^{100}}=10^{10^{10^2}}</math> | {10,{10,100,2},2} = 10 {2} 10 {2} 100 | |
Googolplexian | <math>10^{10^{10^{100}}}</math> | {10,{10,{10,100,2},2},2} = 10 {2} 10 {2} 10 {2} 100 | |
Giggol group | <math>\ ^{10}10</math> | {10,10,4} = 10 {4} 10 | |
Giggol | <math>\ ^{100}10</math> | {10,100,4} = 10 {4} 10: 10 to the power of itself 100 times. | |
Giggolplex | 10 {4} 10 {4} 10: 10 tetrated to a giggol | ||
Giggolduplex | 10 {4} 10 {4} 10 {4} 100. | ||
Gaggol group | Gaggol | {10,100,5} = 10 {5} 100: 10 pentated to 100. | |
gaggolplex | 10 {5} gaggol = 10 {5} 10 {5} 100: 10 tetrated to itself a gaggol times. | ||
geegol | {10,100,6}=10 {6} 100 | ||
geegolplex | {10,geegol, 6} | ||
gygol | {10,100,7} | ||
gygolplex | {10,gygol,7} | ||
goggol | {10,100,8} | ||
goggolplex | {10,goggol,8} | ||
gagol | {10,100,9} | ||
gagolplex | {10,gagol,9} | ||
Infinity Scrapers | Tridecal | {10,10,10} = 10 {10} 10 = 10 decated to 10 | |
boogol | {10,10,100} = 10 {100} 10 |
Bowers notation extends beyond even these numbers and includes the Corporal {10,100,1,2}, Grand Tridecal {10,10,10,2}, Tetratri = {3,3,3,3}, General = {10,10,10,10}, Pentadecal = {10,10,10,10,10}, Iteral = {10,10,10,10,10,10,10,10,10,10}. Still larger numbers require an extended array notation and include the Emperal Group, Hyperal Group, Dutritri and Dutridecal. The largest number mentioned is the Guapamongaplex which is still considerably less than infinity, but is much larger than even Graham's number.
Illion group
Jonathan Bowers has also created a number of names for large numbers that extend the -illion family of much smaller numbers than those mentioned above. Here is a list of the names of large -illion numbers on the short scale.
Name | Short scale value |
---|---|
Million | <math>{10}^6</math> |
Billion | <math>{10}^9</math> |
Trillion | <math>{10}^{12}</math> |
Quadrillion | <math>{10}^{15}</math> |
Quintillion | <math>{10}^{18}</math> |
Sextillion | <math>{10}^{21}</math> |
Septillion | <math>{10}^{24}</math> |
Octillion | <math>{10}^{27}</math> |
Nonillion | <math>{10}^{30}</math> |
Decillion | <math>{10}^{33}</math> |
Undecillion | <math>{10}^{33}</math> |
Duodecillion | 1039 |
Tredecillion | 1042 |
Quattuordecillion | 1045 |
Quindecillion | 1048 |
Sexdecillion | 1051 |
Septendecillion | 1054 |
Octodecillion | 1057 |
Novemdecillion | 1060 |
Vigintillion | 1063 |
Trigintillion | 1093 |
Googol (Ten Duotrigintillion) |
10100 |
Quadragintillion | 10123 |
Quinquagintillion | 10153 |
Sexagintillion | 10183 |
Septuagintillion | 10213 |
Octogintillion | 10243 |
Nonagintillion | 10273 |
Centillion | 10300 |
Platillion | <math>{10}^{6000}</math> |
Myrillion | <math>{10}^{6000}</math> |
Micrillion | 10^ 3000003 |
Nanillion | 10^ 3billion3 |
Picillion | 10^ 3trillion3 |
Femtillion | 10^ 3quadrillion3 |
Attillion | 10^ 3 quintillion3 |
Zeptillion | 10^ 3sextillion3 |
Yoctillion | 10^ 3septillion3 |
Xonillion | 10^ 3octillion3 |
Vecillion | 10^ 3nonillion3 |
Mecillion | 10^ 3decillion3 |
Duecillion | 10^ 3undecillion3 |
Trecillion | 10^ 3doedecillion3 |
Tetrecillion | 10^ 3tridecillion3 |
Pentecillion | 10^ 3quattuordecillion3 |
Hexecillion | 10^ 3quindecillion3 |
Heptecillion | 10^ 3sexdecillion3 |
Octecillion | 10^ 3septendecillion3 |
Ennecillion | 10^ 3octodecillion3 |
Icosillion | 10^ 3novemdecillion3 |
Triacontillion | 10^ (3x10^90+3) |
Googolplex | 10^10^100 |
Tetracontillion | 10^ (3x10^120+3) |
Pentacontillion | 10^ (3x10^150+3) |
Hexacontillion | 10^ (3x10^180+3) |
Heptacontillion | 10^ (3x10^210+3) |
Octacontillion | 10^ (3x10^240+3) |
Ennacontillion | 10^ (3x10^270+3) |
Hectillion | 10^ (3x10^300+3) |
Killillion | 10^ (3x10^3000+3) |
Megillion | 10^ (3x10^3million +3) |
Gigillion | 10^ (3x10^3billion +3) |
Terillion | 10^ (3x10^3trillion +3) |
Petillion | 10^ (3x10^3quadrillion +3) |
Exillion | 10^ (3x10^3quintillion +3) |
Zettillion | 10^ (3x10^3sextillion +3) |
Yottillion | 10^ (3x10^3septillion +3) |
Xennillion | 10^ (3x10^3octillion +3) |
Vekillion | 10^ (3x10^3nonillion +3) |
Duekillion | 10^ (3x10^3undecillion +3) |
Trekillion | 10^ (3x10^3doedecillion +3) |
Tetrekillion | 10^ (3x10^3tridecillion +3) |
Pentekillion | 10^ (3x10^3quattuordecillion +3) |
Hexekillion | 10^ (3x10^3quindecillion +3) |
Heptekillion | 10^ (3x10^3sexdecillion +3) |
Octekillion | 10^ (3x10^3septendecillion +3) |
Ennekillion | 10^ (3x10^3octodecillion +3) |
Twentillion | 10^ (3x10^(3x10^60) +3) |
Triatwentillion | 10^ (3x10^(3x10^69) +3) |
Thirtillion | 10^ (3x10^(3x10^90)+3) |
Googolduplex | 10^10^10^100 |
Fortillion | 10^ (3x10^(3x10^120)+3) |
Fiftillion | 10^ (3x10^(3x10^150)+3) |
Sixtillion | 10^ (3x10^(3x10^180)+3) |
Seventillion | 10^ (3x10^(3x10^210)+3) |
Eightillion | 10^ (3x10^(3x10^240)+3) |
Nintillion | 10^ (3x10^(3x10^270)+3) |
Hundrillion | 10^ (3x10^(3x10^300)+3) |
Thousillion | 10^ (3x10^(3x10^3000)+3) |
Manillion | <math>{10}^{{3 * {{10}^{3 * {{10}^{30,000}}}}} + 3}</math> |
Lakhillion | <math>{10}^{{3 * {{10}^{3 * {{10}^{300,000}}}}} + 3}</math> |
Crorillion | <math>{10}^{{3 * {{10}^{3 * {{10}^{30,000,000}}}}} + 3}</math> |
Awkillion | <math>{10}^{{3 * {{10}^{3 * {{10}^{300,000,000}}}}} + 3}</math> |
Bentrizillion | <math>{10}^{6 * {{10}^{6 * {{10}^{6 * {{10}^{6 * {{10}^9}}}}}}}}</math> |
Googol, giggol and gaggol groups
Jonathan Bowers defines the googol, giggol, and gaggol groups as being lower than the infinity scrapers. Here's a list of the names of numbers in these groups:
Name | Value |
---|---|
Googol | 10^100 |
Googolplex | 10^10^100 |
Googolduplex | 10^10^10^100 |
Googoltriplex | 10^10^10^10^100 |
Googolquadriplex | 10^10^10^10^10^100 |
Googolcentplex | Googolduplex and (98 more plexes) |
Googolmilleplex | Googolduplex and (998 more plexes) |
Googolmegaplex | Googolplex and (999,999 more plexes) |
Googolgigaplex | Googolplex and (999,999,999 more plexes) |
Googolteraplex | Googolplex and (999,999,999,999 more plexes) |
Googolpetaplex | Googolplex and (999,999,999,999,999 more plexes) |
Fzgoogolplex | Googolplex to the power of a googolplex |
Giggol | {10,100,5} = 10 {5} 100: 10 pentated to 100 |
Giggolplex | {10,100,5} = 10 {5} 100: 10 pentated to 100 |
Mega | ... |
Gaggol | {10,100,5} = 10 {5} 100: 10 pentated to 100 |
Gaggolplex | 10 {5} gaggol = 10 {5} 10 {5} 100: 10 tetrated to itself a gaggol times |
Megaston | ... |
Tripent | {5,5,5} = 5 {5} 5 = 5 {4} 5 {4} 5 {4} 5 {4} 5 |
Trisept | {7,7,7} = 7 {7} 7 (7 heptated to 7) |
Geegol | {10,100,6}=10 {6} 100 |
Geegolplex | {10,geegol, 6} |
Gygol | {10,100,7} |
Gygolplex | {10,gygol,7} |
Goggol | {10,100,8} |
Goggolplex | {10,goggol,8} |
Gagol | {10,100,9} |
Gagolplex | {10,gagol,9} |
Infinity scrapers
Numbers higher than those in the Gaggol group are referred to by Jonathan Bowers as the infinity scrapers. Here's a list of the names of those numbers that are infinity scrapers:
Name | Value |
---|---|
Tridecal | {10,10,10} = 10 {10} 10 = 10 decated to 10 |
Boogol | {10,10,100} = 10 {100} 10 |
Boogolplex | {10,10,boogol} |
Moser's number | ... |
Graham's number | ... |
Corporal | {10,100,1,2} |
Corporalplex | {10,corporal,1,2} |
Grand Tridecal | {10,10,10,2} |
Tetratri | {3,3,3,3} |
General | {10,10,10,10} |
Generalplex | {10,10,10,general} |
Pentatri | {3,3,3,3,3} |
Pentadecal | {10,10,10,10,10} |
Pentadecalplex | {10,10,10,10,pentadecal} |
Hexatri | {3,3,3,3,3,3} |
Hexadecal | {10,10,10,10,10,10} |
Hexadecalplex | {10,10,10,10,10,hexadecal} |
Iteral | {10,10,10,10,10,10,10,10,10,10} |
Ultatri | {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} |
Iteralplex | {10,10,10,10,10,10,...........,10,10,10} |
The following numbers requre an extended array notation to define. These are defined recursively, using rules such as:
- <math>\left\langle\begin{matrix}a&b\\2&\end{matrix}\right\rangle=\{a,a,\ldots,a\}</math> with a repeated b times.
- <math>\left\langle\begin{matrix}a&b\\k&\end{matrix}\right\rangle=\left\langle\begin{matrix}a&a&\ldots&a\\k-1\end{matrix}\right\rangle</math> with a repeated b times.
Name | Value |
---|---|
Emperal | <math>\left\langle\begin{matrix}10&10\\10&\end{matrix}\right\rangle</math> |
Emperalplex | <math>\left\langle\begin{matrix}10&10\\emperal&\end{matrix}\right\rangle</math> |
Hyperal | <math>\left\langle\begin{matrix}10&10\\10&10\end{matrix}\right\rangle</math> |
Hyperalplex | <math>\left\langle\begin{matrix}10&10\\10&Hyperal\end{matrix}\right\rangle</math> |
Dutritri | <math>\left\langle\begin{matrix}3&3&3\\3&3&3\\3&3&3\end{matrix}\right\rangle</math> |
Dutridecal | <math>\left\langle\begin{matrix}10&10&10\\10&10&10\\10&10&10\end{matrix}\right\rangle</math> |
Xappol | 10 by 10 array of 10's |
Xappolplex | xappol by xappol array of 10's |
Dimentri | 3 x 3 x 3 array of 3's |
Colossal | 10 x 10 x 10 array of 10's |
Colossalplex | colossal x colossal x colossal array of 10's |
Dimendecal | 10x10x10x10x10x10x10x10x10x10 array of 10's |
Gongulus | 100 dimensional array of 10's (10^100 array that is) |
Gongulusplex | gongulus dimensional array of 10's (10^gongulus array) |
Dulatri | (3^3)^2 array of 3's |
Trimentri | 3^(3^3) array of 3's |
Goppatoth | 10 tetrated to 100 array of 10's |
Goppatothplex | {10,goppatoth,4} array of 10's |
Tridecatrix | {10,10,10} array of 10's |
Gongulus | a "10^100 array of 10's" array of 10's. |
Golapulus | a * "10^100 array of tens" array of tens* array of tens. |
Big Boowa | X3, {X3,dutritriX, 2} X |
Great Big Boowa | X3,3,3X |
Wompogulus | 10^10 "100th level" exploded array of 10's |
Wompogulusplex | 10^10 "wompogulusth level" exploded array of 10's!! |
Guapamonga | 10^100 array of B's within "# #" |
Guapamongaplex | 10^100 array of B's within guapamonga-level "# #" |