Googolplex in zeroes

Googolplex is a large number equal to 1010100 = 10googol or one followed by a googol (10100) zeroes.[1] Milton Sirotta originally defined it as "one, followed by writing zeroes until you get tired". Edward Kasner, unsatisfied by this vague definition, redefined it to its current value.[2][3] It is 10100+1 digits long. It seems to be slightly larger than a duotrigintillionplex (101099), but it in fact equals to be duotrigintillionplex10.

Contrary to popular belief, googolplex is neither the largest number nor the largest named number. It is easily beaten by other named numbers such as giggol.

Ten to the power of googolplex is called googolduplex (also called googolplexplex, googolplusplex, and googolplexian).

Writing down the full decimal expansion would take 1094 books of 400 pages each, with 2,500 digits on each page.

Written out in decimal form googolplex is:

1010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

Etymology

The name of this number is derived from googol, but there is no logic to the definition otherwise. Googologists later took this number and backformed an etymology by considering "-plex" to be a new suffix.

Computation

Googolplex can be computed using the following process:

Step 0: 10

Step 1: 10,000,000,000 (ten billion)

Step 2: \((10^{10})^{10} = 10^{100}\)

Step 3: \((10^{100})^{10} = 10^{1,000}\)

...

At every new step, the number of zeroes multiplies by 10.

...

Step 100: Googolplex.

This can be defined recursively:

Googolplex is \(f(100)\) when: \(\left\{ \begin{array}{ll} f(n)=f(n-1)^{10} \\ f(0)=10 \end{array} \right.\)

In other notations

It is approximately 57↑↑3 in up-arrow notation and exactly 10↓↓101 in down-arrow notation.

In Hyper-E notation, it can be written as E100#2 or EE100.

Aarex Tiaokhiao coined the name googolunex for this number.[4]

Names in -illion systems

In Conway and Guy's -illion system, it has a very long name:[5]

ten trillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentilliduotrigintatrecentillion

In Jonathan Bowers' -illion system, a googolplex also has a long name:

ten tretriotriaconto-tretrigintitrecentiduetriaconto-tretrigintitrecentimetriaconto-tretrigintitrecentitriaconto-tretrigintitrecentienneicoso-tretrigintitrecentiocteicoso-tretrigintitrecentihepteicoso-tretrigintitrecentihexeicoso-tretrigintitrecentipenteicoso-tretrigintitrecentitetreicoso-tretrigintitrecentitrioicoso-tretrigintitrecentidueicoso-tretrigintitrecentiicoso-tretrigintitrecentienneco-tretrigintitrecentiocteco-tretrigintitrecentihepteco-tretrigintitrecentihexeco-tretrigintitrecentipenteco-tretrigintitrecentitetreco-tretrigintitrecentitreco-tretrigintitrecentidueco-tretrigintitrecentimeco-tretrigintitrecentimeco-tretrigintitrecentiveco-tretrigintitrecentixono-tretrigintitrecentiyocto-tretrigintitrecentizepto-tretrigintitrecentiatto-tretrigintitrecentifemto-tretrigintitrecentipico-tretrigintitrecentinano-tretrigintitrecentimicro-tretrigintitrecentimilli-doetrigintitrecentillion

According to Landon Curt Noll's The English name of a number, googolplex is also known as:

ten tremilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliamilliatrecentretriginmilliamilliamilliamilliatrecentretriginmilliamilliamilliatrecentretriginmilliamilliatrecentretriginmilliatrecenduotrigintillion

In long scale by Jaden Hartson using Jonathan Bowers' -illion system:

ten thousand triotriaconto-sexsexagintisescentiduetriaconto-sexsexagintisescentimetriaconto-sexsexagintisescentitriaconto-sexsexagintisescentienneicoso-sexsexagintisescentiocteicoso-sexsexagintisescentihepteicoso-sexsexagintisescentihexeicoso-sexsexagintisescentipenteicoso-sexsexagintisescentitetreicoso-sexsexagintisescentitrioicoso-sexsexagintisescentidueicoso-sexsexagintisescentimeicoso-sexsexagintisescentiicoso-sexsexagintisescentienneco-sexsexagintisescentiocteco-sexsexagintisescentihepteco-sexsexagintisescentihexeco-sexsexagintisescentipenteco-sexsexagintisescentitetreco-sexsexagintisescentitreco-sexsexagintisescentidueco-sexsexagintisescentimeco-sexsexagintisescentiveco-sexsexagintisescentixono-sexsexagintisescentiyocto-sexsexagintisescentizepto-sexsexagintisescentiatto-sexsexagintisescentifemto-sexsexagintisescentipico-sexsexagintisescentinano-sexsexagintisescentimicro-sexsexagintisescentimilli-sexsexagintisescentilliard

Approximations in other notations

Notation Lower bound Upper bound
Arrow notation \(10\uparrow10\uparrow100\)
Down-arrow notation \(10\downarrow\downarrow101\)
Steinhaus-Moser Notation 56[3][3] 57[3][3]
Copy notation 9[9[100]] 1[1[101]]
Chained arrow notation \(10\rightarrow(10\rightarrow100)\)
PlantStar's Debut Notation [1,59] [1,60]
H* function H(3H(32)) H(4H(32))
Taro's multivariable Ackermann function A(3,A(3,330)) A(3,A(3,331))
Pound-Star Notation #*((1))*(0,6,8,1,1)*8 #*((1))*(0,6,6,3,4)*7
BEAF {10,{10,100}}
Hyper-E notation E2#3, E100#2
Bashicu matrix system (0)(1)[18] (0)(1)[19]
Hyperfactorial array notation (68!)! (69!)!
Bird's array notation {10,{10,100}}
Graham Array Notation [10,[10,100]]
Strong array notation s(10,s(10,100))
Fast-growing hierarchy \(f_2(f_2(325))\) \(f_2(f_2(326))\)
Hardy hierarchy \(H_{\omega^22}(325)\) \(H_{\omega^22}(326)\)
Slow-growing hierarchy \(g_{\omega^{\omega^{\omega^2}}}(10)\)

 Examples

A googolplex is too large in comparison to anything computable in the universe, but the only exception may be the estimated number of subatomic particles which is 1010115. Without proton decay the Degenerate Era of the universe will most likely last 101500 years, and the Black Hole Era 1010120 years until it reaches its final energy state.

In popular culture

The company Google named their Mountain View headquarters "The Googleplex."[6]

Even though googolplex is easily surpassed by larger numbers, it has nonetheless been used as a benchmark for a large quantity that is difficult to comprehend. In the documentary Cosmos: A Personal Voyage, Carl Sagan discusses the difficulty of writing out a googolplex.[7]

In the Samurai Jack episode "Jack vs. Mad Jack," a bounty of one googolplex, later increased to two googolplex, is placed on Jack.

Numbers near googolplex

Dario Alpern maintains a website listing the known factors of 1010100+ n,[8] where n is an integer from 0 to 999. The case n = 1 has the smallest known factor 316,912,650,057,057,350,374,175,801,344,000,001 (about 3.17×1035), found by Robert Harley. Several larger prime factors are known. 1010100+ 37 is the smallest with no known prime factors. It has no prime factors below 3.5×1014.

A few of those numbers have a huge number of known factors due to their algebraic properties. In particular, 1010100+10 has 57,445 known prime factors,[9] beginning with: 2, 5, 7, 11 (appearing twice in factorization), 13, 19, 23, 503, 607, 739, 809, ...

See also

Sources

External links

Googol plex bang stack numbers

0-stack series: googol · googolplex · googolbang · googolplexplex · googolplexbang · googolbangplex · googolbangbang · googolplexplexplex · googolplexplexbang · googolplexbangplex · googolplexbangbang · googolbangplexplex · googolbangplexbang · googolbangbangplex · googolbangbangbang
1-stack series: googolstack · googolstackplex · googolstackbang · googolstackplexplex · googolstackplexbang · googolstackbangplex · googolstackbangbang · googolplexstack · googolplexstackplex · googolplexstackbang · googolbangstack · googolbangstackplex · googolbangstackbang · googolplexplexstack · googolplexbangstack · googolbangplexstack · googolbangbangstack
2-stack series: googolstackstack · googolstackstackplex · googolstackstackbang · googolstackplexstack · googolstackbangstack · googolplexstackstack · googolbangstackstack
3-stack series: googolstackstackstack

Googol series: googol · googolplex · googolduplex · googoltriplex · googolquadriplex · googolquintiplex · googolsextiplex · googolseptiplex
Grangol series: grangol · grangoldex · grangoldudex · grangoltridex · grangolquadridex · grangolquintidex
Googoldex series: googoldex · googolplexidex · googolduplexidex · googoldexiplex · googoldexiduplex
Greagol series: greagol · greagolthrex · greagolduthrex
Gigangol series: gigangol · gigangoltetrex · gigangoldutetrex
Gorgegol–googondol series: gorgegol · gorgegolpentex · gorgegoldupentex · gulgol · gulgolhex · gulgolduhex · gaspgol · gaspgolheptex · gaspgolduheptex · ginorgol · ginorgoloctex · ginorgolduoctex · gargantuul · gargantuulennex · googondol · googondoldecex
Gugold–grinningold series: gugold · gugolda-suplex · gugolda-dusuplex · graatagold · graatagolda-sudex · graatagolda-dusudex · greegold · greegolda-suthrex · greegolda-dusuthrex · grinningold · grinningolda-sutetrex · grinningolda-dusutetrex
Golaagold–googondold series: golaagold · golaagolda-supentex · golaagolda-dusupentex · gruelohgold · gruelohgolda-suhex · gruelohgolda-dusuhex · gaspgold · gaspgolda-suheptex · gaspgolda-dusuheptex · ginorgold · ginorgolda-suoctex · ginorgolda-dusuoctex · gargantuuld · googondold
Googolgong series: googolgong · googolplexigong · googolduplexigong · googoltriplexigong · googolquadriplexigong · googolquintiplexigong
Grangolgong series: grangolgong · grangoldexigong · grangoldudexigong · grangoltridexigong · grangolquadridexigong · grangolquintidexigong
Greagolgong series: greagolgong · greagolthrexigong · greagolduthrexigong
Gigangolgong series: gigangolgong · gigangoltetrexigong · gigangoldutetrexigong
Gorgegolgong–ginorgolgong series: gorgegolgong · gorgegolpentexigong · gorgegoldupentexigong · gulgolgong · gulgolhexigong · gulgolduhexigong · gaspgolgong · gaspgolheptexigong · gaspgolduheptexigong · ginorgolgong · ginorgoloctexigong · ginorgolduoctexigong · gargantuulgong · gargantuulennexigong · gargantuulduennexigong · googondolgong · googondoldecexigong · googondoldudecexigong
Gugoldagong–grinningoldagong series: gugoldagong · gugolda-suplexigong · gugolda-dusuplexigong · graatagoldagong · graatagolda-sudexigong · graatagolda-dusudexigong · greegoldagong · greegolda-suthrexigong · greegolda-dusuthrexigong · grinningoldagong · grinningolda-sutetrexigong · grinningolda-dusutetrexigong
Golaagoldagong–ginorgoldagong series: golaagoldagong · golaagolda-supentexigong · golaagolda-dusupentexigong · gruelohgoldagong · gruelohgolda-suhexigong · gruelohgolda-dusuhexigong · gaspgoldagong · gaspgolda-suheptexigong · gaspgolda-dusuheptexigong · ginorgoldagong · ginorgolda-suoctexigong · ginorgolda-dusuoctexigong · gargantuuldagong · googondoldagong

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