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At the same time that he suggested “googol” he gave a name for a still larger number: “Googolplex.” A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out. It was suggested that a googolplex should be 1, followed by writing zeros until you get tired.
—Edward Kasner, James R Newman, Mathematics and the Imagination (1940)

Googolplex is the official name of a reference large number, defined as 1010100 = 10googol or one followed by a googol (10100) zeroes.[1][2] Milton Sirotta originally defined it as "one, followed by writing zeroes until you get tired". His uncle, Edward Kasner, unsatisfied by this vague definition, redefined it to its current value.[3][4] It is 10100+1 digits long. The total number of commas in this extremely large number is about 3.33 duotrigintillon commas.

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 10 trigintillion (1094) books of 400 pages each, with 2,500 digits on each page (except for the first, which would have 2,501).

Written out in scientific notation of 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.[5]

DeepLineMadom calls the number troogolplex, and is equal to 10[3]10[3]100 in DeepLineMadom's Array Notation[6]. It should not to be confused with the much larger Bowers' troogolplex.

## Names in -illion systems

In Conway-Wechsler system,[7] googolplex is expressed in short scale as follows by using Fish's result.[8][9]

ten trilli-trestrigintatrecentilli-(32 times)-duotrigintatrecentillion

Full expansion is

ten trillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentilliduotrigintatrecentillion

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

ten tremillia^33trecentretriginmillia^32trecentretriginmillia^31trecentretriginmillia^30trecentretriginmillia^29trecentretriginmillia^28trecentretriginmillia^27trecentretriginmillia^26trecentretriginmillia^25trecentretriginmillia^24trecentretriginmillia^23trecentretriginmillia^22trecentretriginmillia^21trecentretriginmillia^20trecentretriginmillia^19trecentretriginmillia^18trecentretriginmillia^17trecentretriginmillia^16trecentretriginmillia^15trecentretriginmillia^14trecentretriginmillia^13trecentretriginmillia^12trecentretriginmillia^11trecentretriginmillia^10trecentretriginmillia^9trecentretriginmillia^8trecentretriginmillia^7trecentretriginmillia^6trecentretriginmillia^5trecentretriginmillia^4trecentretriginmillia^3trecentretriginmillia^2trecentretriginmilliatrecenduotrigintillion

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-tretrigintitrecentiveco-tretrigintitrecentixono-tretrigintitrecentiyocto-tretrigintitrecentizepto-tretrigintitrecentiatto-tretrigintitrecentifemto-tretrigintitrecentipico-tretrigintitrecentinano-tretrigintitrecentimicro-tretrigintitrecentimilli-doetrigintitrecentillion

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 $$\textrm{E}2\#3, \textrm{E}100\#2$$
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 $$\textrm{s}(10,\textrm{s}(10,100))$$
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 $$g_{\omega^{\omega^{\omega^2}}}(10)$$
CatPrinceHQ's Letter Notation (intended) 10czoyfmseqjdhsonafncmxadaklgkdhnayvnpgpxsdpjjtzzarbfwczrpoqqlvihoxwyaroe

## Examples

A googolplex is too large in comparison to anything computable in the observable universe. As written in the article of Universe size, if we do not limit our discussion in the observable universe, some numbers appear that exceed googolplex. For example, Linde and Vanchurin (2010) estimated possible number of universes in the multiverse as $$10^{10^{10^7}}$$.[10]

Antonio Padilla defined doppelgängion = $$10^{10^{68}}$$ as the largest number of microstates you will ever need to describe a cubic metre of space and clarified that "You're better than one in a googol, but you're not one in a googolplex. The best any of us could be is one in a doppelgängion."[11]

## In popular culture

The company Google named their Mountain View headquarters "The Googleplex."[12]

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.[13]

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

The Japanese trading card game デュエルマスターズ has the card "無量大龍グーゴルプレックス" named after 無量大数 and googolplex[14].

## Numbers near googolplex

Dario Alpern maintains a website listing the known factors of 1010100+ n,[15] 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,[16] beginning with: 2, 5, 7, 11 (appearing twice in factorization), 13, 19, 23, 503, 607, 739, 809, ...

## Sources

1. Edward Kasner, James Roy Newman. Mathematics and the Imagination Originally published by Simon and Shuster, 1940. Dover Edition published in 2001. ISBN 978-1556151040 p.23
2. Conway and Guy. The Book of Numbers. Copernicus. 1995. ISBN 978-0387979939 p.16
3. Googolplex from Wolfram MathWorld
4. Googol and googolplex
5. DeepLineMadom's googology - Numbers I've coined (Retrieved 4 May 2022)
6. Conway and Guy (1995) "The book of Numbers" Copernicus. pp.14-15.
7. Fish Illion name of 10^10^x in Conway-Wechsler system 2021-12-25
8. A. Linde and V. Vanchurin. (2010) How many universes are in the multiverse? Phys. Rev. D 81, 083525. Preprint.
9. Antonio Padilla (2023) "Fantastic numbers and where to find them: A journey to the edge of physics" Penguin Books. ISBN 978-0141992822 p.56