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]
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).
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.
In other notations
In Hyper-E notation, it can be written as E100#2 or EE100.
In Conway and Guy's -illion system, it has a very long name:[4]
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
It is approximately \(57\uparrow\uparrow 3\) in arrow notation.
Aarex Tiaokhiao calls this number googolunex.[5]
Approximations in other notations
| Notation | Approximation |
|---|---|
| Up-arrow notation | \(10 \uparrow 10 \uparrow 100\) (exact) |
| Chained arrow notation | \(10 \rightarrow 10^{100}\) (exact) |
| Hyper-E notation | \(E100\#2\) (exact) |
| Hyperfactorial array notation | \((69!)!\) |
| BEAF | \(\{10,\{10,100\}\}\) (exact) |
| Fast-growing hierarchy | \(f_2(f_2(324))\) |
| Hardy hierarchy | \(H_{\omega^22}(324)\) |
| Slow-growing hierarchy | \(g_{\omega^{\omega^{\omega^2}}}(10)\) (exact) |
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]
Numbers near googolplex
Dario Alpern maintains a website listing the known factors of \(10^{10^{100}} + n\),[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. \(10^{10^{100}} + 37\) is the smallest with no known prime factors.
A few of those numbers have a huge number of known factors due to their algebraic properties. In particular, \(10^{10^{100}} + 10\) has 57445 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
- Googolplex written out
- Googolplex Written Out, a multivolume set of books which consists of 1094 volumes
Originals: googol · googolplex
Cockburn's examples: gargoogolplex · fzgoogolplex · (mega)fugagoogolplex
Googol-n-plex: googolduplex · googoltriplex · -quadriplex · -quinplex · -sextiplex · -septiplex · -octiplex · -noniplex · -deciplex · -centiplex
Googol-103n-plex: googolmilliplex · -megaplex · -gigaplex · -teraplex · -petaplex · -exaplex · -zettaplex · -yottaplex · -xennaplex · -vekaplex · -mekaplex
Bowers' extensions: giggol · gaggol · geegol · boogol · biggol · troogol · goobol · more...
Saibian's extensions: googol-minutia · googolchime · googoltoll · googolgong · grangol · greagol · gigangol · gugold · graatagold · gugolthra · throogol · godgahlah · tethrathoth · more...
Miscellany: googolbang (10100!) · googolminex (10^-10100) · googolteen · googolty · great googol(plex) · gooprol · little googol/googolbit · zootzootplex