Googolplex

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Googolplex is the number 10^10¹⁰⁰. lt can also be written as 10^googol, or as a one followed by a googol zeros.

1 Etymology
2 How big is a googolplex?
3 In popular culture
4 See also
5 External links
6 References

Etymology

In about 1920, Edward Kasner's nine-year-old nephew Milton Sirotta coined the term googol; Milton then proposed the further term googolplex to be "one, followed by writing zeroes until you get tired". Kasner decided to adopt a more formal definition "because different people get tired at different times and it would never do to have Carnera [a champion boxer] be a better mathematician than Dr. Einstein, simply because he had more endurance".^[1]

How big is a googolplex?

One googol is greater than the number of elementary particles in the observable universe, which has been variously estimated from 10⁷⁹ up to 10⁸⁵. Since a googolplex is one followed by a googol zeroes, it would not be possible to write down or store a googolplex in decimal notation, even if all the matter in the known universe were converted into paper and ink or disk drives. Thinking of this another way, consider printing the digits of a googolplex in unreadable, one-point font. TeX one-point font is .3514598 mm per digit, which means it would take about 3.5 × 10⁹⁶ meters to write in one-point font. The known universe is estimated at 7.4 × 10²⁶ meters in diameter, which means the distance to write the digits would be about 4.7 × 10⁶⁹ times the diameter of the known universe. The time it would take to write such a number also renders the task implausible: if a person can write two digits per second, it would take around 1.1 × 10⁸² billion years to write down a googolplex. Thus in the physical world it is difficult to give examples of numbers that compare closely to a googolplex. In analyzing quantum states and black holes, physicist Don Page writes that "determining experimentally whether or not information is lost down black holes of solar mass ... would require more than 10^{10^76.96} measurements to give a rough determination of the final density matrix after a black hole evaporates".^[2] In a separate article, Page shows that the number of states in a black hole with a mass roughly equivalent to the Andromeda Galaxy is in the range of a googolplex.^[3] In pure mathematics, the magnitude of a googolplex is not as large as some of the specially defined extraordinarily large numbers, such as those written with tetration, Knuth's up-arrow notation, Steinhaus-Moser notation, or Conway chained arrow notation. Even more simply, one can name numbers larger than a googolplex with fewer symbols, for example,

<math>9^{9^{9^{9^{9^9}}}}</math>,

is much larger. This last number can be expressed more concisely as ⁶9 using tetration, or 9⇈6 using Knuth's up-arrow notation. Some sequences grow very quickly; for instance, the first two Ackermann numbers are 1 and 2²=4; but then the third is ^³33, a power tower of threes more than seven trillion high. Yet, much larger still is Graham's number, perhaps the largest natural number mathematicians actually have a use for. A googolplex is a huge number that can be expressed compactly because of nested exponentiation. Other procedures (like tetration) can express large numbers even more compactly. The natural question is: what procedure uses the smallest number of symbols to express the biggest number? A Turing machine formalizes the notion of a procedure or algorithm, and a busy beaver is the Turing machine of size n that can write down the biggest possible number [1]. The bigger n is, the more complex the busy beaver, hence the bigger the number it can write down. For n=1, 2, 3, 4 and 5 the numbers expressible are not huge, but research as of 2006 shows that for n=6 the busy beaver can write down a number at least as big as <math>1.29\times10^{865}</math>. [2] It is an open question whether the seventh busy beaver can express a googolplex.

In popular culture

Googolplex (an extremely large movie theater) appears in the The Ziff Who Came to Dinner episode of The Simpsons, broadcast on 14 March, 2004, and in the video game The Simpsons Hit & Run
Google (a deliberate misspelling) refers to its headquarters as the Googleplex.
The Googleplex Star Thinker in the Seventh Galaxy of Light and Ingenuity appears (an earlier "playful" misspelling) in the Douglas Adams 1979 novel, The Hitchhiker's Guide To The Galaxy.
The band Clutch released an album in 2002 called Live at the Googolplex.
In episode IX of Carl Sagan's Cosmos: A Personal Voyage television series, Sagan, illustrating the advantages of scientific notation over decimal notation, starts writing a googolplex on a roll of paper. He unwinds the roll of paper throughout Cambridge University's campus, before finally giving up.
In Back to the Future Part III, Emmett Brown, in lamenting the loss of the girl of his dreams, says, "Clara was one in a million. One in a billion. One in a googol-plex."
Googol and Googolplex are also several times mentioned by the protagonist, the 9 year-old Oskar, of Jonathan Safran Foer's novel "Extremely Loud and Incredibly Close".
In the Loonatics Unleashed Season 1 Episode "The Comet Cometh" Tech creates a device which amplifies Ace's laster vision by "a full factor of google." Rev begins to explain what it is only to be interrupted by Danger Duck in boredom.