Googolplex

Googolplex is the number $10^{\,\!10^{100}}$.

It can also be written as ${10}^{\rm {googol}}$, or

$10^{\scriptscriptstyle 10\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000}$,

or as a 1 followed by a googol (10¹⁰⁰) zeroes. Note that $10^{\,\!10^{100}}$ is the same as $10^{\,\!(10^{100})}$ because exponentiation works from the top down. The term googol was coined by Milton Sirotta, nephew of mathematician Edward Kasner. Googolplex was coined by Kasner to define an especially large number by extension from his nephew's idea.

How big is a googolplex?

A googol is greater than the number of elementary particles in the known universe, which has been variously estimated from 10⁷² up to 10⁸⁷. 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, 1-point font. TeX 1pt font is .3514598mm per digit, which means it would take about $3.5\times 10^{96}$ meters to write in one point font. The known universe is estimated at $7.4\times 10^{26}$ meters in diameter, which means the distance to write the digits would be about $4.7\times 10^{69}$ 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 $1.1\times 10^{82}$ times the age of the universe 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 black hole evaporates." [1] In a separate article [2], 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.

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,

$9^{9^{9^{9^{9^{9}}}}}$,

is much larger. This last number can be expressed more concisely as ${\ ^{6}9}$ using tetration, or $9\uparrow \uparrow 6$ using up-arrow notation.

Some sequences grow very quickly; for instance, the first two Ackermann numbers are 1 and 4, but then the third is ${\ ^{7625597484987}3}$, 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, and a busy beaver is the Turing machine of size n that can write down the biggest possible number [3]. 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 $1.29\times 10^{865}$ [4]. It is an open question whether the seventh busy beaver can express a googolplex.

In popular culture

• Googolplex is the name of multiplex cinema in the fictional town of Springfield in the television series The Simpsons.
• Google refers to its headquarters as the "Googleplex".
• In Back to the Future Part III, googolplex was used by Dr. Emmett Brown to describe his girlfriend, Clara Clayton:

  Clara is one in a million. One in a billion. One in a googolplex!

• The band Clutch released an album in 2002 called Live at the Googolplex.
• In Veronica Mars episode Look Who's Stalking, Veronica states Dick Casablancas' odds of getting to fool around with her as "a googolplex to one."
• In Douglas Adams' The Hitchhiker's Guide to the Galaxy, the Googolplex Star Thinker is a supercomputer that can "calculate the trajectory of every single dust particle throughout a five-week Dangrabad Beta sand blizzard".

See also

• Googol
• Large numbers
• Names of large numbers

External links

• Who Can Name the Bigger Number? http://www.scottaaronson.com/writings/bignumbers.html
• Comparing googolplex to numbers similar in size: http://home.earthlink.net/~mrob/pub/math/numbers-15.html#l_p1_1000e100
• The Biggest Numbers in the Universe: http://www.strangehorizons.com/2001/20010402/biggest_numbers.shtml
• Known prime factors of googolplex + n (0 <= n <= 999): http://www.alpertron.com.ar/GOOGOL.HTM
• A googolplex as a compressed file: http://selenic.com/googolplex/
• Another Googolplex page: http://www.procrastinators.org/googolplex.html
• A humorous C program to count to a googolplex: http://www.fpx.de/fp/Fun/Googolplex/
• The Challenge of Large Numbers http://www.fortunecity.com/emachines/e11/86/largeno.html
• Googolplex is "inconceivable" but still "describable": http://jimvb.home.mindspring.com/hamlet.htm
• Weisstein, Eric W. "Googolplex". MathWorld.
• googolplex at PlanetMath.