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# Googolplex

[goo-gawl-pleks, -gol-, -guhl-]

A googolplex is the number 10googol, which means it's a 1 followed by a googol of zeros (i.e. 10100 zeros).

1 googolplex
= 10googol
= $10^\left\{10^\left\{100\right\}\right\},!$
= 1010,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
In the documentary Cosmos: A Personal Voyage, astrophysicist and broadcast personality Carl Sagan estimated that writing a googolplex in numerals (i.e., "10,000,000,000...") would be physically impossible, since doing so would require more space than the known universe occupies.

In 1938, 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 [the boxer champion] Carnera be a better mathematician than Dr. Einstein, simply because he had more endurance".

## How big is a googolplex?

One googol is presumed to be greater than the number of elementary particles in the observable universe, which has been variously estimated from 1079 up to 1081. A googol is also greater than the number of Planck times elapsed since the Big Bang which is estimated at around 8 × 1060.

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 0's. Indeed, if you had an unlimited supply of ink and paper, you would need around 1020 times the current age of universe to fully write down a googolplex.

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 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 101076.96 measurements to give a rough determination of the final density matrix after a black hole evaporates". 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.

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,

999999, 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 gigantic 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 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 2008 shows that for n=6 the busy beaver can write down a number at least as big as $4.640times10^\left\{1439\right\}$.

• 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
• 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