Googolplex

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

1 googolplex

= 10^googol

= $10^{10^{100}}\,\!$

= 10^{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}

In the PBS science show Cosmos: A Personal Voyage, Episode 9: "The Lives of the Stars", astrophysicist and broadcast personality Carl Sagan estimated that writing a googolplex in numerals (i.e., "10,000,000,000...") would be physically improbable and impractical, 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 0s. Indeed, if you had an unlimited supply of ink and paper, you would need around 10²⁰ times the current age of the 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 .35145989 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 10^1076.96 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.

"A googolplex is precisely as far from infinity as is the number one." — Carl Sagan, Cosmos

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 [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 2008 shows that for n=6 the busy beaver can write down a number at least as big as $4.640\times 10^{1439}$. [2]

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
• 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

See also

• Large numbers
• Names of large numbers

External links

• Weisstein, Eric W. "Googolplex". MathWorld.
• googolplex at PlanetMath.