Googolplex

A googolplex is the large number 10^googol, or equivalently, 10¹⁰¹⁰⁰ or 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}. Written out in ordinary decimal notation, it is 1 followed by 10¹⁰⁰ zeroes; that is, a 1 followed by a googol of zeroes. Its prime factorization is 2^googol ×5^googol.

History

In 1920, Edward Kasner's nine-year-old nephew, Milton Sirotta, coined the term googol, which is 10¹⁰⁰, and 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 [be] a better mathematician than Dr. Einstein, simply because he had more endurance and could write for longer". It thus became standardized to 10^(10100) = 10¹⁰¹⁰⁰, due to the right-associativity of exponentiation.

Size

A typical book can be printed with 10⁶ zeros (around 400 pages with 50 lines per page and 50 zeros per line). Therefore, it requires 10⁹⁴ such books to print all the zeros of a googolplex (that is, printing a googol zeros). If each book had a mass of 100 grams, all of them would have a total mass of 10⁹³ kilograms. In comparison, Earth's mass is 5.972 × 10²⁴ kilograms, the mass of the Milky Way galaxy is estimated at 2.5 × 10⁴² kilograms, and the total mass of all the stars in the observable universe is estimated at 2 × 10⁵² kg.

To put this in perspective, the mass of all such books required to write out a googolplex would be vastly greater than the masses of the Milky Way and the Andromeda galaxies combined (by a factor of roughly 2.0 × 10⁵⁰), and greater than the mass of the observable universe by a factor of roughly 7 × 10³⁹.

In pure mathematics

In pure mathematics, there are several notational methods for representing large numbers by which the magnitude of a googolplex could be represented, such as tetration, hyperoperation, Knuth's up-arrow notation, Steinhaus–Moser notation, or Conway chained arrow notation.

In the physical universe

In the PBS science program Cosmos: A Personal Voyage, Episode 9: "The Lives of the Stars", astronomer and television personality Carl Sagan estimated that writing a googolplex in full decimal form (i.e., "10,000,000,000...") would be physically impossible, since doing so would require more space than is available in the known universe. Sagan gave an example that if the entire volume of the observable universe is filled with fine dust particles roughly 1.5 micrometers in size (0.0015 millimeters), then the number of different combinations in which the particles could be arranged and numbered would be about one googolplex.

Writing the number would ultimately hasten the heat death of the universe: if a person can write two digits per second, then writing a googolplex would take about 1.58×10⁹² years, which is about 1.1×10⁸² times the accepted age of the universe, and each digit written would result in an increase of entropy by the second law of thermodynamics.

10⁹⁷ is a high estimate of the elementary particles existing in the visible universe (not including dark matter), mostly photons and other massless force carriers.

Mod n

The residues (mod n) of a googolplex, starting with mod 1, are:

0, 0, 1, 0, 0, 4, 4, 0, 1, 0, 1, 4, 3, 4, 10, 0, 1, 10, 9, 0, 4, 12, 13, 16, 0, 16, 10, 4, 24, 10, 5, 0, 1, 18, 25, 28, 10, 28, 16, 0, 1, 4, 24, 12, 10, 36, 9, 16, 4, 0, ... (sequence A067007 in the OEIS)

This sequence is the same as the sequence of residues (mod n) of a googol up until the 17th position.

See also

• Graham's number
• Names of large numbers
• Orders of magnitude (numbers)
• Skewes's number

External links

• Weisstein, Eric W.. "Googolplex". http://mathworld.wolfram.com/Googolplex.html. 
• googolplex at PlanetMath.org.
• Padilla, Tony; Symonds, Ria. "Googol and Googolplex". Numberphile. Brady Haran. http://www.numberphile.com/videos/googolplex.html.