Googolplex
Template:Two other uses A googolplex is the number 10googol, which can also be written as the number 1 followed by a googol of zeros (i.e., 10100 zeros).
- 1 googolplex
- = 10googol
- = 10(10100)
- = 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
- = 1.e+100
History
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 Carnera be a better mathematician than Dr. Einstein, simply because he had more endurance and could write for longer.".[1] It thus became standardized to 10googol
Size
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 numerals (i.e., "10,000,000,000...") would be physically impossible, since doing so would require more space than the known universe occupies.
An average book of 60 cubic inches can be printed with 5 x 105 '0's (5 characters per word, 10 words per line, 25 lines per page, 400 pages), or 8.3 × 103 '0's per cubic inch. The observable (i.e. past light cone) universe contains 6 × 1083 cubic inches (1.3 × π × (14 × 109 light year in inches)3). This implies that if the universe is stuffed with paper printed with '0's, it could contain only 5.3 × 1087 '0's—far short of a googol of '0's. In fact there are only about 2.5 × 1089 elementary particles in the observable universe so even if you used an elementary particle to represent each digit you still would have to make the universe's mass about a trillion times larger. Therefore a googolplex can not be written out since a googol of '0's can not fit into the observable 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 about 1.51 × 1092 years, which is 1.1 × 1082 times the age of the universe, to write a googolplex. [2]
Thinking of this another way, consider printing the digits of a googolplex in unreadable, one-point font. TeX one-point font is 0.35145989 mm per digit,[3] so it would take about 3.5 × 1096 meters to write a googolplex in one-point font. The observable universe is estimated to be 8.80 × 1026 meters, or 93 billion light-years, in diameter,[4] so the distance required to write the necessary zeroes is longer than the estimated universe; however, text wrapping would make this task possible.
One googol is also presumed to be greater than the number of hydrogen atoms in the observable universe, which has been variously estimated to be between 1079 and 1081.[5] A googol is also greater than the number of Planck times elapsed since the Big Bang, which is estimated at about 8 × 1060.[citation needed]
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".[6]
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.[2]
In pure mathematics, the magnitude of a googolplex could be related to other forms of large number notation such as tetration, Knuth's up-arrow notation, Steinhaus-Moser notation, or Conway chained arrow notation.
Some sequences grow very quickly; for instance, the first two Ackermann numbers are 1 and 22 = 4; but then the third is 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 ever talk about.
See also
References
- ^ Kasner, Edward (2001). Mathematics and the imagination. Mineola, NY: Dover Publications.
- ^ a b Page, Don, "How to Get a Googolplex", 3 June 2001.
- ^ Metric typographic units 23 February 2003.
- ^ Lineweaver, Charles (2005). "Misconceptions about the Big Bang". Scientific American. Retrieved 2008-11-06.
{{cite web}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - ^ Mass, Size, and Density of the Universe Article from National Solar Observatory, 21 May 2001.
- ^ Page, Don N., "Information Loss in Black Holes and/or Conscious Beings?", 25 Nov. 1994, for publication in Heat Kernel Techniques and Quantum Gravity, S. A. Fulling, ed. (Discourses in Mathematics and Its Applications, No. 4, Texas A&M University, Department of Mathematics, College Station, Texas, 1995)