Googolplex
A googolplex is the number 10googol, i.e. 10(10100), which can also be written as the number 1 followed by a googol zeros (i.e. 10100 zeros).
In pure mathematics, the magnitude of a googolplex could be related[vague] to other forms of large number notation such as tetration, Knuth's up-arrow notation, Steinhaus-Moser notation, or Conway chained arrow notation.
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 provides.
An average book of 60 cubic inches can be printed with 5×105 zeroes (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 math 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 one were to use an elementary particle to represent each digit, one would run out of particles well before reaching a googol of digits.
Consider printing the digits of a googolplex in unreadable, one-point font (0.353 mm per digit). It would take about 3.5×1096 metres 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,[1] so the distance required to write the necessary zeroes is 4.0×1069 times as long as the estimated 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 about 1.1×1082 times the age of the universe, to write a googolplex.[2]
A Planck space has a volume of a Planck length cubed, which is the smallest measurable volume, which is approximately 4.222×10−105 m3 = 4.222×10−99 cm3. Thus 2.5 cm3 contain about a googol Planck spaces. Hold your thumb and index finger about an inch apart and there will be about a googol Planck spaces between them. There are only about 3×1080 cubic metres in the observed universe, making the number of Planck spaces in the universe to be about 7.1×10184 quantum spaces in the observed universe, so a googolplex is far larger than even the number of the smallest spaces in the observed universe.
Scale
In pure mathematics
In pure mathematics,the magnitude of a googolplex could be related[vague] to other forms of large number notation such as tetration, Knuth's up-arrow notation, Steinhaus-Moser notation, or Conway chained arrow notation, though neither googol nor googolplex are anywhere near the largest representable or even specifically named numbers. For example Graham's number, though finite, is unimaginably larger than other well-known large numbers such as a googol, googolplex, and even larger than Skewes' number and Moser's number. Indeed, the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies at least one Planck volume.
In the physical universe
One googol is 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.[3] 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 to the vastly greater 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".[4] The end of the Universe via Big Freeze without proton decay is subject to be 101076 years into the future, which is still short of a googolplex.
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]
(~ googolplex) years—scale of an estimated Poincaré recurrence time for the quantum state of a hypothetical box containing a black hole with the mass within the presently visible region of our universe.[5] This time assumes a statistical model subject to Poincaré recurrence. A much simplified way of thinking about this time is in a model where our universe's history repeats itself arbitrarily many times due to properties of statistical mechanics, this is the time scale when it will first be somewhat similar (for a reasonable choice of "similar") to its current state again.
(~ googolplex) years—scale of an estimated Poincaré recurrence time for the quantum state of a hypothetical box containing a black hole with the estimated mass of the entire universe, observable or not, assuming a certain inflationary model with an inflaton whose mass is 10−6 Planck masses.[5]
If the entire volume of the observable universe (taken to be 3 × 1080 m3) were packed solid with fine dust particles about 1.5 microns in size, then the number of different ways of ordering these particles (that is, assigning the number 1 to one particle, then the number 2 to another particle, and so on until all particles are numbered) would be approximately one googolplex.
Other uses in popular culture
- The "Googleplex" is the Google company headquarters, located at 1600 Amphitheatre Parkway in Mountain View, Santa Clara County, California, near San Jose.
- The "Googolplex" theatre is where Homer takes Bart, Lisa and the Flanders family in the Simpsons episode The Ziff Who Came to Dinner. In the episode, Bart and Homer look up at the features playing and the list rises to infinity in the sky.
- The Googolplex is also the name of a mall on the Disney Channel TV series Phineas and Ferb.
- In the movie Back to the Future Part III, Dr. Emmett "Doc" Brown refers to his love interest, Clara Clayton, as being "..one in a googolplex."
- In an episode of the TV series Harvey Birdman: Attorney at Law, the title character is gathering files to be photocopied, noting, "[...]this in duplicate, this in triplicate, this in googolplexicate..."
See also
References
- ^ Lineweaver, Charles (2005). "Misconceptions about the Big Bang". Scientific American. Retrieved 6 November 2008.
{{cite web}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - ^ a b Page, Don, "How to Get a Googolplex", 3 June 2001.
- ^ 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)
- ^ a b Information Loss in Black Holes and/or Conscious Beings?, Don N. Page, Heat Kernel Techniques and Quantum Gravity (1995), S. A. Fulling (ed), p. 461. Discourses in Mathematics and its Applications, No. 4, Texas A&M University Department of Mathematics. arXiv:hep-th/9411193. ISBN 0-9630728-3-8.
External links