In 1938, Edward Kasner's nine year old nephew, Milton Sirotta, coined the term googol which is 10100, 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).
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 standard 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.
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 zeros per cubic inch. The observable (i.e. past-facing light-cone) universe contains 6×1083 cubic inches (4/3 × π × (14×109 light years in inches)3). Therefore if the universe were stuffed with paper that were printed with zeros, then the universe could contain only 5.3×1087 zeros — far short of a googol of zeros. With only about 2.5×1089 elementary particles in the observable universe, even if an elementary particle represented each digit, far too few particles would be needed to represent a googolplex.
Printing digits of a googolplex would be unreadable, one-point font (0.353 mm per digit). Writing a googolplex in that font would take about 3.5×1097 meters. The observable universe is estimated to be 8.80×1026 metres, or 93 billion light-years, in diameter,; the distance required to write the necessary zeroes therefore is 4.0×1069 times as long as the estimated universe.
Writing the number takes too long: if a person can write two digits per second, then writing a googolplex would take around about 1.51×1092 years, which is about 1.1×1082 times the age of the universe.
A Planck space has a volume of a Planck length cubed, which is the smallest measurable volume, at approximately 4.222×10−105 m3 = 4.222×10−99 cm3. Therefore 2.5 cm3 contain about a googol Planck spaces. Only about 3×1080 cubic metres exist in the observable universe, giving about 7.1×10184 Planck spaces in the entire observable universe; therefore, a googolplex is far larger than even the number of the smallest measurable spaces in the observable universe.
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, Knuth's up-arrow notation, Steinhaus-Moser notation, or Conway chained arrow notation.
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. A googol is also greater than the number of Planck times elapsed since the Big Bang, which is estimated at about 8×1060. 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 1076.96 measurements to give a rough determination of the final density matrix after a black hole evaporates". The end of the Universe via Big Freeze without proton decay is expected to be around 101075 years into the future, which is still short of a googolplex.
If the entire volume of the observable universe (taken to be 3 × 1080 m3) were packed solid with fine dust particles about 1.5 micrometres 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.[clarification needed]
- Edward Kasner & James R. Newman (1940) Mathematics and the Imagination, page 23, NY: Simon & Schuster
- Lineweaver, Charles; Tamara M. Davis (2005). "Misconceptions about the Big Bang". Scientific American. Retrieved 2008-11-06.
- 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.
- convert age of the universe to Planck times – Wolfram|Alpha, 8 August 2011
- 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)
- Weisstein, Eric W., "Googolplex", MathWorld.
- googolplex at PlanetMath
- Padilla, Tony; Symonds, Ria. "Googol and Googolplex". Numberphile. Brady Haran.
- Googolplex written out