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A googolplex is the number 10^googol, i.e. \( 10^{(10^{100})} \). 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.

History

In 1938, Edward Kasner's nine-year-old nephew, Milton Sirotta, coined the term googol, 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 \( 10^{(10^{100})} \).
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 standard form (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 zeros per cubic inch. The observable (i.e. past light cone) universe contains 6×10⁸³ cubic inches (⁴/₃ × π × (14×10⁹ light years 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 zeros—far short of a googol of zeros. 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,^[2] 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.^[3]

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. Thus 2.5 cm³ contain about a googol Planck spaces. There are only about 3×10⁸⁰ cubic metres in the observable universe, giving about 7.1×10¹⁸⁴ Planck spaces in the entire observable universe, so a googolplex is far larger than even the number of the smallest measurable spaces in the observable 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.^[4] A googol is also greater than the number of Planck times elapsed since the Big Bang, which is estimated at about 8×10⁶⁰.^[5] 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 \( 10^{(10^{76.96})} \) measurements to give a rough determination of the final density matrix after a black hole evaporates".[6] The end of the Universe via Big Freeze without proton decay is subject to be \( 10^{(10^{76})} \) 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.[3]

\( 10^{(10^{(10^{(10^{2.08})})})}\) years (~ googolplex\( ^{60.1132217} \) 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.[7] 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.

\( 10^{(10^{(10^{(10^{(10^{1.1})})})})}\) years (~ googolplex\( ^{1,941,887,670,000} \) 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.^[7]

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.

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Googolplex