# Googolplex

Googolplex is the number 1010100, or 10googol.

## 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 [the boxer champion] Carnera be a better mathematician than Dr. Einstein, simply because he had more endurance".[1]

## 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 only contain 5.3 × 1087 '0's—far short of a googol of '0's. Therefore a googolplex can not be written out since a googol of '0' 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 1.1 × 1082 times the age of the universe (which is about 1.37 × 1010 years) to write a googolplex.

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, so it would take about 3.5 × 1096 meters to write a googolplex in one-point font. The visible universe is estimated to be 1.48 × 1027 meters in diameter,[2] so the distance required to write the necessary zeroes is larger than the estimated universe.

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. 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 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".[3]

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.[4]

In pure mathematics, the magnitude of a googolplex is not as large as some of the specially defined extraordinarily large numbers, such as those written with tetration, Knuth's up-arrow notation, Steinhaus-Moser notation, or Conway chained arrow notation. Even more simply, one can name numbers larger than a googolplex with fewer symbols; for example,

99999,

is much larger than a googolplex.

This last number can be expressed more concisely as 59 using tetration, or 9↑↑5 using Knuth's up-arrow notation.

Even larger still is the "googol multiplex", which was defined by Paul Doyle (University of Maryland, College Park, 1981), using Knuth's up-arrow notation as Gp↑↑Gp, where "Gp" = a googolplex.

Some sequences grow very quickly; for instance, the first two Ackermann numbers are 1 and 22 = 4; but then the third is 333, 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 have a use for.

A googolplex is a gigantic number that can be expressed compactly because of nested exponentiation. Other procedures (like tetration) can express large numbers even more compactly. The natural question is: what procedure uses the smallest number of symbols to express the biggest number? A Turing machine formalizes the notion of a procedure or algorithm, and a busy beaver is the Turing machine of size n that can write down the biggest possible number [1]. The bigger n is, the more complex the busy beaver, hence the bigger the number it can write down. For n = 1, 2, 3, 4 and 5 the numbers expressible are not huge, but research as of 2008 shows that for n = 6 the busy beaver can write down a number at least as big as 4.640 × 101439.[5]

## References

1. ^ Kasner, Edward (2001). Mathematics and the imagination. Mineola, NY: Dover Publications.
2. ^ Britt, Robert Roy, "Universe Measured: We're 156 Billion Light-years Wide!", Space.com, 24 May 2004.
3. ^ [http://arxiv.org/pdf/hep-th/9411193 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)
4. ^ Page, Don, "How to Get a Googolplex", 3 June 2001.
5. ^ Marxen, Heiner, "Busy Beaver", 27 Oct., 2008.