The Magic of Googolplex

A “googol” is the number 10^100. A “googolplex” is an even larger number 10^[(10)^100]. You can find it, and other large numbers discussed in the book:”The Joy of Mathematics” by T. Pappus, published by Wide World Publ./Tetra (1989), or “Mathematical Mysteries” by Calvin C. Clawson, published by Perseus Books.

They are related to another number, actually several numbers, that bear their author’s name: “Skews” numbers. Skews(1937) = 10^{10}^10^[10]^29 (approximately), which is less than Skews(1955) = 10^[10]^(10)^1000. These, and other large numbers arise out of number theory. They are related to a function pi(n) [‘pi’ here is not = 3.1415926…] but is a function that estimates the number of prime numbers less than the number ‘n’, which is related in turn to Gauss’s Zeta function.

In 1938, prominent American mathematician 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 boxing champion Carnera at that time be a much better mathematician than Dr. Einstein, simply because he had more endurance and could write for longer.” (What a quote by justifying a mathematical number).

Kasner is perhaps best remembered today for introducing the term “googol” In or about 1938, in order to pique the interest of children, Kasner sought a name for a very large number: one followed by a hundred zeros. On a walk in the New Jersey Palisades with his nephews, Milton (1929–1980) and Edwin Sirotta, Kasner asked for their ideas. Nine-year-old Milton suggested “googol.”

In 1940, with James R. Newman, Kasner co-wrote a non-technical book surveying the field of mathematics, called Mathematics and the Imagination. It was in this book that the term “googol” was first introduced. Now, it’s currently used everywhere possible. Making it the most popular among mathematicians and the sort, even in pop culture.

How big Googolplex could be?

Just one simple googol is presumed to be greater than the number of elementary particles in the observable universe, which has been variously estimated from 1079 up to 1081. A googol is also greater than the number of Planck times elapsed since the Big Bang which is estimated at around 8 × 1060.

Since a googolplex is one followed by a googol zeroes, it would not be possible to write down or store a googolplex in decimal notation, even if all the matter in the known universe were converted into 0’s. Indeed, if you had an unlimited supply of ink and paper, you would need around 1020 times the current age of universe to fully write down a googolplex.

Thinking of this another way, consider printing the digits of a googolplex in unreadable, one-point font. TeX one-point font is .3514598 mm per digit, which means it would take about 3.5 × 1096 meters to write in one-point font. The known universe is estimated at 7.4 × 1026 meters in diameter, which means the distance to write the digits would be about 4.7 × 1069 times the diameter of the known 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 to write down a googolplex.

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”. 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.

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, 999999, is much larger. This last number can be expressed more concisely as 69 using tetration, or 9↑↑6 using Knuth’s up-arrow notation. 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 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 4.640times10^{1439}.