1.2 - Googol, googolplex and various extensions
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The infamous large number, coined in 1920 by Edward Kasner's 9-year-old nephew, Milton Sirotta. This is a prime example of large number, and becoming the basics of the large number study, along with googolplex. It is equal to 10^100 (or 1 followed by 100 zeroes). As well as the name Google which influenced by this name, that is the most popular search engine, to date.
History and definition
The name "googol" and "googolplex" are actually coined by Milton Sirotta. The (former) name was most likely influenced by the American comic strip, named Barney Google and Snuffy Smith. (in 1937)
The word "googolplex" (not to be confused with Googleplex) however, has a different story and definition behind it. It says the original definition of it was "One, followed by writing zeros until it gets tired". However, due to its vague and somewhat ill-defined definition, Edward Kasner, was not too pleased with that definition. As a result, he redefined that word to make it similar to googol, which is "1 followed by googol zeros" (or 10^10^100).
Number properties
As mentioned before, a googol is 10^100 or 10*10*10*10...10*10*10*10 (100 10's) or 1 followed by 100 zeros. Its full decimal expansion is:
10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
It is also equal to 10 duotrigintillion in short scale, or 10 sexdecilliard in long scale, and it is 101 digits long.
It can also be expressed in other notations, such as:
{10,100} in BEAF and Bird's array notation
E100 in scientific notation and Hyper-E notation (can also called E2#2)
s(10,100) in strong array notation
10→100 in chained arrow notation
g(ω^ω^2)(10) in slow growing hierarchy
As for googolplex, it is equal to 10^10^100, which is equal to 10^googol, or 1 followed by googol zeros. That means, the full decimal expansion of googolplex has googol+1 digits. It is impossible to write out googolplex fully in our life, as we imagined. And also, the number itself is impossible to fit the entire universe!
Googolplex can be expressed or approximated with various notations/functions.
{10,{10,100}} in BEAF and Bird's array notation (not {10,10,100} which is WAY LARGER than googolplex!!!, as we will see in the BEAF/Bird's array notation article)
EE100 in scientific notation
E100#2 or E2#3 in Hyper-E notation
s(10,s(10,100)) in strong array notation
10→(10→100) in chained arrow notation (once again, do not misinterpret this number as "10→10→100", again, much much LARGER than googolplex)
g(ω^ω^ω^2)(10) in slow growing hierarchy
Comparison of googol
For beginners, this number is already mind-blowingly huge. Since it is equal to 1 followed by 100 zeros, compared to the smaller (short scale) scale, more popular -illions, such as million, billion, trillion, quadrillion, etc., the -illions are pretty much left in the dust by googol, since even a quadrillion (in terms of short scale), is just 1 followed by 15 zeros. Googol belongs to Class 2 (between 10^6 to 10^1,000,000).
So, to get to even googol, one must raised the quadrillion to the 6th power or the 7th power as googol is between the 2 bounds (quadrillion^6 & quadrillion^7, which corresponds to 10^90 and 10^105, respectively)
For experts, it's just a tiny dot as larger googolisms (large numbers) dwarfed googol, as we will see later. (including googolplex)
Googol is also larger than even total number of elementary particles in the observable universe (known as the Eddington number), which is 136*2^256, which is only 80 digits long. So, googol is 20 orders of magnitude (yes, not times, which is a mere multiplication) larger than Eddington number!
For comparison purposes, I'll add some powers of whole numbers (from 2 to 16) closest to googol (in numerical order) as well as their boundaries.
duotrigintillion (in short scale) or googolchunk (10^99)
13^89 (approximately 1.38343371454*10^99)
8^110 = 2^330 (approximately 2.1872507248*10^99)
6^128 = 36^64 (approximately 4.0119919145*10^99)
14^87 (approximately 5.16581803314*10^99)
3^209 (approximately 5.228080143*10^99)
7^118 (approximately 5.2670655303*10^99)
2^332 = 4^166 = 16^83 (approximately 8.749*10^99)
5^143 (approximately 8.968310171*10^99)
15^85 (approximately 9.284467915*10^99)
11^96 (approximately 9.412343651*10^99)
googol (10^100)
70! (where ! means factorial) (~ 1.197857167*10^100) (=70*69*68*67...3*2*1)
9^105 = 3^210 (approximately 1.5684240429*10^100)
2^333 = 8^111 (approximately 1.7498005798*10^100)
13^90 (approximately 1.79846382889*10^100)
12^93 (approximately 2.31129767502*10^100)
4^167 = 2^334 (approximately 3.49960115965*10^100)
7^119 (approximately 3.68694587128*10^100)
googolty or googolbunch (10^101)
16^84 = 2^336 (approximately 1.39984046386*10^101)
tretrigintillion (in short scale) or septendecillion (in long scale) (10^102)
Comparison of googolplex
As for googolplex, it is equal to 10^10^100, which is definitely larger than even the largest known prime (which is 2^82589933-1, as of 2022)! Googolplex falls under Class 3 (between 10^10^6 and 10^10^10^6).
Here are the numbers and functions that is closer to googolplex (in numerical order):
duotrigintillionplex (10^10^99)
triotriacontillion (10^(3*10^99)+3)
tretriotriacontillion (10^(9*10^99)+3)
googolplex (10^10^100)
quadritriotriacontillion (10^(12*10^99)+3 = 10^(1.2*10^100)+3)
decitriotriacontillion (10^(30*10^99)+3 = 10^(3*10^100)+3)
10^10^101
centitriotriacontillion (10^(300*10^99)+3 = 10^(3*10^101)+3)
googolbang, a factorial of googol, googol! ~10^10^101.9
tretrigintillionplex or fzgoogol, googol to the power of itself (10^100)^(10^100) = (10^10^102)
(70!)!
Extending the googol and googolplex
Nowadays, many people still think that googolplex is the largest number, but it's not. Some of them extended it to the limits, but of course... it's still NOWHERE near Graham's number! That's why some of them tried to add 1 to googolplex to name it the "largest number", but it FAILED, as there are more powerful functions than just adding numbers to googolplex.
Below are the larger examples of naive extensions of googolplex:
googolplex×2 (googolplex + googolplex)
googolplex×10 (10^(10^100+1)) (adding 1 zero to googolplex)
googolplex×10^10 (10^(10^100+10))
...you know the drill...
googolplex^2 (googolplex×googolplex) (a.k.a gargoogolplex) (10^(2*10^100))
googolplex^3 (googolplex×googolplex×googolplex) (googolplex cubed) (10^(3*10^100))
googolplex^10 (googolplex to the 10th power) (10^10^101) (a lot better than multiplying it by 10)
googolplex^100 (googolplex to the 100th power) (10^10^102) (equal to fzgoogol)
...eternities later...
googolplex^googolplex (10^(10^(10^100+100)))
When raising the power of googolplex to itself, it is now larger than 10^googolplex. But these numbers will be left in the dust as we will see soon.
The -plex
Some of the googologists observed that the -plex, when applied to googol, yielding 10^googol, and some of them extend it to other non-negative numbers (negative numbers, have -minex suffix on the other hand).
example: 100-plex = 10^100 = googol
Well... can we use nested "-plexes" or extend it by using Latin + SI unit prefixes? Yes! Thus, the n-plexplex or -duplex suffix is created which means 10^10^n, but it's more convenient for the latter than the former, as it is shorter.
So, googolplexplex is also known as googolduplex (10^10^10^100), googolplexplexplex is googoltriplex (10^10^10^10^100), and so on.
Basically, n-plexplex...(where there are x "-plex"es)...plex = n-x-plex.
Click or tap here for list of googol-n-plexes (up to googol-googol-plex).
In Hyper-E notation, which was invented by Sbiis Saibian that will have its own page, googol-n-plex can be expressed as E100#(n+1). For example, googolplex is equal to E100#2. Note: In the googol-n-plex page, I expressed the numbers in terms of that notation to avoid the page from getting longer.
Conclusion
As we can see, these numbers are already beyond our understanding visually, but it is still a tiny step to learn about googology. So, for larger numbers, we will proceed into the next section, which highlights the use of -illions in our life.