No, Graham’s number is much larger than a googolplex.
So let us take a deeper look
Graham’s number is an enormous number that was introduced by the mathematician Ronald Graham in 1971. It is so large that it is practically impossible to write it in standard notation. In fact, if you were to write every single digit of Graham’s number, the resulting number would be too large to fit into the observable universe!
To answer the question at hand, Graham’s number is not equal to a googolplex. In fact, Graham’s number is much larger than a googolplex. A googolplex is equal to ten to the power of a googol, which itself is ten to the power of 100. In comparison, Graham’s number is so much larger that it is difficult to even conceptualize.
According to mathematician Clifford Pickover, “Graham’s number so utterly dwarfs the total number of particles in the observable universe that it may as well be infinite for all practical purposes.” Here are some other interesting facts about Graham’s number:
- Graham’s number is actually the upper bound of a problem in Ramsey theory, a branch of mathematics that deals with patterns and structure in complex systems.
- Graham’s number is so large, it was considered the largest number ever used in a mathematical proof until 2018.
- The number of digits in Graham’s number is so large that if you wrote one digit on each proton in the observable universe, you still wouldn’t have enough protons to write all of them down.
- Graham’s number is sometimes used in number theory and computer science as a benchmark for testing the performance of algorithms and supercomputers.
While the concept of Graham’s number may seem mind-boggling to many, it serves an important purpose in mathematics and computer science. As mathematician John L. Casti notes, “While Graham’s number may seem like an amusing intellectual oddity, it serves to illustrate the essentially unbounded nature of pure mathematics, and the seemingly infinite distance between even the most abstract concepts and the world of everyday experience.”
Here is a table showcasing the relative size of Graham’s number compared to other well-known large numbers:
| Number | Size |
|---|---|
| Googol | 10^100 |
| Googolplex | 10^(10^100) |
| Graham’s number | f_f_64(3, 3, 64) where f is a recursive function called Knuth’s up-arrow notation |
As you can see, Graham’s number is in a league of its own when it comes to size.
Associated video
Mathematicians Tony Padilla and Matt Parker discuss arrow notation, used to represent very large numbers and particularly in combinatorics problems. They discuss the concept of Graham’s number and its development as the maximum possible number of people needed to be in committees with certain conditions on connections, using arrow notation to show how the number increases and its scale, which lies between 6 and Graham’s number. Despite being smaller than infinity, which is currently used in mathematical proofs, Graham’s number is shockingly large, with only its last 500 digits known and its first digit unknown. The video ends with an interesting anecdote about Graham, who was a mathematician and circus performer.
I found further information on the Internet
Graham’s number is vastly larger than one googolplex and even than a googolplex googolplexes.
Graham’s number is also bigger than a googolplex, which Milton initially defined as a 1, followed by writing zeroes until you get tired, but is now commonly accepted to be 10 googol =10 (10100). A googleplex is significantly larger than the 48th Mersenne prime.
A Googolplex is defined as 10 Googol. A Googolplexian is defined as 10 Googolplex. Intuitively, it seems to me that Graham’s number is larger (maybe because of it’s complex definition).
HAHAHAHAHAHAHAHA!!!
Not even close. Not even remotely close. Not even funny, how not even close these two numbers are. They aren’t in the same ballpark, the same world, the same universe, the same multi-universe.
Googolplexian is “ten to the power googolplex”, while googolplex is “ten to the power googol”, and googol is “ten to the power one hundred”. In other words, googol is 1 followed by a hundred zeroes, googolplex is 1 followed by a googol zeroes, and googolplexian is 1 followed by a googolplex of zeroes.
Large?
That’s nothing. Less than microscopic. Negligible. Peanuts, smashed to bits with a sledgehammer and then crushed in a particle accelerator. A single quark, compared with a googolplexian universes, is still far from representing just how tiny googolplexian is compared to Graham’s number.
Seriously, I’m not exaggerating.
Look:
[math]\displaystyle \text{googol} = 10^{100} %3C \left(3^3
ight)^{100} = 3^{300} %3C 3^{3^{3^3}} = 3\uparrow\!\uparrow 4[/math]The mighty goo…
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Written out in ordinary decimal notation, it is 1 followed by 10100 zeroes; that is, a 1 followed by a googol of zeroes.