system
Given the following post, respond as if you're commenting on Reddit:
human
"Googol", "Googolplex", "Graham's Number"... outside pure mathematics, is there any real significance in referring to numbers so huge that it doesn't even make sense to try to express them conventionally?
gpt
Googol, Googolplex, and tinou's order of the Monster group are completely different than Graham's number. You can't even count the digits of Graham's number.

To give you an idea, imagine the following sequence...

1. 3

2. 3^3^3 ~= 7.6 trillion

3. 3^3^3^3.... {7.6 trillion times} = !!!

4. 3^3^3^3... { as many times as the previous number } = oh god why

That's a pretty goddamn huge number. Remember that exponentiation is *left-associative*, so 3^3^3^3 is not (3^3^3)^3, but rather 3^(3^3^3).

In other words, every additional exponent has about the same *number of digits* as the *entire previous value*.

Not getting it? Remember how the value of 3^3^3 was about 7.6 trillion? Let's add one more exponent, making 3^3^3^3. The value of this number is about 3 trillion *digits long*.


 3^3^3^3^3 is so long that Wolfram|Alpha doesn't even bother telling me how long it is. There are literally not enough atoms in the universe to write out *how many digits* there are in that number. The "number of digits" itself has about 10^1600000000000 digits, and there are only ~10^80 atoms in the universe.


That was with 5 exponents. We have 7.6 trillion exponents for the third entry on the list.

How long does the list go, you ask?

The list goes on for a total of 7.6 trillion entries

After that process, you now have g1, succinctly written as 3↑↑↑↑3. One up-arrow denotes the repeated exponentiation. Two of those tiny up-arrows are responsible for the two levels of recursion. One remaining up-arrow is because 3↑3 = 3^3.

So that whole process was because of four up-arrows. Remember that.

... because g2 is 3↑↑... {g1 times} ...↑↑3.

... and g3 is 3↑↑... {g2 times} ...↑↑3.

Now repeat *that* series until you reach

... g64 is 3↑↑... {g63 times} ...↑↑3.

***THAT***, ladies and gentleman, is Graham's number.