What always gets me about Graham's Number is that you can't even write down how many digits it has using the entire observable universe ... but "the last 12 digits are ...262464195387".
It's so big that going from the number itself to the number of digits (i.e. taking its logarithm) is not even something worth talking about. It's like blowing a kiss to an exploding nuclear bomb.
You just get dizzy even thinking about it. The observable universe is just not really that big when talking about this stuff, there are not even 100 orders of magnitude in the universe from smallest to biggest. Anything that you can have a conception of (even stuff like the number of possible lottery tickets or possible books that could ever be written etc) is just indistinguishable from mundane everyday sizes when dealing with googology. You're trying to roar but it comes out as a whisper. You have to explicitly design these. Most of the time if you think "but wouldn't it be comparable if we counted the XYZ) the answer is no.
It's like an adult equivalent of a kid saying "I get that galaxies are big, but are they also bigger than really big houses? How about a house with its own swimming pool? Really really big mansion of a big movie star?"
Edit: Tone is difficult in writing, I don't mean to be condescending, I'm getting dizzy myself when thinking of this and remember my frustration and amazement when first hearing about these.
Another fun property of this is that you can use it across bases. For example, how many bits would I need to represent 34812923 in binary? log2(34812923) = 25.05 -> round up to 26 bits.
Thank you for posting this. The significance of this statement was lost on me until you pointed it out. Going by MathOverflow, it's slightly more complicated than that, but it's pretty close.
And, as it always goes with the Gramham's number, all your physical metaphors for how ridiculous small metaphors go are so ridiculous small that if I try to convey the factor with a metaphor it will make almost no difference.
Or, in other words, blowing a kiss against a nuclear blow? That absolutely hugely more relevant than taking the log of the Graham's number. I can't even say how hugely more relevant it is without an almost equally irrelevant comparison.
(And yes, I am certain you know this. It's a pedantic clarification, I hope it made the point clearer.)
Yes, I agree, but your correction also doesn't make that much of a difference, even after realizing this, we're still far.
On the other hand there is just maybe a page of explanation needed for a somewhat educated layperson to understand the construction of Graham's number.
So on one hand it's unexplicably big to the point where it's even unexplicable how unexplicably big it is etc, on the other hand it can be compactly described. It's an example of the power of thought and reason and deliberate design, making so much of so little.
I swear mathematicians are witches.