A Grahamic Number: Graham's Number

This probably seems more or less out of place, but have you ever heard of Graham's number?  It's huge, but that's an understatement.  It's big enough of a number that there's a somewhat infamous (at least among anyone who has tried to write or talk about it) linguistic problem that occurs whenever it pops up.  This problem is pretty simple; it's just really hard to explain how big it is.  People like to use 'unimaginable' with it a lot, but the thing is, most people can't really imagine comparatively insignificant numbers in exactitude, either.  Can you picture a million of anything?  Probably not accurately.  So anyway, there's a pretty easy solution to this.  Just call it "Grahamic," with slight explanation, and you're good to go.  There's no way to explain how big the number really is without outlining the impossibilities in calculating it in its entirety, or explaining how it works, of course.  But "Grahamic" is the only way to refer to the size of Graham's Number.  Using this colloquially would never fail to get me a laugh, like "Ah, man, I just took a Grahamic shit," or "Dude, your dad is a Grahamic fatass."

Interestingly enough, unless you defined the word as 'describing anything that is so extensive in size or other measurement that it is inherently difficult to explain,' there really isn't anything besides Graham's number itself that you could literally call Grahamic.  I suppose you could say that numbers close to Graham's number are Grahamic, but there isn't really much reason to talk about those numbers anyway.  And every bigger number of any importance is so much bigger than even Graham's number that it puts our newly formed adjective to shame.  Nah, it puts putting the number to shame to shame, and that's an understatement of an understatement.

But I haven't even explained the thing yet, so it doesn't quite do to start talking about even more massive numbers.  Graham's number isn't the hardest concept to grasp, even if its application might be.  So, anyway, the number's sheer size makes it necessary to talk about it with extensive notations.  There is a system known as Knuth's up-arrow notation, which is often used for this purpose.  I'd like to explain it concisely, but since I have no idea who will read this, I'll have to explain some other things first.  This is going to sound pretty stupid, but to explain Knuth's up-arrow notation by explaining addition.

So what is addition?  Addition is an iteration of the successor function, of course!  Wait, what?  Okay, so this might sound kind of out there to anyone who isn't already familiar with this, but it's simple.  The successor function adds 1 to whatever number is applied to the function, in layman's terms.  It's S(n) = n+1.  How does this apply to Knuth's up-arrow notation?  We'll get there.  See, addition is an iteration of the successor function (that is, it is iterated zeration), in that a + b = S(S(S(...S(a)...))), with b copies of S.  All that means is that, say, 6 + 3 = S(S(S(6))).  Not hard to grasp.  And you surely already know that multiplication is iterated addition.  6 x 3 = 6 + 6 + 6.  And if you've ever done any algebra, you'll know that exponentiation is iterated multiplication, so 6 ^ 3 = 6 x 6 x 6.  Many people aren't aware that this continues.  Tetration is iterated exponentiation, and pentation is iterated tetration.  

Now, Knuth's notation uses arrows.  Lots of them.  But it uses them in a simple way.  Saying a↑b is the same as saying a^b.  Saying a↑↑b is the same as tetrating b by a.  ↑↑↑ is pentation, and so on.  With arrows beyond a small number, you can just put the number of arrows in a superscript above the arrow (unfortunately I cannot figure how to type superscripts in this editor, so I'll just write it out), so a↑(superscript 4)b would be hexation.  Now, I can tell you how Graham's number is written in this notation.

I screencapped this from WP because I'm lazy, but whatever.  You can ignore most of this, I'll explain how to form the number.  g subscript 1 is the starting term here, and it is 3 hexated to itself.  g sub 1 is a really, really, really massive number in itself.  The number of digits in g sub 1 alone is in the trillions.  So, G, which is Graham's number, is equal to g sub 64.  g sub 2 is 3↑↑↑...↑3, with g sub 1 arrows, and g sub 3 is 3↑↑↑...↑3 with g sub 2 arrows.  That's how fucking massive this thing gets.  G is g sub sixty-four.  That's 64 iterations of this crazy explosive mutation of a number scheme, when g sub 1 is already huge as hell.

In fact, Graham's number, written out normally, can't even fit in the universe, even with the smallest possible writing(even if one digit took up the space of a Planck volume).  In fact, the number of digits of Graham's number can't fit in the universe.  Or the number of digits of that number, or the number of digits of that one.  This repeats so many times that the number of digits of the number of repetitions cannot fit inside the universe.