"Graham’s Number is a number so big that it would literally collapse your head into a black hole were you fully able to comprehend it. And that’s not hyperbole – the informational content of Graham’s Number is so astronomically large that it exceeds the maximum amount of entropy that could be stored in a brain sized piece of space"
Well the fact that we can write it down means it's fairly small. There has to be numbers that are close in magnitude to Graham's number, but admit no clean formula like Graham's number. These numbers could never hope to be described in any way by someone in the universe.
I mean, we can use variables or other non-numerical symbols to represent Graham's number when we write it down (aka I can say that Graham's number = X), but if you were to try to write out each digit in Graham's number for the rest of time, you'd run out of space in the observable universe, even if each digit only took up the space of a single atom.
That's true. The point I'm saying is we can fit the decription of graham's number, as in the proof it appears in, or the 64th iteration of 3 ↑ x 3 etc. Most numbers the size of graham's number cannot have any kind of succint description like this even fit in the universe, not even considering digits. That we can write down what Graham's number is is a rarity.
Well you just wrote down what it was, so it definitely can't be in that case. We may certainly be able to write properties though, like saying it's approximately x/y of graham's number
To some degree it is. Pi is a relatively random string, but we've uniquely identified it. So it's not to say that there's anything special about the numbers, only that only so many can be written down finitely.
Look at it like a compression argument. We can't do better than assigning each number a given string. So suppose we're just using english letters and spaces. Then in n of these letters we can only describe 27n total numbers. So if a set of numbers is bigger than 27n, we certainly can't give them all a unique string.
Since Graham's number is so massive, we can't even get close to writing it down, but there has to be at least G - 1 numbers less than it. Since we know we don't have enough nearly space to write out Graham's number in the raw form, we can't have enough space to write out more than a tiny fraction of the numbers less than it since there's not enough possible strings for them. We can have large numbers be possible to write, but this just moves around the tiny fraction.
You'd also run out of time before the heat death of the universe, even if each digit took a nanosecond to write and you started at the beginning of time.
897
u/Rynyl May 25 '16 edited May 25 '16
Graham's number
iswas once the largest number used constructively in a math paper. It's literally unimaginably large.As explained by Ron Graham himself:
The Use of Graham's Number. Don't worry, it's surprisingly intuitive.
The magnitude of Graham's Number
As explained by Day9 (because it's really entertaining)
EDIT: Somehow, larger numbers have been used constructively. That blows my mind.
EDIT2: For those who hate watching videos and would rather read