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Graham's Number
February 15, 2018 at 8:47 am
Recently, some of us here have been having an intense debate about infinity, and while infinity is certainly a fascinating topic about a usually counter-intuitive concept, we're so familiar with the word itself that we nowadays go "meh" when we hear about it.
So let's instead have a thread about one number that is nevertheless such a monstrosity (as Rev. described it in another thread I created recently) that you need new strange symbols to come up with a shorthand notation for it, and even then it's just crazily big that infinity itself seems so timid in comparison. And yet mathematicians have made use of it! That number is called Graham's number.
For a fun stimulating read (with lots of fun pictures), here's a Wait But Why article for those interested in mathematics, have not read/heard much about this specific number, and willing/ready to be mindblown. Just take your time to absorb what the author is saying once he reaches Graham's number, and take a break and reread that section later if you have to.
https://waitbutwhy.com/2014/11/1000000-g...umber.html
Note: If you want to start reading from the earlier numbers, feel free to read Part 1 of that article (you'll find the link right there in the first paragraph).
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RE: Graham's Number
February 15, 2018 at 9:49 am
(This post was last modified: February 15, 2018 at 9:51 am by polymath257.)
I was thinking about numbers like this on the other thread.
Another one, even larger than Graham's number, it TREE(3).
See
https://joshkerr.com/tree-3-is-a-big-num...390da86d93
Another link:
https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem
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RE: Graham's Number
February 15, 2018 at 11:14 am
Skewes Number for the win . . .
The granting of a pardon is an imputation of guilt, and the acceptance a confession of it.
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RE: Graham's Number
February 15, 2018 at 12:28 pm
(This post was last modified: February 15, 2018 at 1:17 pm by polymath257.)
(February 15, 2018 at 11:14 am)vorlon13 Wrote: Skewes Number for the win . . .
Actually, this is much, much less that Graham's number. In fact, it is less than 10^(10^10^(964))). This is much smaller than the *first* level for Graham's number
Another link for large numbers:
https://en.wikipedia.org/wiki/Large_numbers
To create *really* large numbers, use Conwway's chained arrow notation:
https://en.wikipedia.org/wiki/Conway_cha...w_notation
For example, Graham's number is between 3->3->64->2 and 3->3->65->2
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RE: Graham's Number
February 15, 2018 at 6:23 pm
TREE(!G)
"The first principle is that you must not fool yourself — and you are the easiest person to fool." - Richard P. Feynman
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RE: Graham's Number
February 15, 2018 at 6:32 pm
(This post was last modified: February 15, 2018 at 6:33 pm by polymath257.)
TREE^TREE(G!) (G!)
This could go on forever.....
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RE: Graham's Number
February 15, 2018 at 6:38 pm
(February 15, 2018 at 6:32 pm)polymath257 Wrote: TREE^TREE(G!) (G!)
This could go on forever.....
Damn TREE. Taking all the glory away from Graham's number.
Whatever, it's still mind boggling to me how huge these numbers are that you can't even use exponents to even get anywhere close to representing 0.0000000000000000000000000000000000000000000000000000000001 % of any of these numbers. Even saying "not getting anywhere close" undermines how far away it is.
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RE: Graham's Number
February 15, 2018 at 9:07 pm
(February 15, 2018 at 6:38 pm)Grandizer Wrote: (February 15, 2018 at 6:32 pm)polymath257 Wrote: TREE^TREE(G!) (G!)
This could go on forever.....
Damn TREE. Taking all the glory away from Graham's number.
Whatever, it's still mind boggling to me how huge these numbers are that you can't even use exponents to even get anywhere close to representing 0.0000000000000000000000000000000000000000000000000000000001 % of any of these numbers. Even saying "not getting anywhere close" undermines how far away it is.
You can't use exponents to get close to the logarithm of the logarithm of the logarithm of such numbers!
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RE: Graham's Number
February 16, 2018 at 8:43 am
It was definitely a wtf moment when I read that the volume of the entire universe, all 90,000,000,000 ly diameter of it, couldn't hold enough ink to print that number. And apparently that analogy is out by a very large factor.
It's completely, utterly insane.
It's amazing 'science' always seems to 'find' whatever it is funded for, and never the oppsite. Drich.
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RE: Graham's Number
February 16, 2018 at 9:13 am
(February 16, 2018 at 8:43 am)Succubus Wrote: It was definitely a wtf moment when I read that the volume of the entire universe, all 90,000,000,000 ly diameter of it, couldn't hold enough ink to print that number. And apparently that analogy is out by a very large factor.
It's completely, utterly insane.
It is actually *so* much worse than this.
Graham's number is the result of a tower of exponents of 3. So, for example,
3^3=27,
3^3^3 =3^27=7625597484987
3^3^3^3=3^7625597484987
This last (with only four 3's) is large enough that it is immeasurably more than the number of hydrogen molecules that would fit into the universe if they were all packed side-by side.
Now, Graham's number is the end result of a sequence of 64 steps, each *immeasurably* more than the previous one (much worse than comparing 1 to the number from four 3's above).
The *first* stage of this 64 stage of this process is found by constructing an exponential tower of 3's in four steps. The *third* step in this construction of the *first* stage is an exponential tower of size the number above. That is the number of 3's in the tower.
So, not only is is is not possible to write out Graham's number with enough ink to fill the universe, it isn't even possible to write out the tower of 3's in the third step of the first stage in the construction.
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