gladiolas wrote:I know about those algorithms which generate the number pi. But how do we know this isn't the *actual* value of the ratio of a circle's circumference to its diameter?
Some of the algorithms are derived directly from the definition of a (Euclidean) circle, and the rest of them can be proven to converge on the same number.
Klear wrote:Hmmm... this reminds me of a discussion I had with my friend once. Given some sensible mapping of numbers to letters, there is the whole Hamlet contained somewhere within pi, right? So what we were wondering is whether Hamlet is likely to be found sooner than Graham number-th digit of pi, or not. None of us had any idea how it even begin to estimate that. I just reasoned that one of the numbers must be quite extremely smaller than the other.
gladiolas' post seems to suggest that Graham's number wins by far. Is that so? I was leaning towards that option as well...
Graham's number is so much larger than anything you could possibly wrap your head around that it is very, very likely that the whole of all the text produced in the history of humanity, once encoded with some appropriate scheme, appears not once but millions of times in the first Graham's number of digits of a normal number
(which pi is believed but not proven to be).
The expected position of a subsequence N characters long in a random string from an alphabet of B characters is B^N. So, converting a normal number to base-128 and using ASCII, we can expect the 250,000 or so characters of Hamlet
to show up somewhere around the 128250000
th position. This seems like a lot, but relative to Graham's number it's basically indistinguishable from the number of thumbs I have on my left hand.
Unless stated otherwise, I do not care
whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
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