"Existential Angst" <email@example.com> wrote in message news:firstname.lastname@example.org... > "Clark Smith" <email@example.com> wrote in message > news:firstname.lastname@example.org... >> On Mon, 26 Nov 2012 15:08:17 -0500, Existential Angst wrote: >> >>> Would be the digits of e, pi, et al? >>> If that's the case, no need for fancy pyooter algorithms? >>> >>> Inneresting article on pi, randomness, chaos. >>> http://www.lbl.gov/Science-Articles/Archive/pi-random.html >> >> Is it not the case that the digits of e, pi et al. can't strictly >> be random, if it is only because they are highly compressible? I.e. >> because there small, compact formulas that spit out as many digits as you >> want in a completely deterministic way? > > Deterministic?? y = mx + b is deterministic..... Any curve you can > graph is deterministic, but I think Bailey and Crandall would certainly > not use the word dterministic here, as in "predictive". Yeah, the formula > or whatever "determines" the next digit, but the user of the formula > doesn't know what that next digit will be, formula or no formula. To wit: > > "This result derives directly from the discovery of an ingenious formula > for pi that Bailey, together with Canadian mathematicians Peter Borwein > and Simon Plouffe, found with a computer program in 1996. Named the BBP > formula for its authors, it has the remarkable property that it permits > one to calculate an arbitrary digit in the binary expansion of pi without > needing to calculate any of the preceding digits. Prior to 1996, > mathematicians did not believe this could be done."
Oh, the very excellent wiki article http://en.wikipedia.org/wiki/Pi shows this formula under the section Spigot Algorithm.... super-neat. Note that i in this case is not the complex i, but simply the series counter.
Some very inneresting observations on pi, its calculation, the practical limit in accuracy (39 digits). The section on memorizing pi is an eye-opener. Note also the youtube of some guy who apparently verifiably memorized pi to 20,000 digits, altho that is far short of the guiness record of 68,000 digits. Hard to even comprehend that, altho the article does explain some of the techniques. -- EA
> > Which, apropos of your point, is an even worse scenario, formula-wise, for > randomness, yet Bailey/Crandall don't think "formula-ization" of > randomness precludes true randomness. > > "The digit-calculation algorithm of the BBP formula yields just the kind > of chaotic sequences described in Hypothesis A. Says Bailey, "These > constant formulas give rise to sequences that we conjecture are uniformly > distributed between 0 and 1 -- and if so, the constants are normal." > > In addition, pi et al meet "casino-type" tests of randomness, which of > course are not proofs of randomness. > > > My point was: > Even random number generators can be suspect, from what I read some time > ago. I just thought it mildly interesting -- esp in light of Hypothesis > A -- that if true randomness is *intrinsic* to the mathematical fabric of > irrationals like e, pi etc, then generators are semi-moot, from a true > "need" pov. But still innersting and perhaps important from a "how do > they do it" pov. > > This randomness thing may be a kind of consequence of going from analog to > digital, ie, altho you can graph y = sinx, AND you can graph random > numbers, one is predictive while the other is essentially an ad-hoc > descriptive, with no notions of derivatives or integrals applying to the > curve whatsoever -- except for the trivial case of a random plot (y = the > random value, x = the trial "count"), that dYavg/dX = 0. > > Anyway, I always thought e, pi et al were random sequences. Apparently > others do, as well. > But proly it will never be proven, one way or the other. > -- > EA >