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3.1415927 is really just a shorthand for 31415927/10000000, so shouldn't it count as 17 characters rather than 9?
by finnw 14y ago
3.1415927 is really just a shorthand for 31415927/10000000, so shouldn't it count as 17 characters rather than 9?
- guard-of-terra 14y agoNo it isn't. It isn't "a shorthand". It is a notation. Of those two notations, no one is "a shorthand" of another.
- beefman 14y agoI think the point is that McLoone is using two different notations and measuring an artifact of this. There's nothing special about base-10 denominators.
- guard-of-terra 14y agoHe's comparing apples to apples. Kolmogorov complexity for different ways to approximate п is more or less flat at bang for the buck.
- finnw 14y agoStrictly speaking it is not Kolmogorov complexity (That has an upper bound, namely the constant size of a program that can output arbitrarily many digits of Pi.)
- beefman 14y agoKolmogorov complexity requires a standard Turing machine to measure -- switching notations isn't allowed. Rational approximations to Pi (or any other irrational number) vary substantially in terms of accuracy/size, which is why many standard libraries include functions for computing convergents.
- marshray 14y agoPerhaps we could say he's comparing Shannon entropy per symbol for the two systems. It seems that what's special about rationals with power-of-10 (with power > 1) denominators is that we have a readily available shorthand, uh notation, for them. Notation can be very significant. The transition from Roman numberals to positional notation with 0 took a thousand or so years. But boy did it ever make long division easier!
- nandemo 14y agoStill, it is a rational approximation of pi.
- incompatible 14y agoYes, isn't saying that rational approximations of pi are useless the same as saying that all approximations of pi are useless? Or is there some meaningful irrational approximation of pi that can be made?