Re: Re: algebraic numbers

*To*: mathgroup at smc.vnet.net*Subject*: [mg106225] Re: [mg106192] Re: algebraic numbers*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Tue, 5 Jan 2010 01:48:15 -0500 (EST)*References*: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net>*Reply-to*: drmajorbob at yahoo.com

Oops, big typo that time. I meant RootApproximant, not RootApproximate! With that corrected, I find that just about EVERY random real is algebraic. roots = First@ Last@Reap@ Do[x = RootApproximant@RandomReal[]; Root == Head@x && Sow@x, {10^2}]; Length@roots 100 (It's very slow, hence the small sample.) When RootApproximant fails, I suspect it's a limitation of the algorithm, not a property of the real. Bobby On Mon, 04 Jan 2010 15:17:56 -0600, DrMajorBob <btreat1 at austin.rr.com> wrote: > Computer reals are precisely equal to, and in one-to-one correspondence > with, a miniscule subset of the rationals. Every one of them has a > finite binary expansion. > > x = RandomReal[] > digitForm = RealDigits@x; > Length@First@digitForm > rationalForm = FromDigits@digitForm > {n, d} = Through[{Numerator, Denominator}@rationalForm] > d x == n > > 0.217694 > > 16 > > 1088471616079187/5000000000000000 > > {1088471616079187, 5000000000000000} > > True > > A number can't get more rational or algebraic (solving a FIRST degree > polynomial with integer coefficients) than that. > > If computer reals are THE reals, why is it that RandomReal[{3,4}] can > never return Pi, Sqrt[11], or ANY irrational? > > OTOH, how often does RootApproximate@RandomReal[] succeed? > > Never, essentially: > > Reap@Do[x = RootApproximate@RandomReal[]; > RootApproximate =!= Head@x && Sow@x, {10^8}] > > {Null, {}} > > Bobby > > On Mon, 04 Jan 2010 05:01:55 -0600, Noqsi <jpd at noqsi.com> wrote: > >> On Jan 3, 1:37 am, DrMajorBob <btre... at austin.rr.com> wrote: >>> Mathematica Reals may not be Rational, but computer reals certainly >>> are. >>> (I shouldn't have capitalized "reals" in the second case.) >> >> Only in the shallow sense that there is a low entropy mapping between >> computer "reals" and rational numbers in the intervals they represent. >> But computer "reals" don't behave arithmetically like rationals or the >> abstract "reals" of traditional mathematics. This fact often causes >> confusion. >> > > -- DrMajorBob at yahoo.com