Re: Re: algebraic numbers
- To: mathgroup at smc.vnet.net
- Subject: [mg106220] Re: [mg106192] Re: algebraic numbers
- From: DrMajorBob <btreat1 at austin.rr.com>
- Date: Tue, 5 Jan 2010 01:47:15 -0500 (EST)
- References: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net>
- Reply-to: drmajorbob at yahoo.com
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
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