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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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