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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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