       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>

```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, 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

```

• Prev by Date: Re: Bind double-[ to keyboard shortcut
• Next by Date: Re: Re: Re: algebraic numbers
• Previous by thread: Re: Re: algebraic numbers
• Next by thread: Re: Re: Re: algebraic numbers