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Re: Re: Re: Re: algebraic numbers

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106139] Re: [mg106099] Re: [mg106054] Re: [mg106011] Re: [mg105989] algebraic numbers
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Sat, 2 Jan 2010 05:07:42 -0500 (EST)
  • References: <200912290620.BAA02732@smc.vnet.net> <201001011036.FAA05287@smc.vnet.net> <4558BE3F-EFFF-4990-9451-CBA3D767CF15@mimuw.edu.pl>

On 1 Jan 2010, at 20:41, Andrzej Kozlowski wrote:

>
> On 1 Jan 2010, at 19:36, DrMajorBob wrote:
>
>> "If so, you will indeed have recognized the number x as algebraic, from
>> its first N figures."
>>
>> No... you will have identified an algebraic number that agrees with x, to 
>> N figures.
>>
>> OTOH, every computer Real is rational, so they're all algebraic.
>>
>> Bobby
>
> Well, actually Mathematica does not agree with your last assertion:
>
> Head[1.12]
>
> Real
>
> Element[1.12, Rationals]
>
> False
>
> Element[1.12, Reals]
>
> True
>
> In fact it seems clear that the designers of Mathematica have decided to interpret all approximate numbers (with head Real) as approximations to irrationals rather than as finite expansions of rationals. The only rationals in Mathematica are indeed the ones that have the head Rational, i.e. fractions.
>
> Andrzej Kozlowski
>
>

I forgot to add why I think this is justified. Note that the function

RandomReal[]

0.691808

Returns an approximate number between 0 and 1. This is supposed to 
represent a real number from the interval between 0 and 1 returned by a 
random variable with uniform distribution. But clearly as the rationals 
are a set of measure 0, such a number ought always to be an irrational. 
Thus it is natural to regard all approximate reals as (approximations 
to) irrationals.
I don't think this is an issue of any real importance but I think the 
seemingly strange looking behaviour of Element is mathematically well 
justified.

Andrzej Kozlowski


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