       Re: Re: Re: algebraic numbers

• To: mathgroup at smc.vnet.net
• Subject: [mg106246] Re: [mg106220] Re: [mg106192] Re: algebraic numbers
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Wed, 6 Jan 2010 05:59:02 -0500 (EST)
• References: <200912290620.BAA02732@smc.vnet.net> <hhpl0g\$9l1\$1@smc.vnet.net> <201001050647.BAA24123@smc.vnet.net> <E44EA2F2-1274-43E8-93DE-DC5BD31884A5@mimuw.edu.pl> <op.u52ai6jwtgfoz2@bobbys-imac.local>

```Well, you are obviously misunderstanding  what I am trying to explain
but I have no desire to spend any more time on it. I give up.

Perhaps you should try to explain yourself why Mathematica gives

In:= Element[1.2, Rationals]

Out= False

In:= Element[1.2, Reals]

Out= True

http://en.wikipedia.org/wiki/Computable_number

(but that's the last time I posting anything to do with any logic or
mathematics here.)

Andrzej Kozlowski

On 5 Jan 2010, at 22:31, DrMajorBob wrote:

> RandomReal[] returns numbers from a countable set of rationals.
>
> Or call them reals, if you must; it still selects from a countable set
of possibilities... not from the uncountable unit interval in the reals.
>
> The range of RandomReal[] is a set of measure zero, just like the
algebraic numbers.
>
> Bobby
>
> On Tue, 05 Jan 2010 02:08:24 -0600, Andrzej Kozlowski
<akoz at mimuw.edu.pl> wrote:
>
>>
>> On 5 Jan 2010, at 15:47, DrMajorBob wrote:
>>
>>> If computer reals are THE reals, why is it that RandomReal[{3,4}]
can
>>> never return Pi, Sqrt, or ANY irrational?
>>
>> It can't possibly do that because these are computable real numbers
the set of computable real numbers if countable and has measure 0.
Computable numbers can never be the outcome of any distribution that
selects numbers randomly from a real interval.
>>
>> The most common mistake people make about real numbers is to think
that numbers such as Sqrt or Pi as being in some sense typical
examples of an irrational number or a transcendental number but they are
not. They are very untypical because they are computable: that is, there
exists a formula for computing as many of their digits as you like. But
we can prove that the set of all reals with this property is countable
and of measure 0. So Sqrt is a very untypical irrational and Pi a
very untypical transcendental. So what do typical real look like? Well,
I think since a "typical" real is not computable we cannot know all of
its digits and we cannot know any formula for computing them. But we can
know a finite number of these digits. So this looks to me very much like
the Mathematica concept of Real - you know a specified number of
significant digits and you know that there are infinitely many more than
you do not know. It seems to me the most natural way to think about
non-computable reals.
>>
>> Roger Penrose, by the way, is famous for arguing that our brain is
somehow able to work with non-computable quantities, although of course
not by using digital expansions. But this involves quantum physics and
has been the object of a heated dispute since the appearance of "The
emperor's New Mind".
>>
>
>
> --
> DrMajorBob at yahoo.com

```

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