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Re: Re: Re: algebraic numbers
*To*: mathgroup at smc.vnet.net
*Subject*: [mg106264] Re: [mg106220] Re: [mg106192] Re: algebraic numbers
*From*: DrMajorBob <btreat1 at austin.rr.com>
*Date*: Wed, 6 Jan 2010 06:02:44 -0500 (EST)
*References*: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net>
*Reply-to*: drmajorbob at yahoo.com
I completely understand that Mathematica considers 1.2 Real, not
Rational... but that's a software design decision, not an objective fact.
If we consider something that's not representable in binary, it even makes
a kind of sense:
RealDigits[1/3.]
{{3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3}, 0}
But 1.2 _is_ representable in binary, that's the way it is represented in
the computer, and there's no doubt about the digits, whatsoever:
RealDigits[1.2]
{{1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 1}
Even 1/3. and Sqrt[2.] are stored as members of a countable set of
rationals... and representations of that sort are countable themselves
(since they're all algebraic).
But... I agree there's no point debating it.
Bobby
On Tue, 05 Jan 2010 16:04:20 -0600, Andrzej Kozlowski <akoz at mimuw.edu.pl>
wrote:
> 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[1]:= Element[1.2, Rationals]
>
> Out[1]= False
>
> In[2]:= Element[1.2, Reals]
>
> Out[2]= True
>
> and you might also read
>
> 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[11], 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[2] 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[2] 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
>
--
DrMajorBob at yahoo.com
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