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Re: Re: Re: Bug Report - Two numerical values for a same variable
*To*: mathgroup at smc.vnet.net
*Subject*: [mg54354] Re: [mg54300] Re: [mg54271] Re: Bug Report - Two numerical values for a same variable
*From*: DrBob <drbob at bigfoot.com>
*Date*: Sat, 19 Feb 2005 02:31:55 -0500 (EST)
*References*: <00ed01c512b0$2f242850$6400a8c0@Main> <curpbn$r28$1@smc.vnet.net> <200502150438.XAA29728@smc.vnet.net> <200502161936.OAA19223@smc.vnet.net> <d3d3aacf7f18939828890ce85676bd26@mimuw.edu.pl> <opsmcn3rqciz9bcq@monster> <07c9f53bde86ce72650298f7c2a6ccbc@mimuw.edu.pl> <opsmcqckt8iz9bcq@monster> <ede82a021e2cd1f6f2eb6181c05014d8@mimuw.edu.pl>
*Reply-to*: drbob at bigfoot.com
*Sender*: owner-wri-mathgroup at wolfram.com
I see your point, but it's beyond me why I should have seen it without seeing it documented or explained anywhere. Where documentation says, "You can enter a complex number in the form x + I y", it could also say, "Constructs entered in the form Complex[a,b] are meaningless unless a and b are real constants, not to include symbolic quantities such as E and Pi."
I take your point (not entirely explicit) that otherwise, every use of a Complex number would require checking its parts, with large performance implications.
Still, your explanation is unofficial and hence (so far as I know) legitimately subject to question by anyone who doesn't understand it or isn't convinced.
My apologies, if you believe otherwise.
Bobby
On Thu, 17 Feb 2005 19:38:51 +0100, Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
> *This message was transferred with a trial version of CommuniGate(tm) Pro*
> O.K. I will try to explain it very slowly. Sigh.
>
> Complex[Pi,E] does not have any meaning in Mathemaitca at all. Neither
> does Complex[Sqrt[2],Sqrt[3]] etc. Complex[a,b] only has a meaning when
> a and b are real numbers (exact or approximate).
>
> The reason, that I tried to explain was this. There is no point
> defining just Complex[Pi,E] and a few other obvious cases. Either you
> are going to allow all real numerics or you have to disallow all of
> them except those that are actually real numbers. This is the crucial
> point. If you were to allow real numeric expressions in Complex[a,b]
> then you would have to be able to determine whether an arbitrary
> numeric expression is real or not. While it is easy to do for Pi and E
> it is equally easy to produce a radical (for example), of which it is
> highly not trivial to decide if it is real or not. If you don't believe
> it I can construct one for you, but really ...
> So, in the case of such a radical m there would be no way to decide
> whether Complex[m,m] is meaningful or meaningless. Further more as you
> started manipulating such expressions things would get worse and worse.
> So, the only thing to do is to disallow all non-numbers in
> Complex[a,b]. I am therefore not just saying that Complex[Pi,E] is not
> numeric, I am saying it has no meaning in Mathematica at all, it is
> just a compound symbol. On the other hand Pi+I E is numeric but is not
> a complex number in Mathematica's sense. It is not a complex number
> just as Pi is not a number, but both are numeric. All numbers in
> Mathematica are atoms, but not all numerics are Stoms. Pi is a numeric
> but it is also a Symbol, and hence it is an Atom. On the other hand
> Pi+I E is numeric but it is not an Atom because it is not a Symbol and
> is not a number; it is an expression with Head Plus.
>
>
> Of course all of these statements refer to the design of the language.
> To me they seem perfectly logical and natural. To you presumably not.
> Well, then there is nothing else to say.
>
>
> Andrzej Kozlowski
>
>
> On 17 Feb 2005, at 18:18, DrBob wrote:
>
>> All that was very puzzling. You're saying Pi+E I is numeric but
>> Complex[Pi,E] isn't because...
>>
>> Umm...
>>
>> I've got no idea what your reasoning is.
>>
>>>> Complex[a,b] when a and b are numeric quantities, such
>>>> as Pi or E or others is not numeric
>>
>> Because... because why?
>>
>>>> Complex[a,b] where a and b are non real is meaningless
>>
>> But Pi and E _are_ real.
>>
>> Are you saying Mathematica doesn't know that?
>>
>>>> Complex[Pi,E] (unlike Pi+ I E to which it is not equal [to]
>>
>> Why aren't they equal?
>>
>> We're really back to "things fall down" to explain gravity.
>>
>> Bobby
>>
>> On Thu, 17 Feb 2005 17:50:28 +0100, Andrzej Kozlowski
>> <akoz at mimuw.edu.pl> wrote:
>>
>>> *This message was transferred with a trial version of CommuniGate(tm)
>>> Pro*
>>> On 17 Feb 2005, at 17:29, DrBob wrote:
>>>
>>>> Your "structure" argument is too vague to be useful.
>>>
>>> Whether useful or not it is true.
>>>
>>>
>>>> NumericQ[Pi + E*I]
>>>> NumericQ[Complex[Pi, E]]
>>>> True
>>>> False
>>>>
>>>> The last result, at least, seems unambiguously wrong.
>>>
>>>
>>> It is not only right but it is the only sensible possibility.
>>> In Mathematica Complex[a,b] when a and b are numeric quantities, such
>>> as Pi or E or others is not numeric but meaningless. It has to be
>>> meaningless because Complex[a,b] where a and b are non real is
>>> meaningless. However, if a and b are numeric but not numbers
>>> Mathematica would have to use FullSimplify or high precision
>>> arithmetic
>>> to determine if they are real or have non zero imaginary parts (and it
>>> may not be able to do so anyway).
>>> So until it was determined that a and b have zero imaginary parts
>>> Complex[a,b] would have to be like the Schroedinger cat that is
>>> neither
>>> dead nor alive: neither meaningful nor meaningless. That's is
>>> definitely not the way to make a computer algebra program work. So
>>> Complex[Pi,E] (unlike Pi+ I E to which it is not equal
>>>
>>> In[24]:=
>>> Complex[Pi, E] == Pi + I*E
>>>
>>> Out[24]=
>>> Complex[Pi, E] == I*E + Pi )
>>>
>>> is meaningless. Hence it is not numeric and it is you and not
>>> Mathematica that is "unambiguously wrong".
>>>
>>> Andrzej Kozlowski
>>>
>>>
>>>
>>>
>>
>>
>>
>> --
>> DrBob at bigfoot.com
>> www.eclecticdreams.net
>>
>>
>
>
>
>
--
DrBob at bigfoot.com
www.eclecticdreams.net
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