Re: "Assuming"
- To: mathgroup at smc.vnet.net
- Subject: [mg85620] Re: "Assuming"
- From: dh <dh at metrohm.ch>
- Date: Fri, 15 Feb 2008 06:51:56 -0500 (EST)
- References: <fp0m11$5u7$1@smc.vnet.net> <fp192b$gju$1@smc.vnet.net> <fp2khg$pka$1@smc.vnet.net>
Hi David,
we all learned not to divide by zero, but things are a bit more subtle
than this.
You claim that (a^2 - 1)/(a - 1) should simplify to If[a == 1,
Indeterminate, a + 1] and not to (a+1), but you do not give a good
reason for this.
Obviously (a^2 - 1)/(a - 1) is not defined for a==1, but note that the
left and right limit exist and are identical. Now what harm is done if
we replace the first function by a second one (a+1) that agrees with the
first one where it is defined, but that is also defined for a==1?
This seems to me a better aproach than your proposal, especially if we
consider the implications on practical calculations.
Daniel
David W.Cantrell wrote:
> dh <dh at metrohm.ch> wrote:
>> HI Markus,
>>
>> I think you are fooling yourself.
>
> Rather, I think you, Daniel, are fooling yourself.
>
>> Consider Simplify[(a^2 - 1)/(a - 1)]
>>
>> As a^2 - 1)== (a+1)(a-1) this will give (a+1).
>
> Naively interpreted (as, say, a beginning calculus student should do)
>
> (a^2 - 1)/(a - 1) should simplify to something equivalent to
>
> If[a == 1, Indeterminate, a + 1]
>
> Whether a CAS should interpret something like (a^2 - 1)/(a - 1) in a more
> sophisticated fashion is debatable, IMO.
>
>> This is true in general, for a==1 understood as limit.
>
> But of course Simplify should not -- and in fact does not -- "understand as
> limit" automatically.
>
> Here's a simple example:
>
> In[2]:= Simplify[Sin[a]/a, a == 0]
> Out[2]= Indeterminate
>
> That behavior is correct. OTOH, if Mathematica had been designed to
> "understand as limit" automatically under Simplify, then Out[2] should have
> been 1 instead.
>
>> Therefore, this has nothing to do with assumptions.
>
> But it arguably _should_ have had something to do with assumptions; that
> was part of the point Markus was making.
>
>> On the other hand, (a^2 - 1)/(a - 1) /. a -> 1 can not be calculated as
>>
>> it gives 0/0. Note that "/. and ->" mean replacement, not limit. If
>>
>> you want a limit, you have to say so: Limit[(a^2 - 1)/(a - 1), a -> 1]
>>
>> hope this helps, Daniel
>
> Compare the following:
>
> In[8]:= Assuming[x == 0, {x/x^2, x/x, x^2/x}]
> Out[8]= {1/x, 1, x}
>
> In[9]:= x = 0; {x/x^2, x/x, x^2/x}
> Out[9]= {Indeterminate, Indeterminate, Indeterminate}
>
> Of course, in Mathematica, Out[9] is correct. And arguably, Out[8] should
> have been the same as Out[9].
>
> Here's a portion of a response of mine which appeared some years ago in
> this newsgroup. It's relevant to what Markus was asking about.
>
> -------------------------------------------------------
>
> In a sense, there is something wrong. These simplifications are what David
> Stoutemyer called "misdemeanors" in his article "Crimes and Misdemeanors in
> the Computer Algebra Trade", _Notices of the American Mathematical Society_
> 38:7 (1991) 778-785.
>
> In case anyone doesn't see why simplifying, say, x^0 to 1 is a misdemeanor:
> In Mathematica, 0^0 is regarded as being Indeterminate. Thus, without
> knowing that x is nonzero, Mathematica should not bluntly simplify x^0 to
> 1. Rather, it would seem that, in Mathematica, x^0 should simplify to, say,
> If[x == 0, Indeterminate, 1]. But such a result is normally considered to
> be too cumbersome to be practical; the "misdemeanor" simplification is
> considered preferable. All CASs known to me make such simplifications.
>
> -------------------------------------------------------
>
> David W. Cantrell
>
>
>> markusbinder wrote:
>>
>>> Hi,
>>> I would appreciate some comments about the Assuming-function. Several
>>> examples using Mathematica 6 as follows:
>>> In[1]:= Assuming[a > 0, Simplify[a == -1]]
>>> Out[1]= False
>>> In[2]:= Assuming[a \[Element] Integers, Simplify[a == -1]]
>>> Out[2]= a == -1
>>> In[3]:= Assuming[a \[Element] Integers && a > 0 && a < 2, Simplify[a^2<
>>> 1.1]]
>>> Out[3]= True
>>> In[4]:= Assuming[a \[Element] Reals && a > 0 && a < 2, Simplify[a^2 <
>>> 1.1]]
>>> Out[4]= a < 1.04881
>>> So far, everything seems pretty reasonable, but
>>> In[5]:= Assuming[a \[Element] Integers && a > 0 && a < 2, Simplify[(a^2
>>> - 1)/(a - 1)]]
>>> Out[5]= 1 + a
>>> whereas
>>> In[6]:= (a^2 - 1)/(a - 1) /. a -> 1
>>> Out[6]= Indeterminate
>>> I consider this as blunder (or bug?); I comprehend working with domains
>>> can be pretty tricky and of course I don't expect
>>> In[7]:= Simplify[(a^2 - 1)/(a - 1)]
>>> to yield
>>> Out[7]= "Beware, a != 1 neccessary!"
>>>
>>> instead of
>>>
>>> Out[7]= 1 + a.
>>>
>>> Am I tricked by some personal misconception of how Mathematica deals
>>> with Assuming plus fractions as shown above, or is this a mere flaw?
>>> Thanks in advance for your corrections & suggestions.
>>>
>>> Best regards
>>> Markus Binder
>