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MathGroup Archive 2007

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Re: Re: simplification of 0/0 to 1?


I think if Mathematica followed your suggestion things might become  
just too good to be true! Note that:

Simplify[(2*Cos[x]^2 + 2*Sin[x]^2 - 2)/(Cos[x]^2 + Sin[x]^2 - 1)]
2

while

Simplify[2*Cos[x]^2 + 2*Sin[x]^2 - 2]/Simplify[Cos[x]^2 + Sin[x]^2 - 1]

would return 1 or 0 (or whatever the user specified, according to  
your suggestion).

At least the fact that you get Indeterminate with an error message  
alerts you to what is going on, without that it would be easy to  
"prove" that 1==2 or whatever you wished (and hide it so deeply in an  
obscure "one-liner" nobody might notice). And since

Implies[1 == 2, "The moon is made of green cheese"]
True

you could then prove anything you have ever wished to prove :-)

Andrzej Kozlowski




On 12 Jun 2007, at 14:27, Murray Eisenberg wrote:

> The issue of what a piece of software should do about 0/0 -- even when
> it arises in precisely that form, without any CAS first simplifying
> algebraic expressions to reach that form -- is a sticky one.  The  
> issue
> was a mildly hot one many years ago in connection with the programming
> language APL (from which parts of the Mathematica language are
> descended).  Aside from the possibility of signaling an error,  
> there are
> two other "good" possibilities for 0/0: 0 and 1.  So I proposed in APL
> that the handling be user-specifiable.
>
> The simple answer is that there is no good way to handle 0/0 that will
> satisfy everyone.
>
> dimitris wrote:
>> Hi fellas.
>> This appeared in another forum as part of a question
>> what another CAS does.
>> Just of curiosity I check Mathematica's performance (5.2).
>> The result was poor!
>>
>> Here is the expression
>>
>>
>> In[16]:=
>> o = (Log[2]*Cos[Pi/12] - Log[2]*Sin[Pi/12] - 2*Cos[Pi/12] + 2*Sin[Pi/
>> 12] + Sqrt[2] +
>>     2*Log[Cos[Pi/12] - Sin[Pi/12]]*Cos[Pi/12] - 2*Log[Cos[Pi/12] -
>> Sin[Pi/12]]*Sin[Pi/12])/
>>    (Log[2]*Cos[Pi/12] - Log[2]*Sin[Pi/12] + 2*Log[Cos[Pi/12] -
>> Sin[Pi/
>> 12]]*Cos[Pi/12] -
>>     2*Log[Cos[Pi/12] - Sin[Pi/12]]*Sin[Pi/12])
>>
>>
>> Out[16]=
>> (Sqrt[2] + (-1 + Sqrt[3])/Sqrt[2] - (1 + Sqrt[3])/Sqrt[2] - ((-1 +
>> Sqrt[3])*Log[2])/(2*Sqrt[2]) +
>>    ((1 + Sqrt[3])*Log[2])/(2*Sqrt[2]) - ((-1 + Sqrt[3])*Log[-((-1 +
>> Sqrt[3])/(2*Sqrt[2])) + (1 + Sqrt[3])/(2*Sqrt[2])])/
>>     Sqrt[2] + ((1 + Sqrt[3])*Log[-((-1 + Sqrt[3])/(2*Sqrt[2])) + (1 +
>> Sqrt[3])/(2*Sqrt[2])])/Sqrt[2])/
>>   (-(((-1 + Sqrt[3])*Log[2])/(2*Sqrt[2])) + ((1 + Sqrt[3])*Log[2])/
>> (2*Sqrt[2]) -
>>    ((-1 + Sqrt[3])*Log[-((-1 + Sqrt[3])/(2*Sqrt[2])) + (1 + Sqrt[3])/
>> (2*Sqrt[2])])/Sqrt[2] +
>>    ((1 + Sqrt[3])*Log[-((-1 + Sqrt[3])/(2*Sqrt[2])) + (1 + Sqrt[3])/
>> (2*Sqrt[2])])/Sqrt[2])
>>
>> Watch now a really bad performance!
>>
>> In[17]:=
>> (Simplify[#1[o]] & ) /@ {Numerator, Denominator}
>>
>> Out[17]=
>> {0, 0}
>>
>> That is Mathematica simplifies succesfully both the numerator
>> and denominator to zero. So, you wonder what goes wrong?
>>
>> Try now to simplify the whole expression!
>>
>> In[19]:=
>> Simplify[o]
>>
>> Out[19]=
>> 1
>>
>> A very weird result to my opinion!
>> Simplification of 0/0 to 1?
>> I think no simplification or some
>> warning messages would be much better
>> than 1!
>>
>> Note also that
>>
>> In[20]:=
>> RootReduce[o]
>>
>> Out[20]=
>> 1
>>
>> Dimitris
>>
>>
>
> -- 
> Murray Eisenberg                     murray at math.umass.edu
> Mathematics & Statistics Dept.
> Lederle Graduate Research Tower      phone 413 549-1020 (H)
> University of Massachusetts                413 545-2859 (W)
> 710 North Pleasant Street            fax   413 545-1801
> Amherst, MA 01003-9305
>



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