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

*To*: mathgroup at smc.vnet.net*Subject*: [mg77556] Re: [mg77543] Re: [mg77510] simplification of 0/0 to 1?*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Wed, 13 Jun 2007 07:20:48 -0400 (EDT)*References*: <200706110822.EAA20741@smc.vnet.net> <200706120527.BAA14286@smc.vnet.net> <1582EDB4-119B-4BE5-A6FA-E9D6452843A2@mimuw.edu.pl>

I forgot to note that, according to Mathematica, 1 == Indeterminate is not False, so one can't hope to become famous simply by evaluating: Implies[Simplify[(Cos[x]^2 + Sin[x]^2 - 1)/(Cos[x]^2 + Sin[x]^2 - 1)] == Simplify[Cos[x]^2 + Sin[x]^2 - 1]/ Simplify[Cos[x]^2 + Sin[x]^2 - 1], "The Riemann hypothesis is false."] If your convention were to be adopted then, (as I showed below), there would be a chance. Andrzej Kozlowski Andrzej Kozlowski On 12 Jun 2007, at 16:31, Andrzej Kozlowski wrote: > > 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 >> >

**References**:**simplification of 0/0 to 1?***From:*dimitris <dimmechan@yahoo.com>

**Re: simplification of 0/0 to 1?***From:*Murray Eisenberg <murray@math.umass.edu>