There was a thread on this issue a few months ago, just on the simplication of algebraic expressions, and I don't want to repeat all that. Briefly, people tend to think this is easy when they look at examples which only involve square roots of positive reals, since there is an "obvious" convention that by default we mean the positive root. It's a lot more complicated when you deal with general algebraic numbers which have several ways of being embedded into CC. Even for square roots of negative reals: you might suggest taking the root with positive imaginary part, but then sqrt(-2)*sqrt(-3) equals -sqrt(6) and not +sqrt(6).
John 2008/6/1 Henryk Trappmann <[EMAIL PROTECTED]>: > > >> But coercing symbolic constants into RR or CC is not a simple, (or >> even well-defined?) matter. Just think of many-valued nested >> radicals; or if a=sqrt(2), b=sqrt(3), c=sqrt(6), would a*b-c >> simplify/coerce to 0? This is not stratightforward at all. > > Is it? > I just would evaluate the expression in RR. > And then sqrt(2)*sqrt(3)-sqrt(6) is not 0. > > Btw. I just realized that SymbolicRing does assert that > not (sqrt(5+2*sqrt(2)*sqrt(3))-sqrt(2)-sqrt(3)).is_zero() > > which is wrong. While your example > (sqrt(2)*sqrt(3)-sqrt(6)).is_zero() > > is properly recognized. > > I guess for radical expressions there is an algorithm that can decide > zeroness. But if we also involve powers of radical expressions and > perhaps logarithms this becomes more difficult. Take for example: > > sqrt(2)**log(9,2) = sqrt(2)**(2*log(3,2))=2**log(3,2)=3 > which strangely enough is recognized by Sage: > (sqrt(2)**log(9,2)-3).is_zero() > > I would be very grateful if anyone could point me to literature about > those decidability problems (whether for radicals, algebraic numbers, > or numbers involving log and exp, or even sin, cos, etc for which I > think there are some undecidability results). > > > > --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
