Alpha shows the domain of y = x^(1/3) <http://www.wolframalpha.com/input/?i=domain+of+y+%3D+x%5E%281%2F3%29> to be non-negative. However, it shows the domain of y = cube root of x <http://www.wolframalpha.com/input/?i=domain+of+y+%3D+cube+root+of+x> to be all reals.
Most high school texts present a cube root function whose domain is all reals. A few years ago this became an issue in an Alg 2 class. I was having the students find domains and ranges of functions, and a student pointed out that Alpha contradicted our text and wondered why. This turned into a discussion of 'what is the cube root of -1?' Both Google calc and GeoGebra evaluate (-1)^(1/3) as -1. Alpha gives the complex expression <http://www.wolframalpha.com/input/?i=%28-1%29%5E%281%2F3%29&lk=4&num=1&lk=4&num=1> . Both Mathematica and Sage leave (-1)^(1/3) unevaluated <http://sagecell.sagemath.org/?z=eJzT0DXUjNMw1DfWBAAKsgHy&lang=sage> unless you ask for the numeric approximation <http://sagecell.sagemath.org/?z=eJzT0DXUjNMw1DfW1MvT0AQAFPUC3w==&lang=sage> . I find it interesting that Alpha and Mathematica give different evaluations for (-1)^(1/3). And then, Alpha evaluates cube root of -1 <http://www.wolframalpha.com/input/?i=cube+root+of+-1> as -1! Also note the different uses of the radical sign. In the cases of 'cube root' there's an extra little hook on the radical that is not present for '^(1/3)'. This is very interesting. I think there are some issues here pertinent to our secondary curriculum. We tell the kids about the Fundamental Theorem of Algebra, but then as they go into Analysis and Calculus, we pretend it doesn't exist. All attention gets focused on the reals. - Michel On Fri, Jun 20, 2014 at 12:47 PM, 'Gerald Smith' via sage-edu < [email protected]> wrote: > Guys, please keep in mind that exponentials of negative bases are not > really functions. They give a mix of real and multivalued complex results. > You cannot treat x^(1/3) as a synonym for cuberoot (x) if x,0. The latter > is a well defined function and the former is not. Perhaps this might be a > good reason to introduce a new function cuberoot(x) and evaluate it (using > for example Newton''s Formula). > Cheers, > MathBear > P.S. If you let the independent variable be the exponent and have a > negative base, then for any interval of x however short, the function will > flip flop infinitely often between real and multivalued complex results. > The output is completely discontinuous and very messy. Just try graphing > it! There is a good reason why exponential finctions of a real variable > are restricted to positive bases. > > > On Thursday, June 19, 2014 11:39 AM, David Smith <[email protected]> > wrote: > > > I could live with nthroot -- but is cuberoot (or curt or cbrt) so > different from sqrt? Anyway, it would be nice to have functions that > resemble ones students see routinely without generating error messages. > I'm very much aware of the difficulties involved in reaching that goal, and > I'll be patient. > > Still waiting for a solution on arcsec, as well. There the issue is > primarily plotting, although "integral" fails, even with a proper domain. > numerical_integral works as it should. > > In the great scheme of things, these are certainly peripheral issues -- > and they wouldn't be issues at all if textbook authors didn't feel obliged > to drag in every function they know about. As author, I'll guilty of that > too (fear of the adoption committee), but in my defense, we only do these > things in exercises. > > On Wednesday, June 18, 2014 8:48:36 PM UTC-4, kcrisman wrote: > > Just to chime in, as someone who has dealt with this question a lot > (though, perhaps ironically, never in a classroom situation): > > I would be very against a "cuberoot" function, but an "nthroot" function > where it was really clear what input was allowed could fly. I appreciate > Greg's rationale. Note however - what is the 0.1 power? Is that the same > as the 1/10 power? This is a tricky floating point question to interpret. > > I don't think that the slowdown would be too bad since it would primarily > be for pedagogical purposes for plotting. > > David, I don't know how this would work with integrals, though - we'd have > to see if Maxima had something equivalent. Perhaps it could do a temporary > set of the Maxima domain to real somehow, if that is what allows Maxima to > do the "right thing" in this context, I don't know. > > Thanks for fighting the good fight on trying to resolve this once and for > all! > - kcrisman > > -- > You received this message because you are subscribed to the Google Groups > "sage-edu" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/sage-edu. > For more options, visit https://groups.google.com/d/optout. > > > -- > You received this message because you are subscribed to the Google Groups > "sage-edu" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/sage-edu. > For more options, visit https://groups.google.com/d/optout. > -- =================================== "What I cannot create, I do not understand." - Richard Feynman =================================== "Computer science is the new mathematics." - Dr. Christos Papadimitriou =================================== -- You received this message because you are subscribed to the Google Groups "sage-edu" group. 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