kcrisman wrote: > >> Thus you have constructed a nice expression for 1: >> >> sage: sol[2].subs(a=1).right() >> (2/(3*sqrt(3)) + 10/27)^(1/3) - 2/(9*(2/(3*sqrt(3)) + 10/27)^(1/3)) + 1/3 >> >> Quiz: how to simplify that expression to 1 within SAGE? I've tried simplify, >> and radical_simplify, but neither succeeds... >> >> Paul Zimmermann > > Ted, I've been there - running into that sort of "nice" formula with > Sage in the class, presumably via Maxima. As cool as it is, it is > also confusing and disheartening to my students, who have no > experience to interpret it with. > > More to the point, I would softly plead a different view on > (-1)^(1/3). As a mathematician, I think getting the "usual" primitive > sixth root of unity is wonderful (though perhaps even making that > choice is suspect, why not its complex conjugate?), but as a pedagogue > it would cause me to question using Sage in any environment where I > needed numerical solutions to simple algebraic equations, such as any > HS class or many (though not all) freshman non-major math courses. > > I definitely want my students to know how to do it by hand - in fact, > they *should* solve x^3=-1 by hand. But as Ted points out, getting a > useful numerical approximation for x^3=-3 by typing (-3)^(1/3) is a > legitimate need, even for students who aren't ready for complex > numbers (and such students definitely exist). And yet, > > sage: (-3)^(1/3) > (-1)^(1/3)*3^(1/3) > sage: n(_) > 0.721124785153704 + 1.24902476648341*I > > Even using the decimal point to ensure we get a numerical solution, > which does seems like something a student could understand why they > should do when describing symbolic versus numerical computation, > doesn't help: > > sage: (-3.0)^(1/3) > 0.721124785153704 + 1.24902476648341*I > > and Sage fails something even my eight dollar CVS calculator can (to > my surprise) do! This seems problematic.
For the record, in Mathematica: In[1]:= (-1)^(1/3) Out[1]= (-1)^(1/3) In[2]:= % // N Out[2]= 0.5+ 0.866025 I In[3]:= (-3)^(1/3) // N Out[3]= 0.721125+ 1.24902 I In previous versions of Mathematica, there was a "RealOnly" package which defined odd roots as negative and printed "Nonreal" anytime a complex number was unavoidable. The idea was that you could simplify things for high school students or in situations in which you knew you were only interested real numbers. That package has apparently been deprecated now in version 6.0, being replaced by the functionality to "Reduce" an equation over the reals, etc. For details, you can see the package at http://library.wolfram.com/infocenter/MathSource/6771/ Jason --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-support@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-support URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
