(Note: I have a lot of cube-root related emails to reply to today. I promise that I will get to everyone, eventually.)
The phenomenon which Michel Paul has spoken of is extremely interesting. The way I would address the perplexed (or perhaps, intrigued) student---provided that they know what complex numbers are---is to say: If I ask for the cube root of 8, there are three numbers z in the complex plane such that z^3 = 8. Specifically, they are z = 2cos(0) + i2sin(0) = 2 + 0i = 2 z = 2cos(120) + i2sin(120) = -1 + i sqrt(3) z = 2cos(240) + i2sin(240) = -1 - i sqrt(3) However, of these three, the first is real, and so normally that's "the one we want" (at least, usually.) If I ask for the cube root of -8, there are three numbers z in the complex plane such that z^3 = -8. Specifically, they are z = -2cos(0) - i2sin(0) = -2 - 0i = -2 z = -2cos(120) - i2sin(120) = 1 - i sqrt(3) z = -2cos(240) - i2sin(240) = 1 + i sqrt(3) Again, of these three, the first is real, and so normally that's "the one we want" (at least, usually.) Now it is noteworthy that I can get all three with solve( x^3 == 8, x ) or solve( x^3 == -8, x ). However, I think asking nth_real_root( -8, 3 ) to return -2 is no more exotic than asking nth_real_root( 8, 3 ) to return 2. Does this make sense? ---Greg p.s. I'm not sure how to address it with students who are not exposed to the complex numbers yet. Perhaps I'd say, well... (2)(2)(2) = 8 so we write cuberoot(8) = 2 (-2)(-2)(-2) = -8 so we write cuberoot(-8) = -2 On Fri, Jun 20, 2014 at 3:35 PM, michel paul <[email protected]> wrote: > Alpha shows the domain of y = x^(1/3) to be non-negative. However, it shows > the domain of y = 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. > > Both Mathematica and Sage leave (-1)^(1/3) unevaluated unless you ask for > the numeric approximation. > > I find it interesting that Alpha and Mathematica give different evaluations > for (-1)^(1/3). > > And then, Alpha evaluates 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 a topic in the > Google Groups "sage-edu" group. > To unsubscribe from this topic, visit > https://groups.google.com/d/topic/sage-edu/0vGw_Pd6v_Y/unsubscribe. > To unsubscribe from this group and all its topics, 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. 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