On Thu, 2007-04-12 at 03:05 +0700, Andrey G. Grozin wrote:
> And as for teaching junior school children, I am sure many things are
> done wrong. A classical example is x^(1/3). School children (and even
> teachers) beleive that it is real and negative for x<0.
(-8)^(1/3)=-2 is "wrong" in the same sense that 8^(1/3)=2 is also
"wrong". If you're thinking in terms of complex numbers, in both cases
you are choosing just one of the 3 solutions; I do not think
any of those two statements is wrong, if you are restricted to a real
domain.
> Maxima uses a more consistent definition - a cut along a negative
> real half-axis, with an additional rule that when we are exactly on
> the cut, the value from its upper side is used. So, for x<0 the result
> is complex. I'd say that here (as very often) maxima is right, and the
> school education is wrong.
As has been already pointed out by others, Maxima gives -2 for
(-8)^(1/3) :) And there is no right or wrong choice of branch cut; the
fact remains that we are dealing with a multiple-valued function.
To school kids, before telling them about branches, I'd rather teach
them that y=x^(1/3) is the inverse of the function y=x^3 and the graphs
of the two are symmetric under a reflection through y=x. (which is what
you currently get from Maxima for plot2d(x^3,[x,-2,2]) and
plot2d(x^(1/3),[x,-8,8]) and I think we should keep it that way).
Regards,
Jaime