[R-sig-teaching] Plot f(x) = x^(1/5)

[Albyn Jones]

I think R is doing just what it should do, and the TI has been hacked to
make HS teachers happy :-)

> log(as.complex(-2))/5
[1] 0.1386294+0.6283185i
> a <- log(as.complex(-2))/5
> exp(a)^5
[1] -2-0i

> exp(a)
[1] 0.9293165+0.675188i
> (as.complex(-2))^(1/5)
[1] 0.9293165+0.675188i
> 2^(1/5)*cos(pi/5)
[1] 0.9293165
> 2^(1/5)*sin(pi/5)
[1] 0.675188

albyn


On Tue, May 17, 2016 at 8:10 AM, Richard M. Heiberger <rmh at temple.edu>
wrote:

> This problem is an example of FAQ 7.31.  Floating point numbers
> inside the computer can not represent odd fractions exactly.
>
> > seq(1, 15, 2)
> [1]  1  3  5  7  9 11 13 15
> > 1/seq(1, 15, 2)
> [1] 1.00000000 0.33333333 0.20000000 0.14285714 0.11111111 0.09090909
> 0.07692308
> [8] 0.06666667
> > print(1/seq(1, 15, 2), digits=18)
> [1] 1.0000000000000000000 0.3333333333333333148 0.2000000000000000111
> [4] 0.1428571428571428492 0.1111111111111111049 0.0909090909090909116
> [7] 0.0769230769230769273 0.0666666666666666657
> > ## except for 1, none of these odd fractions is exactly represented
> inside the computer.
> > ## therefore the power requested is not the exact fraction, and the
> result is not capable of calculation
> > (-2)^(1/seq(1, 15, 2))
> [1]  -2 NaN NaN NaN NaN NaN NaN NaN
> >
>
> Rich
>
> On Tue, May 17, 2016 at 10:23 AM, bob at statland.org <bob at statland.org>
> wrote:
> > Forwarded message:
> >> >
> >> > > System is working correctly. A negative number cannot be raised to a
> >> > > fractional power:
> >> > >
> >> > > > (-2)^(1/5)
> >> > > [1] NaN
> >
> > Well, maybe we should say that _R_ can't raise a negative number to a
> > fractional power.  Neither I nor the TI calculators have any trouble
> > doing it;-) I'd say R is funtioning as designed, but it was designed
> > to respond INcorrectly;-)
> >
> > This R exchange
> >
> >> (-2)^5
> > [1] -32
> >
> > shows that -2 is a fifth root of -32 while this exchange
> >
> >> (-32)^(1/5)
> > [1] NaN
> >
> > shows that R cannot find that root.  The various suggestions for
> > dealing with this amount to asking R a different question which we
> > know has the same answer as the intended question (which R won't
> > answer).
> >
> > On one level you could view this as a coding/implementation issue.  I
> > have not looked at R's code, but the usual computer way to handle
> > exponents involves taking the log of the argument.  This does not
> > return the correct answer when the argument is negative.  That's
> > annoying.  The TI graphing calculators were developed with an
> > incredible amount of input from secondary math. teachers.  They
> > complained loudly about calculators returning wrong answers or
> > non-answers to problems to which students knew the right answers.  TI
> > did a LOT of work on this.  I wish R (and lots of scientific software)
> > would do likewise.
> >
> > On another level, involving exponentiation is not entirely avoidable.
> > For rational numbers like 1/5 we can (and usually do) interpret
> > (-32)^(1/5) as a name for a real number that when raised to the fifth
> > power gives -32.  Another name for one such number is -2.  But if we want
> > to use an irrational exponent, say (-32)^pi we can't interpret it that
> > way.  (How do we multiply pi numbers together?)  So eventually we have
> > to either exponentiate or have a funciton that is undefined at many
> > points.  At lesat in theory.  As far as computers and calculators are
> > concerned, they cannot represent irrational numbers anyway --
> > eveything is a rational approximation.
> >
> > So I think the defect is in R, not in the original posted question.
> >
> > ------->  First-time AP Stats. teacher?  Help is on the way! See
> >
> http://courses.ncssm.edu/math/Stat_Inst/Stats2007/Bob%20Hayden/Relief.html
> >       _
> >      | |          Robert W. Hayden
> >      | |          614 Nashua Street #119
> >     /  |          Milford, New Hampshire 03055  USA
> >    |   |
> >    |   |          email: bob@ the site below
> >   /  x |          website: http://statland.org
> >  |     /
> >  ''''''
> >
> > _______________________________________________
> > R-sig-teaching at r-project.org mailing list
> > https://stat.ethz.ch/mailman/listinfo/r-sig-teaching
>
> _______________________________________________
> R-sig-teaching at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-sig-teaching
>

	[[alternative HTML version deleted]]