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

[Richard M. Heiberger]

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.
>
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