On Tue, 3 Sep 1996 12:18:44 -0700, Harley Greenberg
<hgg@bigdog.engr.arizona.edu> wrote:

> I tried graphing [ y = x^(1/3) ] with the hp but it only drew the
> answer in the first quadrant.  Why did the hp do this?

In article <322d4e5b.11239018@nntp.ix.netcom.com>,
gracemon@ix.netcom.com (Kevin Luu) answered:

> I'm not postive about this but I think that algebraically, if you
> put in negative values for x, then you'll get complex numbers, since
> the eqn. ( y = x^(1/3) ) = ( the cube root of x )

Yes, you can be positive!  However, it's not exactly because of
being a "cube root," since cube roots of negative numbers are
simply the negative of the cube roots of their absolute values.

The reason that the HP48 can not get a "real" value for x^(1/3)
for any negative x is that the parentheses require the fraction
1/3 = .333333333333 to be computed first, after which the HP48
is obliged to evaluate x^.333333333333, which actually no longer
represents the "real" cube root at all -- it now represents raising
x to a non-integer real power, which for negative x will produce
a "principal result" in the first quadrant of the complex plane.

E.g. if "polar" display is active, then '(-8)^(1/3)' -> (2,@60)
where "@" here means the "angle symbol" (right-shift SPC).  Since plotting
requires "real" results, nothing will be plotted for any negative X.

To get the "real" cube root, better to use 'Y=XROOT(3,X)',
which will give you the complete graph in both quadrants I and III.

Going a little further:

Calculators which have a general "power" function (y^x) but no corresponding
general "root" function (Xroot of Y), like many older HP's, have always
presented two problems when trying to calculate "cube roots":

  o Accuracy:           ( 8000)^(1/3) -> 19.9999999999 (not 20)
  o Negative arguments: (-8000)^(1/3) -> Error (if no "complex" mode)

Only when HP decided to offer an XROOT function were these fundamental
limitations finally transcended in the HP calculator product line.

Now, consider Casio (you read my last post about Casio?):

  Good old Casio has offered *both* the "power" and complementary "root"
  functions for much longer than has HP, which originally offered only
  the "power" function, as mentioned above.

  Therefore, Casio has been able to get the exact real cube root of -8000
  for a very long time.  But that's not all -- would you believe, Casio
  has even been able to do (-8000)^(1/3) -> -20 (*exactly*), and even
  (-3200000)^.2 -> -20 (hey, no errors?), whereas (-3200000)^.3 -> Error.

  How has Casio managed to do that?  Yes, they are "fudging" these answers
  too, just as I mentioned in an earlier post.  When most Casio's are asked
  to do (x^y), if the power is a non-integer, the Casio takes its reciprocal
  to see whether it is "close" to an integer (same definition as before),
  and if so, it uses the "root" function instead, magically enabling a
  negative number to be raised to a non-integer power, *if* that power
  "looks like" the reciprocal of an odd integer.  Conversely, even the
  Casio "general root" function looks at the reciprocal of any non-integer
  root to see whether it might use a "power" instead!  And of course,
  even though these answers cannot possibly be calculated *exactly*
  by any calculator, the Casio's still deliver miraculously perfect
  exact answers anyway, by virtue of "fudging" them if they "look"
  close to a simpler answer, just as we reviewed last time, and
  with the same side-effect that they sometimes wrongly change a
  correct non-perfect-looking result by fudging it to "look better"
  when it shouldn't (has anybody else done that lately? :)

  How come, if they don't allow HP calculators on certain tests, they
  nonetheless allow these "cheating" Casio calculators to be used? :)

  Okay, now try (-8000)^(2/3) -> sorry, even Casio's can't fudge that one;
  you'll have to figure it out for yourself!

-----------------------------------------------------------
With best wishes from:  John H Meyers  ( jhmeyers@mum.edu )