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cube roots of negative numbers

Posted on 2008-08-16 09:36 By alla (Maple Rating 2 90)
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Why does the following give me an imaginary number?

 (-.008)^(1/3);

Similarly, in plotting y = x1/3 does Maple fail to give me any points where x < 0?

Alla

 

 

Comments

Axel Vogt's picture

x^3 = - 0.008

Comment on 2008-08-16 10:33 from Axel Vogt (Maple Rating 4 948) 
x^3 = - 0.008 has 3 solutions, 2 of them are imaginary, 1 is real
and you want especially one of them
see the 'surd' command (which is a bit difficult to follow in the help pages):

"Note the differences among the outputs of the surd, ^, and root commands":

(-8.0)^(1/3); root(-8.0, 3); surd(-8.0, 3);;
                     1.000000000 + 1.732050807 I
                     1.000000000 + 1.732050807 I
                             -2.000000000
For plotting: Maple does not plot imaginary number, if not been explicitely told like

plot( [Re(root(x,3)),Im(root(x,3))], x= -1 ..1, color=[red, blue]);
 

principal value

Comment on 2008-08-16 11:04 from Joe Riel (Maple Rating 4 1248)

The reason is that Maple computes the principal value of the expression z^(1/3), which, for negative real z, lies in the upper right quadrant of the complex plane.  To see that, you may use conformalplot:

plots[conformalplot](z^(1/3), z = -1..1)

conformal

Comment on 2008-08-16 20:44 from jakubi (Maple Rating 4 899)

I guess that you mean

plots[conformal](z^(1/3), z = -1..1);

A nice plot of the cubic roots is got with 'DynamicSystems' (not obvious though):

with( DynamicSystems ):
NewSystem(1/s^3):
RootLocusPlot(%);
Tim Vrablik's picture

To answer your plot question

Comment on 2008-08-18 09:25 from Tim Vrablik (Maple Rating 1 69)

All of the aforementioned hopefully helped explained why Maple does what it does...although it does make sense, it's not the most intuitive operation.

To get the plot that you asked for though, use the command plot(surd(x,3),x=-5..5).

Hope that helps.

Successful plot

Comment on 2008-08-18 16:56 from alla (Maple Rating 2 90)

My original question stemmed from the fact that Maple wouldn't give me any negative x-points in the graph of y= x^(1/3).  It was suggested that I use

 

          plot(Re(x^(1/3)) etc.

 

and this did give me points when x < 0.  But something was still not right since the graph didn't look as I expected.  It was only when I tried Tim's suggestion:

 

         plot(surd(x,3) etc.

 

that the plot came out OK.

 

Alla

 

 

RealDomain

Comment on 2008-08-18 18:14 from Joe Riel (Maple Rating 4 1248)

Another possibility is

with(RealDomain):
plot(x^(1/3), x = -1..1);

Note that Re(x^(1/3)) is incorrect, as you noticed.  It returns the real part of a complex root, which is different from the real root.

Limitations of RealDomain

Comment on 2008-08-18 19:58 from alla (Maple Rating 2 90)

Two comments on RealDomain:

1) I gather from the Help files that Real Domain is geared to precalculus math, so the commands available for it are limited.  That environment, then, would not be suitable for much of calculus, right?

2) What's the most efficient way to switch out of RealDomain?  Exiting the worksheet and re-opening it as well as Restart are options, but they reset a lot of stuff that you might prefer not be reset.  Any other options?

Alla

unwith

Comment on 2008-08-18 20:23 from Joe Riel (Maple Rating 4 1248)

For 2,

unwith(RealDomain):

 
To use RealDomain with just a few commands, it is probably better to do so within a use environment


use RealDomain in
   plot(x^(1/3), x = -1..1);
end use;
gkokovidis's picture

Limitations of RealDomain

Comment on 2008-08-18 20:24 from gkokovidis (Maple Rating 4 672)

Do not load the package if you are only looking for a limited subset of the features.  For example, your plot that uses the RealDomain features can be called with the line below.  Then you do not have to execute a worksheet restart.

>use RealDomain in plot(x^(1/3), x = -1..1) end;

 

Regards,
Georgios Kokovidis
Dräger Medical

Plotting inconsistency

Comment on 2008-08-19 18:31 from alla (Maple Rating 2 90)

 

 

 

 

 

 

 

 

 

 

 

I want to plot y = x^(2/5), so I tried

 

 

 

use RealDomain in

   plot(x^(2/5), x = -1..1,y = -1..1);

end use;

 

This works OK.

 

But if define

 

     f36 := x -> x^(2/5);

 

Then

 

use RealDomain in

   plot(f36(x), x = -1..1,y = -1..1);

end use;

 

does not give me x-points for x < 0.

 

Why?


Alla

 

 

acer's picture

because

Comment on 2008-08-19 18:55 from acer (Maple Rating 4 1348)

You created your procedure f36() outside of the use of RealDomain. So the `^` bound within f36 is the usual global :-`^` and not the RealDomain:-`^` export. The procedure f36 is getting its binding of `^` when it is created, not when it is used (plotted).

Compare,

restart:
use RealDomain in
  f36 := x -> x^(2/5);
  plot(f36(x), x = -1..1,y = -1..1);
end use;
plot(f36(x), x = -1..1,y = -1..1);
lprint(eval(f36));

restart:
f36 := x -> x^(2/5);
use RealDomain in
  plot(f36(x), x = -1..1,y = -1..1);
end use;
lprint(eval(f36));

restart:
use RealDomain in
  plot(:-`^`(x,2/5), x = -1..1,y = -1..1);
end use;

acer

binding

Comment on 2008-08-19 19:04 from Joe Riel (Maple Rating 4 1248)

When Maple creates the procedure f36 := x -> x^(2/5), it uses the definition of ^ at that time.  To see this do

restart;
f := x -> x^(2/5):
use RealDomain in g := x -> x^(2/5); end use:
eval(f);
                                                           (2/5)
                                                     x -> x

eval(g);
                                             x -> RealDomain:-`^`(x, 2/5)

The solution is as shown above, create the procedure when RealDomain is 'active'.  You can then use it outside RealDomain, that is,

plot(g, -1..1);

does what you want.

Why doesn't limit work here?

Comment on 2008-08-20 13:56 from alla (Maple Rating 2 90)

Why doesn't this work?

> use RealDomain in

   f39 := x -> 4*x^(2/5)-2*x;

   limit(f39(x),x=0,real);

end use;

 

>

I'm getting zero, but answer should be "undefined," no?

 

Alla

 

why?

Comment on 2008-08-20 18:39 from pagan (Maple Rating 2 99)

Why do you think that the result should be undefined, when computed under RealDomain?

RealDomain or not

Comment on 2008-08-20 18:55 from alla (Maple Rating 2 90)

I first tried limit without RealDomain & got the wrong answer.  Since using RealDomain had cured other problems in this thread, I decided to try it, but got the same wrong answer.  The real question is: How can I get the right answer?

Alla

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