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Re: Testing the 'type' of a root returned by Solve

  • To: mathgroup at
  • Subject: [mg54671] Re: [mg54626] Testing the 'type' of a root returned by Solve
  • From: "David Park" <djmp at>
  • Date: Fri, 25 Feb 2005 01:19:41 -0500 (EST)
  • Sender: owner-wri-mathgroup at


sols=N[Solve[20*Feet^3 == (4*Pi*r^3)/5, r]]
(If[FreeQ[#1, Complex], Real, Complex] & ) /@ %
{Complex, Complex, Real}

The following is a little beyond the question. I am always a little dubious
about mixing data values and units into equations before they are solved. If
is often better to obtain a general and exact solution if possible. So for
your problem we could use...

sols = Solve[V == (4*Pi*r^3)/5, r]
{{r -> -(((-(5/Pi))^(1/3)*V^(1/3))/2^(2/3))},
  {r -> ((5/Pi)^(1/3)*V^(1/3))/2^(2/3)},
  {r -> ((-1)^(2/3)*(5/Pi)^(1/3)*V^(1/3))/2^(2/3)}}

Since V is positive we can test with...

If[FreeQ[#, Complex], Real, Complex] & /@ N[sols]
{Complex, Real, Complex}

Then we can substitute the 'data' into the the second real solution and
obtain an exact answer.

Simplify[sols[[2]] /. V -> 20*Foot^3, Foot > 0]
{r -> (5^(2/3)*Foot)/Pi^(1/3)}

David Park
djmp at

From: Mike Witt [mailto:mwNOSPAM at]
To: mathgroup at

If I solve an equation in which the solutions turn out
to be functions of some variable, I can't figure out how
to pick out one of the roots based on whether the root is
real or complex.

The problem is that Head[] reports that the roots are
all "Times" because of the variable in them.

The following notebook demonstrates. Can someone tell me
the right way to do this (or point me to the right place
in the book or help pages?)

For private email remove the NOSPAM.



(************** Content-type: application/mathematica **************
                     CreatedBy='Mathematica 5.0'

                    Mathematica-Compatible Notebook

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application, such as Mathematica, MathReader or Publicon. The data
for the notebook starts with the line containing stars above.

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one of the following:

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(*CacheID: 232*)

(*NotebookOptionsPosition[      2544,         85]*)
(*NotebookOutlinePosition[      3174,        107]*)
(*  CellTagsIndexPosition[      3130,        103]*)

I want to be able to select the 'real' root using the 'Head' \
function, like this:\
\>", "Text",


    \(Head /@ \((r /. \ \((Solve[20 == \(4\ \[Pi]\ r\^3\)\/5, \ r] //
              N)\))\)\)], "Input"],

    \({Complex, Complex, Real}\)], "Output"]
}, Open  ]],

But when I add a variable, then the 'type' of every root becomes \
'Times' because of the product of the variable and the number. Is 'Head' the
wrong way to do this?\
\>", "Text",


    \(Head /@ \((r /. \((\
            Solve[20  Feet\^3 == \(4\ \[Pi]\ r\^3\)\/5, \ r] //
              N)\))\)\)], "Input"],

    \({Times, Times, Times}\)], "Output"]
}, Open  ]]
FrontEndVersion->"5.0 for X",
ScreenRectangle->{{0, 1280}, {0, 1024}},
WindowSize->{520, 600},
WindowMargins->{{Automatic, 24}, {69, Automatic}}

Cached data follows.  If you edit this Notebook file directly, not
using Mathematica, you must remove the line containing CacheID at
the top of  the file.  The cache data will then be recreated when
you save this file from within Mathematica.



Cell[1754, 51, 123, 4, 52, "Text"],

Cell[1902, 59, 119, 2, 51, "Input"],
Cell[2024, 63, 58, 1, 27, "Output"]
}, Open  ]],
Cell[2097, 67, 207, 5, 71, "Text"],

Cell[2329, 76, 141, 3, 51, "Input"],
Cell[2473, 81, 55, 1, 27, "Output"]
}, Open  ]]

End of Mathematica Notebook file.

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