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Asked: 2026-05-24T00:57:15+00:00

I would like to plot a simple interval on the number line in Mathematica.

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I would like to plot a simple interval on the number line in Mathematica. How do I do this?

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  1. Editorial Team

    Here’s another attempt that draws number lines with the more conventional white and black circles, although any graphics element that you want can be easily swapped out.

    It relies on LogicalExpand[Simplify@Reduce[expr, x]] and Sort to get the expression into something resembling a canonical form that the replacement rules can work on. This is not extensively tested and probably a little fragile. For example if the given expr reduces to True or False, my code does not die gracefully.

    numLine[expr_, x_Symbol:x, range:{_, _}:{Null, Null}, 
  Optional[hs:_?NumericQ, 1/30], opts:OptionsPattern[]] := 
 Module[{le = {LogicalExpand[Simplify@Reduce[expr, x]]} /. Or -> List,
   max, min, len, ints = {}, h, disk, hArrow, lt = Less|LessEqual, gt = Greater|GreaterEqual},
  If[TrueQ@MatchQ[range, {a_, b_} /; a < b],
   {min, max} = range,
   {min, max} = Through[{Min, Max}@Cases[le, _?NumericQ, \[Infinity]]]];
  len =Max[{max - min, 1}]; h = len hs;
  hArrow[{x1_, x2_}, head1_, head2_] := {{Thick, Line[{{x1, h}, {x2, h}}]},
                                         Tooltip[head1, x1], Tooltip[head2, x2]};
  disk[a_, ltgt_] := {EdgeForm[{Thick, Black}], 
    Switch[ltgt, Less | Greater, White, LessEqual | GreaterEqual, Black], 
    Disk[{a, h}, h]};
  With[{p = Position[le, And[_, _]]}, 
       ints = Extract[le, p] /. And -> (SortBy[And[##], First] &); 
       le = Delete[le, p]];   
  ints = ints /. (l1 : lt)[a_, x] && (l2 : lt)[x, b_] :> 
     hArrow[{a, b}, disk[a, l1], disk[b, l2]];
  le = le /. {(*_Unequal|True|False:>Null,*)
     (l : lt)[x, a_] :> (min = min - .3 len; 
       hArrow[{a, min}, disk[a, l], 
        Polygon[{{min, 0}, {min, 2 h}, {min - Sqrt[3] h, h}}]]),
     (g : gt)[x, a_] :> (max = max + .3 len; 
       hArrow[{a, max}, disk[a, g], 
        Polygon[{{max, 0}, {max, 2 h}, {max + Sqrt[3] h, h}}]])};
  Graphics[{ints, le}, opts, Axes -> {True, False}, 
   PlotRange -> {{min - .1 len, max + .1 len}, {-h, 3 h}},
   GridLines -> Dynamic[{{#, Gray}} & /@ MousePosition[
                           {"Graphics", Graphics}, None]], 
   Method -> {"GridLinesInFront" -> True}]
  ]

    (Note: I had originally tried to use Arrow and Arrowheads to draw the lines – but since Arrowheads automatically rescales the arrow heads with respect to the width of the encompassing graphics, it gave me too many headaches.)

    OK, some examples:

    numLine[0 < x], 
numLine[0 > x]
numLine[0 < x <= 1, ImageSize -> Medium]

    enter image description here
    enter image description here
    enter image description here

    numLine[0 < x <= 1 || x > 2, Ticks -> {{0, 1, 2}}]

    enter image description here

    numLine[x <= 1 && x != 0, Ticks -> {{0, 1}}]

    enter image description here

    GraphicsColumn[{
  numLine[0 < x <= 1 || x >= 2 || x < 0],
  numLine[0 < x <= 1 || x >= 2 || x <= 0, x, {0, 2}]
  }]

    enter image description here

    Edit: Let’s compare the above to the output of Wolfram|Alpha

    WolframAlpha["0 < x <= 1 or x >= 2 or x < 0", {{"NumberLine", 1}, "Content"}]
WolframAlpha["0 < x <= 1 or x >= 2 or x <= 0", {{"NumberLine", 1}, "Content"}]

    output of the above

    Note (when viewing the above in a Mathematica session or the W|A website) the fancy tooltips on the important points and the gray, dynamic grid lines. I’ve stolen these ideas and incorporated them into the edited numLine[] code above.

    The output from WolframAlpha is not quite a normal Graphics object, so it’s hard to modify its Options or combine using Show. To see the various numberline objects that Wolfram|Alpha can return, run WolframAlpha["x>0", {{"NumberLine"}}] – “Content”, “Cell” and “Input” all return basically the same object. Anyway, to get a graphics object from 

    wa = WolframAlpha["x>0", {{"NumberLine", 1}, "Content"}]

    you can, for example, run

    Graphics@@First@Cases[wa, GraphicsBox[__], Infinity, 1]

    Then we can modify the graphics objects and combine them in a grid to get

    aligned

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