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POHow can I make a "working" repeating decimal representation of a rational number?
 

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  1. POHow can I make a "working" repeating decimal representation of a rational number?
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    <p>I've figured out how to display the repeating part of a repeating decimal using OverBar.</p> <p><code>repeatingDecimal</code> doesn't actually work as a repeating decimal. I'd like to make a variation of it that looks <em>and behaves</em> like a repeating decimal.</p> <hr> <h3>Question</h3> <p>How could I make a <em>working</em> repeating decimal representation (possibly using <code>Interpretation[]</code>)?</p> <hr> <h3>Background</h3> <p>Please excuse me if I ramble. This is my first question and I wanted to be clear about what I have in mind.</p> <p>The following will "draw" a repeating decimal.</p> <pre><code>repeatingDecimal[q2_] := Module[{a}, a[{{nr__Integer}, pt_}] := StringJoin[ Map[ToString, If[pt &gt; -1, Insert[{nr}, ".", pt + 1], Join[{"."}, Table["0", {Abs[pt]}], {nr}]]]]; (* repeating only *) a[{{{r__Integer}}, pt_}] := Row[{".", OverBar@StringJoin[Map[ToString, {r}]]}]; (* One or more non-repeating; more than one repeating digit KEEP IN THIS ORDER!! *) a[{{nr__, {r__}}, pt_}] := Row[{StringJoin[ Map[ToString, If[pt &gt; -1, Insert[{nr}, ".", pt + 1], Join[{"."}, Table["0", {Abs[pt]}], {nr}]]]], OverBar@StringJoin[Map[ToString, {r}]]}]; (* One or more non-repeating; one repeating digit *) a[{{nr__, r_Integer}, pt_}] := Row[{StringJoin[Map[ToString, {nr}]], ".", OverBar@StringJoin[Map[ToString, r]]}]; a[RealDigits[q2]]] </code></pre> <p>So</p> <pre><code>repeatingDecimal[7/31] </code></pre> <p><strong>displays</strong> a repeating decimal properly (shown here as a picture so that the OverBar appears).</p> <p><img src="https://i.stack.imgur.com/sQJik.png" alt="Repeating decimal displayed"></p> <p>Looking under the hood, it's really just an imposter, an image of a repeating decimal ...</p> <pre><code>In[]:= repeatingDecimal[7/31]//FullForm Out[]:= Row[List[".",OverBar["225806451612903"]]] </code></pre> <p>Of course, it doesn't behave like a number:</p> <pre><code>% + 24/31 </code></pre> <p><img src="https://i.stack.imgur.com/fVznm.png" alt="fraction plus repeating decimal"></p> <p>I'd like the addition to yield: 1</p> <hr> <h3>Edit: A cleaned up version of repeatingDecimal</h3> <p>Leonid showed how to wrap Format around the routine and to supply up-values for adding and multiplying repeated decimals. Very helpful! It will take some time for me to be comfortable with up and down values.</p> <p>What follows below is essentially the streamlined version of the code suggested by Mr.Wizard. I set the OverBar above each repeating digit to allow line-breaking. (A single OverBar above Row looks tidier but cannot break when the right screen-margin is reached.) </p> <pre><code>ClearAll[repeatingDecimal] repeatingDecimal[n_Integer | n_Real] := n Format[repeatingDecimal[q_Rational]] := Row @ Flatten[ {IntegerPart@q, ".", RealDigits@FractionalPart@q} /. {{nr___Integer, r_List: {}}, pt_} :&gt; {Table[0, {-pt}], nr, OverBar /@ r} ] repeatingDecimal[q_] + x_ ^:= q + x repeatingDecimal[q_] * x_ ^:= q * x repeatingDecimal[q_] ^ x_ ^:= q ^ x </code></pre> <p>The table below shows some output from <code>repeatingDecimal</code>:</p> <pre><code>n1 = 1; n2 = 15; ClearAll[i, k, r]; TableForm[Table[repeatingDecimal[i/j], {i, n1, n2}, {j, n1, n2}], TableHeadings -&gt; {None, Table[("r")/k, {k, n1, n2}]}] </code></pre> <p><img src="https://i.stack.imgur.com/3rPsO.png" alt="enter image description here"></p> <hr> <h3>Checking the solution: Operating with repeating decimals</h3> <p>Let's now check the addition and multiplication of repeating decimals:</p> <pre><code>a = repeatingDecimal[7/31]; b = repeatingDecimal[24/31]; Print["a = ", a] Print["b = ", b] Print["a + b = ", a, " + ", b, " = ", a + b] Print["7/31 \[Times] 24/31 = " , (7/31)* (24/31)] Print["a\[Times]b = ", a*b, " = \n", repeatingDecimal[a*b]] Print[N[168/961, 465]] </code></pre> <p><img src="https://i.stack.imgur.com/HzlTp.png" alt="repeating decimal addition and multiplication"></p> <p>So addition and multiplication of repeating decimals work as desired. <code>Power</code> also appears to work properly.</p> <p>Notice that 168/961 occupies 465 places to the right of the decimal point. After that, it starts to repeat. The results match those of <code>N[168/961, 465]</code>, except for the <code>OverBar</code>, although line-breaks occur at different places. And, as is to be expected, this jibes with the following:</p> <pre><code>digits = RealDigits[168/961] Length[digits[[1, 1]]] </code></pre> <p><img src="https://i.stack.imgur.com/XKaB3.png" alt="465 digits"></p> <hr> <h3>Some effects of the Format[] wrapper on the behavior of N[] in summing repeated decimals</h3> <p>Mr.Wizard suggested that the <strong>Format</strong> wrapper is superfluous for the cases of Integers and Reals.</p> <p>Let's consider how the following two additions</p> <pre><code>repeatingDecimal[7/31] + repeatingDecimal[24/31] N@repeatingDecimal[7/31] + N@repeatingDecimal[24/31] </code></pre> <p>behave in four different cases:</p> <p><em>Case 1</em>: Results when <code>Format</code> <strong>wrapped</strong> around repeatingDecimals for Reals and Integers and up values are <strong>ON</strong></p> <p><img src="https://i.stack.imgur.com/kDebI.png" alt="Case 1"></p> <p>As expected, the first addition yields an integer, the second a decimal.</p> <p><hr> <em>Case 2</em>: Results when <code>Format</code> <strong>NOT wrapped</strong> around repeatingDecimals for Reals and Integers but up values are <strong>ON</strong></p> <p><img src="https://i.stack.imgur.com/26dQX.png" alt="Case 2"></p> <p>The <code>Format</code> wrapper around Reals and Integers doesn't affect the additions at hand.</p> <p><hr> <em>Case 3</em>: Results when <code>Format</code> <strong>wrapped</strong> around repeatingDecimals for Reals and Integers but up values are <strong>OFF</strong></p> <p><img src="https://i.stack.imgur.com/xHXBJ.png" alt="Case 3"></p> <p>If upvalues are OFF, <code>Format</code> prevents addition from happening.</p> <p><hr> <em>Case 4</em>: Results when <code>Format</code> <strong>NOT wrapped</strong> around repeatingDecimals for Reals and Integers and up values are <strong>OFF</strong></p> <p><img src="https://i.stack.imgur.com/D5z4d.png" alt="Case 4"></p> <p>If upvalues are OFF and Format` <strong>NOT wrapped</strong> around repeatingDecimals for Reals and Integers , the second addition works as expected.</p> <p>All the more reason to remove the Format wrapper for the case of reals and integers.</p> <hr> <p>Anyone have any remarks about the different outcomes in Cases 3 and 4?</p>
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    1. CO@Leonid Shifrin Your suggestions for adding and multiplying repeating decimals via up-values worked like a charm. In the section, **Checking the Solution** I illustrated the operations.
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    2. CO@David Carraher do you actually want the Blue/Red/Gray highlighting, or is that just for development?
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    3. CO@Mr.Wizard The coloring was just for development. When debugging, I found it useful see what patterns were kicking in. Your feedback was useful for me to realize that the original code wasn't air tight.
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