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copied!<p>Here's an attempt. Mr.Wizard did much of the heavy lifting, especially in the base-preserving arithmetic.</p> <pre><code>rd[n_] := rd[n, 10] rd[rd[n_, _], b_] := rd[n, b] Format[rd[n_Integer | n_Real, base_]] := BaseForm[n, base] Format[rd[q_Rational, base_]] := Subscript[Row @ Flatten[{ IntegerString[IntegerPart@q, base], ".", RealDigits[FractionalPart@q, base] /. {{nr___, r_List:{}}, pt_} :> {0~Table~{-pt}, nr, OverBar /@ r} }], base /. 10 -> ""] </code></pre> <p>Base-preserving arithmetic can be implemented using this:</p> <pre><code>Scan[ (#[rd[q1_, b1_], rd[q2_, b2_] | tail___] ^:= rd[ #[q1, q2, tail], If[b1 === b2, b1, 10] ]) &, {Plus, Times, Power} ] </code></pre> <p>Checking to see that conversions to repeating decimals in several bases work. Also checking routines for adding, multiplying, and dividing:</p> <pre><code>Grid[{{"n", "value", "decimal", "rd[n,10]", "rd[n,2]", "rd[n,3]", "rd[n,7]"}, {"a", a = 14/3, N[a], rd[a, 10], rd[a, 2], rd[a, 3], rd[a, 7]}, {"b", b = 2/5, N[b], rd[b, 10], rd[b, 2], rd[b, 3], rd[b, 7]}, {"c", c = 1/2, N[c], rd[c, 10], rd[c, 2], rd[c, 3], rd[c, 7]}, {"a + b", a + b, N[a + b], rd[a, 10] + rd[b, 10], rd[a, 2] + rd[b, 2], rd[a, 3] + rd[b, 3], rd[a, 7] + rd[b, 7]}, {"a + b + c", a + b + c, N[a + b + c], rd[a, 10] + rd[b, 10] + rd[c, 10], rd[a, 2] + rd[b, 2] + rd[c, 2], rd[a, 3] + rd[b, 3] + rd[c, 3], rd[a, 7] + rd[b, 7] + rd[c, 7]}, {"a \[Times] b ", a*b, N[a*b], rd[a, 10]*rd[b, 10], rd[a, 2]*rd[b, 2], rd[a, 3]*rd[b, 3], rd[a, 7]*rd[b, 7]}, {"a \[Times] b \[Times] c ", a*b*c, N[a*b*c], rd[a, 10]*rd[b, 10]*rd[c, 10], rd[a, 2]*rd[b, 2]*rd[c, 2], rd[a, 3]*rd[b, 3]*rd[c, 3], rd[a, 7]*rd[b, 7]*rd[c, 7]}, {"a / b", a/b, N[a/b], rd[a, 10]/rd[b, 10], rd[a, 2]/rd[b, 2], rd[a, 3]/rd[b, 3], rd[a, 7]/rd[b, 7]}}, Dividers -> All] </code></pre> <p><img src="https://i.stack.imgur.com/Pq7wT.png" alt="summary table"></p> <hr> <p><strong>Edit</strong></p> <p>The latest refinements (credit, once again, to Mr.Wizard) support nesting:</p> <pre><code>ClearAll[f, x, y] f := (x/(x + 3 + 2 y) + y)/7 x; f f // FullForm x = 14/3; y = 1/3; f BaseForm[N[f], 10] x = rd[14/3, 10]; y = rd[1/3, 10]; f x = rd[14/3, 2]; y = rd[1/3, 2]; f x = rd[14/3, 5]; y = rd[1/3, 5]; f </code></pre> <p><img src="https://i.stack.imgur.com/xH3Hb.png" alt="nesting"></p>

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