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    <p><a href="http://reference.wolfram.com/mathematica/ref/Convergents.html" rel="nofollow noreferrer">Convergents</a> of <a href="http://reference.wolfram.com/mathematica/ref/ContinuedFraction.html" rel="nofollow noreferrer">continued fractions</a> offer a useful method for getting better and better fractional representations of an irrational number. I've also found them helpful for understanding connections to other ideas by way of the <a href="http://en.wikipedia.org/wiki/Euclidean_algorithm" rel="nofollow noreferrer">Euclidean algorithm</a>. </p> <p>Let's use convergents to approximate pi and the square root of two.</p> <pre><code>ClearAll[approximate]; approximate[r_, nConvergents_: 8, precision_: 10] := With[{c = Convergents[ContinuedFraction[r, nConvergents]]}, TableForm[Transpose[{c, N[r - c, precision]}], TableHeadings -&gt; {None, {Row[{"approximation of ", r}], "error"}}]] </code></pre> <p>Here's are the first 8 convergents for pi:</p> <pre><code>approximate[Pi] </code></pre> <p><img src="https://i.stack.imgur.com/GehVK.png" alt="approximate pi"></p> <p>Here are the first 8 convergents for <code>Sqrt[2]</code>:</p> <pre><code>approximate[Sqrt[2]] </code></pre> <p><img src="https://i.stack.imgur.com/xJip4.png" alt="approximate root 2"></p> <p>The successive error terms shrink and alternate direction as convergence advances. </p> <p>In <code>approximate</code>, you can optionally specify the number of convergents and precision desired.</p> <p>Enjoy.</p> <hr> <p>Here's some additional <a href="http://search.wolfram.com/?query=continued+fraction&amp;x=0&amp;y=0" rel="nofollow noreferrer">documentation</a> about continued fractions, including some lovely demonstrations.</p>
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    1. POhow to obtain the best approximate fraction for a real number in mathematica
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    1. CO@Qiang Li Here is an example of convergents homing in on an irrational number. There are some interesting geometric representations of this in the Cartesian plane. See also: [Continued fraction](http://mathworld.wolfram.com/ContinuedFraction.html)
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    2. COthank you. `ContinuedFraction` is a nice approach for this problem.
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    3. CO+1 Nice, David.
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