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<ul>
<li><a class="reference internal" href="#">fractions – Rational Numbers</a><ul>
<li><a class="reference internal" href="#creating-fraction-instances">Creating Fraction Instances</a></li>
<li><a class="reference internal" href="#arithmetic">Arithmetic</a></li>
<li><a class="reference internal" href="#approximating-values">Approximating Values</a></li>
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<div class="section" id="module-fractions">
<span id="fractions-rational-numbers"></span><h1>fractions – Rational Numbers<a class="headerlink" href="#module-fractions" title="Permalink to this headline">¶</a></h1>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field"><th class="field-name">Purpose:</th><td class="field-body">Implements a class for working with rational numbers.</td>
</tr>
<tr class="field"><th class="field-name">Python Version:</th><td class="field-body">2.6 and later</td>
</tr>
</tbody>
</table>
<p>The Fraction class implements numerical operations for rational numbers based on the API defined by <tt class="xref py py-class docutils literal"><span class="pre">Rational</span></tt> in <tt class="xref py py-mod docutils literal"><span class="pre">numbers</span></tt>.</p>
<div class="section" id="creating-fraction-instances">
<h2>Creating Fraction Instances<a class="headerlink" href="#creating-fraction-instances" title="Permalink to this headline">¶</a></h2>
<p>As with <a class="reference internal" href="../decimal/index.html#module-decimal" title="decimal: Fixed and floating point math"><tt class="xref py py-mod docutils literal"><span class="pre">decimal</span></tt></a>, new values can be created in several ways. One easy way is to create them from separate numerator and denominator values:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="kn">import</span> <span class="nn">fractions</span>
<span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="p">[</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span> <span class="p">]:</span>
<span class="n">f</span> <span class="o">=</span> <span class="n">fractions</span><span class="o">.</span><span class="n">Fraction</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
<span class="k">print</span> <span class="s">'</span><span class="si">%s</span><span class="s">/</span><span class="si">%s</span><span class="s"> = </span><span class="si">%s</span><span class="s">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
</pre></div>
</div>
<p>The lowest common denominator is maintained as new values are computed.</p>
<div class="highlight-python"><pre>$ python fractions_create_integers.py
1/2 = 1/2
2/4 = 1/2
3/6 = 1/2</pre>
</div>
<p>Another way to create a Fraction is using a string representation of <tt class="docutils literal"><span class="pre"><numerator></span> <span class="pre">/</span> <span class="pre"><denominator></span></tt>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="kn">import</span> <span class="nn">fractions</span>
<span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="p">[</span> <span class="s">'1/2'</span><span class="p">,</span> <span class="s">'2/4'</span><span class="p">,</span> <span class="s">'3/6'</span> <span class="p">]:</span>
<span class="n">f</span> <span class="o">=</span> <span class="n">fractions</span><span class="o">.</span><span class="n">Fraction</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
<span class="k">print</span> <span class="s">'</span><span class="si">%s</span><span class="s"> = </span><span class="si">%s</span><span class="s">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-python"><pre>$ python fractions_create_strings.py
1/2 = 1/2
2/4 = 1/2
3/6 = 1/2</pre>
</div>
<p>Strings can also use the more usual decimal or floating point notation of <tt class="docutils literal"><span class="pre">[<digits>].[<digits>]</span></tt>.</p>
<div class="highlight-python"><div class="highlight"><pre><span class="kn">import</span> <span class="nn">fractions</span>
<span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="p">[</span> <span class="s">'0.5'</span><span class="p">,</span> <span class="s">'1.5'</span><span class="p">,</span> <span class="s">'2.0'</span> <span class="p">]:</span>
<span class="n">f</span> <span class="o">=</span> <span class="n">fractions</span><span class="o">.</span><span class="n">Fraction</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
<span class="k">print</span> <span class="s">'</span><span class="si">%s</span><span class="s"> = </span><span class="si">%s</span><span class="s">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-python"><pre>$ python fractions_create_strings_floats.py
0.5 = 1/2
1.5 = 3/2
2.0 = 2</pre>
</div>
<p>There are class methods for creating Fraction instances directly from other representations of rational values such as float or <a class="reference internal" href="../decimal/index.html#module-decimal" title="decimal: Fixed and floating point math"><tt class="xref py py-mod docutils literal"><span class="pre">decimal</span></tt></a>.</p>
<div class="highlight-python"><div class="highlight"><pre><span class="kn">import</span> <span class="nn">fractions</span>
<span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="p">[</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mf">2.0</span> <span class="p">]:</span>
<span class="k">print</span> <span class="s">'</span><span class="si">%s</span><span class="s"> = </span><span class="si">%s</span><span class="s">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">fractions</span><span class="o">.</span><span class="n">Fraction</span><span class="o">.</span><span class="n">from_float</span><span class="p">(</span><span class="n">v</span><span class="p">))</span>
</pre></div>
</div>
<p>Notice that for floating point values that cannot be expressed exactly the rational representation may yield unexpected results.</p>
<div class="highlight-python"><pre>$ python fractions_from_float.py
0.1 = 3602879701896397/36028797018963968
0.5 = 1/2
1.5 = 3/2
2.0 = 2</pre>
</div>
<p>Using <a class="reference internal" href="../decimal/index.html#module-decimal" title="decimal: Fixed and floating point math"><tt class="xref py py-mod docutils literal"><span class="pre">decimal</span></tt></a> representations of the values gives the expected results.</p>
<div class="highlight-python"><div class="highlight"><pre><span class="kn">import</span> <span class="nn">decimal</span>
<span class="kn">import</span> <span class="nn">fractions</span>
<span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="p">[</span> <span class="n">decimal</span><span class="o">.</span><span class="n">Decimal</span><span class="p">(</span><span class="s">'0.1'</span><span class="p">),</span>
<span class="n">decimal</span><span class="o">.</span><span class="n">Decimal</span><span class="p">(</span><span class="s">'0.5'</span><span class="p">),</span>
<span class="n">decimal</span><span class="o">.</span><span class="n">Decimal</span><span class="p">(</span><span class="s">'1.5'</span><span class="p">),</span>
<span class="n">decimal</span><span class="o">.</span><span class="n">Decimal</span><span class="p">(</span><span class="s">'2.0'</span><span class="p">),</span>
<span class="p">]:</span>
<span class="k">print</span> <span class="s">'</span><span class="si">%s</span><span class="s"> = </span><span class="si">%s</span><span class="s">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">fractions</span><span class="o">.</span><span class="n">Fraction</span><span class="o">.</span><span class="n">from_decimal</span><span class="p">(</span><span class="n">v</span><span class="p">))</span>
</pre></div>
</div>
<div class="highlight-python"><pre>$ python fractions_from_decimal.py
0.1 = 1/10
0.5 = 1/2
1.5 = 3/2
2.0 = 2</pre>
</div>
</div>
<div class="section" id="arithmetic">
<h2>Arithmetic<a class="headerlink" href="#arithmetic" title="Permalink to this headline">¶</a></h2>
<p>Once the fractions are instantiated, they can be used in mathematical expressions as you would expect.</p>
<div class="highlight-python"><div class="highlight"><pre><span class="kn">import</span> <span class="nn">fractions</span>
<span class="n">f1</span> <span class="o">=</span> <span class="n">fractions</span><span class="o">.</span><span class="n">Fraction</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">f2</span> <span class="o">=</span> <span class="n">fractions</span><span class="o">.</span><span class="n">Fraction</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="k">print</span> <span class="s">'</span><span class="si">%s</span><span class="s"> + </span><span class="si">%s</span><span class="s"> = </span><span class="si">%s</span><span class="s">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">f1</span><span class="p">,</span> <span class="n">f2</span><span class="p">,</span> <span class="n">f1</span> <span class="o">+</span> <span class="n">f2</span><span class="p">)</span>
<span class="k">print</span> <span class="s">'</span><span class="si">%s</span><span class="s"> - </span><span class="si">%s</span><span class="s"> = </span><span class="si">%s</span><span class="s">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">f1</span><span class="p">,</span> <span class="n">f2</span><span class="p">,</span> <span class="n">f1</span> <span class="o">-</span> <span class="n">f2</span><span class="p">)</span>
<span class="k">print</span> <span class="s">'</span><span class="si">%s</span><span class="s"> * </span><span class="si">%s</span><span class="s"> = </span><span class="si">%s</span><span class="s">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">f1</span><span class="p">,</span> <span class="n">f2</span><span class="p">,</span> <span class="n">f1</span> <span class="o">*</span> <span class="n">f2</span><span class="p">)</span>
<span class="k">print</span> <span class="s">'</span><span class="si">%s</span><span class="s"> / </span><span class="si">%s</span><span class="s"> = </span><span class="si">%s</span><span class="s">'</span> <span class="o">%</span> <span class="p">(</span><span class="n">f1</span><span class="p">,</span> <span class="n">f2</span><span class="p">,</span> <span class="n">f1</span> <span class="o">/</span> <span class="n">f2</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-python"><pre>$ python fractions_arithmetic.py
1/2 + 3/4 = 5/4
1/2 - 3/4 = -1/4
1/2 * 3/4 = 3/8
1/2 / 3/4 = 2/3</pre>
</div>
</div>
<div class="section" id="approximating-values">
<h2>Approximating Values<a class="headerlink" href="#approximating-values" title="Permalink to this headline">¶</a></h2>
<p>A useful feature of Fraction is the ability to convert a floating point number to an approximate rational value by limiting the size of the denominator.</p>
<div class="highlight-python"><div class="highlight"><pre><span class="kn">import</span> <span class="nn">fractions</span>
<span class="kn">import</span> <span class="nn">math</span>
<span class="k">print</span> <span class="s">'PI ='</span><span class="p">,</span> <span class="n">math</span><span class="o">.</span><span class="n">pi</span>
<span class="n">f_pi</span> <span class="o">=</span> <span class="n">fractions</span><span class="o">.</span><span class="n">Fraction</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">))</span>
<span class="k">print</span> <span class="s">'No limit ='</span><span class="p">,</span> <span class="n">f_pi</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="mi">5</span><span class="p">):</span>
<span class="n">limited</span> <span class="o">=</span> <span class="n">f_pi</span><span class="o">.</span><span class="n">limit_denominator</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
<span class="k">print</span> <span class="s">'{0:8} = {1}'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">limited</span><span class="p">)</span>
</pre></div>
</div>
<div class="highlight-python"><pre>$ python fractions_limit_denominator.py
PI = 3.14159265359
No limit = 314159265359/100000000000
1 = 3
6 = 19/6
11 = 22/7
16 = 22/7
21 = 22/7
26 = 22/7
31 = 22/7
36 = 22/7
41 = 22/7
46 = 22/7
51 = 22/7
56 = 22/7
61 = 179/57
66 = 201/64
71 = 223/71
76 = 223/71
81 = 245/78
86 = 267/85
91 = 267/85
96 = 289/92</pre>
</div>
<div class="admonition-see-also admonition seealso">
<p class="first admonition-title">See also</p>
<dl class="last docutils">
<dt><a class="reference external" href="http://docs.python.org/library/fractions.html">fractions</a></dt>
<dd>The standard library documentation for this module.</dd>
<dt><a class="reference internal" href="../decimal/index.html#module-decimal" title="decimal: Fixed and floating point math"><tt class="xref py py-mod docutils literal"><span class="pre">decimal</span></tt></a></dt>
<dd>The decimal module provides an API for fixed and floating point math.</dd>
<dt><tt class="xref py py-mod docutils literal"><span class="pre">numbers</span></tt></dt>
<dd>Numeric abstract base classes.</dd>
</dl>
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