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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <title>9.5. fractions — Rational numbers — Python 2.7.5 documentation</title> <link rel="stylesheet" href="../_static/default.css" type="text/css" /> <link rel="stylesheet" href="../_static/pygments.css" type="text/css" /> <script type="text/javascript"> var DOCUMENTATION_OPTIONS = { URL_ROOT: '../', VERSION: '2.7.5', COLLAPSE_INDEX: false, FILE_SUFFIX: '.html', HAS_SOURCE: true }; </script> <script type="text/javascript" src="../_static/jquery.js"></script> <script type="text/javascript" src="../_static/underscore.js"></script> <script type="text/javascript" src="../_static/doctools.js"></script> <script type="text/javascript" src="../_static/sidebar.js"></script> <link rel="search" type="application/opensearchdescription+xml" title="Search within Python 2.7.5 documentation" href="../_static/opensearch.xml"/> <link rel="author" title="About these documents" href="../about.html" /> <link rel="copyright" title="Copyright" href="../copyright.html" /> <link rel="top" title="Python 2.7.5 documentation" href="../index.html" /> <link rel="up" title="9. 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Numeric and Mathematical Modules</a> »</li> </ul> </div> <div class="document"> <div class="documentwrapper"> <div class="bodywrapper"> <div class="body"> <div class="section" id="module-fractions"> <span id="fractions-rational-numbers"></span><h1>9.5. <a class="reference internal" href="#module-fractions" title="fractions: Rational numbers."><tt class="xref py py-mod docutils literal"><span class="pre">fractions</span></tt></a> — Rational numbers<a class="headerlink" href="#module-fractions" title="Permalink to this headline">¶</a></h1> <p class="versionadded"> <span class="versionmodified">New in version 2.6.</span></p> <p><strong>Source code:</strong> <a class="reference external" href="http://hg.python.org/cpython/file/2.7/Lib/fractions.py">Lib/fractions.py</a></p> <hr class="docutils" /> <p>The <a class="reference internal" href="#module-fractions" title="fractions: Rational numbers."><tt class="xref py py-mod docutils literal"><span class="pre">fractions</span></tt></a> module provides support for rational number arithmetic.</p> <p>A Fraction instance can be constructed from a pair of integers, from another rational number, or from a string.</p> <dl class="class"> <dt id="fractions.Fraction"> <em class="property">class </em><tt class="descclassname">fractions.</tt><tt class="descname">Fraction</tt><big>(</big><em>numerator=0</em>, <em>denominator=1</em><big>)</big><a class="headerlink" href="#fractions.Fraction" title="Permalink to this definition">¶</a></dt> <dt> <em class="property">class </em><tt class="descclassname">fractions.</tt><tt class="descname">Fraction</tt><big>(</big><em>other_fraction</em><big>)</big></dt> <dt> <em class="property">class </em><tt class="descclassname">fractions.</tt><tt class="descname">Fraction</tt><big>(</big><em>float</em><big>)</big></dt> <dt> <em class="property">class </em><tt class="descclassname">fractions.</tt><tt class="descname">Fraction</tt><big>(</big><em>decimal</em><big>)</big></dt> <dt> <em class="property">class </em><tt class="descclassname">fractions.</tt><tt class="descname">Fraction</tt><big>(</big><em>string</em><big>)</big></dt> <dd><p>The first version requires that <em>numerator</em> and <em>denominator</em> are instances of <a class="reference internal" href="numbers.html#numbers.Rational" title="numbers.Rational"><tt class="xref py py-class docutils literal"><span class="pre">numbers.Rational</span></tt></a> and returns a new <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><tt class="xref py py-class docutils literal"><span class="pre">Fraction</span></tt></a> instance with value <tt class="docutils literal"><span class="pre">numerator/denominator</span></tt>. If <em>denominator</em> is <tt class="xref py py-const docutils literal"><span class="pre">0</span></tt>, it raises a <a class="reference internal" href="exceptions.html#exceptions.ZeroDivisionError" title="exceptions.ZeroDivisionError"><tt class="xref py py-exc docutils literal"><span class="pre">ZeroDivisionError</span></tt></a>. The second version requires that <em>other_fraction</em> is an instance of <a class="reference internal" href="numbers.html#numbers.Rational" title="numbers.Rational"><tt class="xref py py-class docutils literal"><span class="pre">numbers.Rational</span></tt></a> and returns a <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><tt class="xref py py-class docutils literal"><span class="pre">Fraction</span></tt></a> instance with the same value. The next two versions accept either a <a class="reference internal" href="functions.html#float" title="float"><tt class="xref py py-class docutils literal"><span class="pre">float</span></tt></a> or a <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><tt class="xref py py-class docutils literal"><span class="pre">decimal.Decimal</span></tt></a> instance, and return a <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><tt class="xref py py-class docutils literal"><span class="pre">Fraction</span></tt></a> instance with exactly the same value. Note that due to the usual issues with binary floating-point (see <a class="reference internal" href="../tutorial/floatingpoint.html#tut-fp-issues"><em>Floating Point Arithmetic: Issues and Limitations</em></a>), the argument to <tt class="docutils literal"><span class="pre">Fraction(1.1)</span></tt> is not exactly equal to 11/10, and so <tt class="docutils literal"><span class="pre">Fraction(1.1)</span></tt> does <em>not</em> return <tt class="docutils literal"><span class="pre">Fraction(11,</span> <span class="pre">10)</span></tt> as one might expect. (But see the documentation for the <a class="reference internal" href="#fractions.Fraction.limit_denominator" title="fractions.Fraction.limit_denominator"><tt class="xref py py-meth docutils literal"><span class="pre">limit_denominator()</span></tt></a> method below.) The last version of the constructor expects a string or unicode instance. The usual form for this instance is:</p> <div class="highlight-python"><pre>[sign] numerator ['/' denominator]</pre> </div> <p>where the optional <tt class="docutils literal"><span class="pre">sign</span></tt> may be either ‘+’ or ‘-‘ and <tt class="docutils literal"><span class="pre">numerator</span></tt> and <tt class="docutils literal"><span class="pre">denominator</span></tt> (if present) are strings of decimal digits. In addition, any string that represents a finite value and is accepted by the <a class="reference internal" href="functions.html#float" title="float"><tt class="xref py py-class docutils literal"><span class="pre">float</span></tt></a> constructor is also accepted by the <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><tt class="xref py py-class docutils literal"><span class="pre">Fraction</span></tt></a> constructor. In either form the input string may also have leading and/or trailing whitespace. Here are some examples:</p> <div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="kn">import</span> <span class="n">Fraction</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="p">)</span> <span class="go">Fraction(-8, 5)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="mi">123</span><span class="p">)</span> <span class="go">Fraction(123, 1)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">()</span> <span class="go">Fraction(0, 1)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="s">'3/7'</span><span class="p">)</span> <span class="go">Fraction(3, 7)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="s">' -3/7 '</span><span class="p">)</span> <span class="go">Fraction(-3, 7)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="s">'1.414213 </span><span class="se">\t\n</span><span class="s">'</span><span class="p">)</span> <span class="go">Fraction(1414213, 1000000)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="s">'-.125'</span><span class="p">)</span> <span class="go">Fraction(-1, 8)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="s">'7e-6'</span><span class="p">)</span> <span class="go">Fraction(7, 1000000)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="mf">2.25</span><span class="p">)</span> <span class="go">Fraction(9, 4)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="mf">1.1</span><span class="p">)</span> <span class="go">Fraction(2476979795053773, 2251799813685248)</span> <span class="gp">>>> </span><span class="kn">from</span> <span class="nn">decimal</span> <span class="kn">import</span> <span class="n">Decimal</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="n">Decimal</span><span class="p">(</span><span class="s">'1.1'</span><span class="p">))</span> <span class="go">Fraction(11, 10)</span> </pre></div> </div> <p>The <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><tt class="xref py py-class docutils literal"><span class="pre">Fraction</span></tt></a> class inherits from the abstract base class <a class="reference internal" href="numbers.html#numbers.Rational" title="numbers.Rational"><tt class="xref py py-class docutils literal"><span class="pre">numbers.Rational</span></tt></a>, and implements all of the methods and operations from that class. <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><tt class="xref py py-class docutils literal"><span class="pre">Fraction</span></tt></a> instances are hashable, and should be treated as immutable. In addition, <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><tt class="xref py py-class docutils literal"><span class="pre">Fraction</span></tt></a> has the following methods:</p> <p class="versionchanged"> <span class="versionmodified">Changed in version 2.7: </span>The <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><tt class="xref py py-class docutils literal"><span class="pre">Fraction</span></tt></a> constructor now accepts <a class="reference internal" href="functions.html#float" title="float"><tt class="xref py py-class docutils literal"><span class="pre">float</span></tt></a> and <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><tt class="xref py py-class docutils literal"><span class="pre">decimal.Decimal</span></tt></a> instances.</p> <dl class="method"> <dt id="fractions.Fraction.from_float"> <tt class="descname">from_float</tt><big>(</big><em>flt</em><big>)</big><a class="headerlink" href="#fractions.Fraction.from_float" title="Permalink to this definition">¶</a></dt> <dd><p>This class method constructs a <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><tt class="xref py py-class docutils literal"><span class="pre">Fraction</span></tt></a> representing the exact value of <em>flt</em>, which must be a <a class="reference internal" href="functions.html#float" title="float"><tt class="xref py py-class docutils literal"><span class="pre">float</span></tt></a>. Beware that <tt class="docutils literal"><span class="pre">Fraction.from_float(0.3)</span></tt> is not the same value as <tt class="docutils literal"><span class="pre">Fraction(3,</span> <span class="pre">10)</span></tt></p> <div class="admonition note"> <p class="first admonition-title">Note</p> <p class="last">From Python 2.7 onwards, you can also construct a <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><tt class="xref py py-class docutils literal"><span class="pre">Fraction</span></tt></a> instance directly from a <a class="reference internal" href="functions.html#float" title="float"><tt class="xref py py-class docutils literal"><span class="pre">float</span></tt></a>.</p> </div> </dd></dl> <dl class="method"> <dt id="fractions.Fraction.from_decimal"> <tt class="descname">from_decimal</tt><big>(</big><em>dec</em><big>)</big><a class="headerlink" href="#fractions.Fraction.from_decimal" title="Permalink to this definition">¶</a></dt> <dd><p>This class method constructs a <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><tt class="xref py py-class docutils literal"><span class="pre">Fraction</span></tt></a> representing the exact value of <em>dec</em>, which must be a <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><tt class="xref py py-class docutils literal"><span class="pre">decimal.Decimal</span></tt></a>.</p> <div class="admonition note"> <p class="first admonition-title">Note</p> <p class="last">From Python 2.7 onwards, you can also construct a <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><tt class="xref py py-class docutils literal"><span class="pre">Fraction</span></tt></a> instance directly from a <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><tt class="xref py py-class docutils literal"><span class="pre">decimal.Decimal</span></tt></a> instance.</p> </div> </dd></dl> <dl class="method"> <dt id="fractions.Fraction.limit_denominator"> <tt class="descname">limit_denominator</tt><big>(</big><em>max_denominator=1000000</em><big>)</big><a class="headerlink" href="#fractions.Fraction.limit_denominator" title="Permalink to this definition">¶</a></dt> <dd><p>Finds and returns the closest <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><tt class="xref py py-class docutils literal"><span class="pre">Fraction</span></tt></a> to <tt class="docutils literal"><span class="pre">self</span></tt> that has denominator at most max_denominator. This method is useful for finding rational approximations to a given floating-point number:</p> <div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="kn">import</span> <span class="n">Fraction</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="s">'3.1415926535897932'</span><span class="p">)</span><span class="o">.</span><span class="n">limit_denominator</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span> <span class="go">Fraction(355, 113)</span> </pre></div> </div> <p>or for recovering a rational number that’s represented as a float:</p> <div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">cos</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">))</span> <span class="go">Fraction(4503599627370497, 9007199254740992)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">limit_denominator</span><span class="p">()</span> <span class="go">Fraction(1, 2)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="mf">1.1</span><span class="p">)</span><span class="o">.</span><span class="n">limit_denominator</span><span class="p">()</span> <span class="go">Fraction(11, 10)</span> </pre></div> </div> </dd></dl> </dd></dl> <dl class="function"> <dt id="fractions.gcd"> <tt class="descclassname">fractions.</tt><tt class="descname">gcd</tt><big>(</big><em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#fractions.gcd" title="Permalink to this definition">¶</a></dt> <dd><p>Return the greatest common divisor of the integers <em>a</em> and <em>b</em>. If either <em>a</em> or <em>b</em> is nonzero, then the absolute value of <tt class="docutils literal"><span class="pre">gcd(a,</span> <span class="pre">b)</span></tt> is the largest integer that divides both <em>a</em> and <em>b</em>. <tt class="docutils literal"><span class="pre">gcd(a,b)</span></tt> has the same sign as <em>b</em> if <em>b</em> is nonzero; otherwise it takes the sign of <em>a</em>. <tt class="docutils literal"><span class="pre">gcd(0,</span> <span class="pre">0)</span></tt> returns <tt class="docutils literal"><span class="pre">0</span></tt>.</p> </dd></dl> <div class="admonition-see-also admonition seealso"> <p class="first admonition-title">See also</p> <dl class="last docutils"> <dt>Module <a class="reference internal" href="numbers.html#module-numbers" title="numbers: Numeric abstract base classes (Complex, Real, Integral, etc.)."><tt class="xref py py-mod docutils literal"><span class="pre">numbers</span></tt></a></dt> <dd>The abstract base classes making up the numeric tower.</dd> </dl> </div> </div> </div> </div> </div> <div class="sphinxsidebar"> <div class="sphinxsidebarwrapper"> <h4>Previous topic</h4> <p class="topless"><a href="decimal.html" title="previous chapter">9.4. <tt class="docutils literal"><span class="pre">decimal</span></tt> — Decimal fixed point and floating point arithmetic</a></p> <h4>Next topic</h4> <p class="topless"><a href="random.html" title="next chapter">9.6. <tt class="docutils literal"><span class="pre">random</span></tt> — Generate pseudo-random numbers</a></p> <h3>This Page</h3> <ul class="this-page-menu"> <li><a href="../bugs.html">Report a Bug</a></li> <li><a href="../_sources/library/fractions.txt" rel="nofollow">Show Source</a></li> </ul> <div id="searchbox" style="display: none"> <h3>Quick search</h3> <form class="search" action="../search.html" method="get"> <input type="text" name="q" /> <input type="submit" value="Go" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form> <p class="searchtip" style="font-size: 90%"> Enter search terms or a module, class or function name. </p> </div> <script type="text/javascript">$('#searchbox').show(0);</script> </div> </div> <div class="clearer"></div> </div> <div class="related"> <h3>Navigation</h3> <ul> <li class="right" style="margin-right: 10px"> <a href="../genindex.html" title="General Index" >index</a></li> <li class="right" > <a href="../py-modindex.html" title="Python Module Index" >modules</a> |</li> <li class="right" > <a href="random.html" title="9.6. random — Generate pseudo-random numbers" >next</a> |</li> <li class="right" > <a href="decimal.html" title="9.4. decimal — Decimal fixed point and floating point arithmetic" >previous</a> |</li> <li><img src="../_static/py.png" alt="" style="vertical-align: middle; margin-top: -1px"/></li> <li><a href="http://www.python.org/">Python</a> »</li> <li> <a href="../index.html">Python 2.7.5 documentation</a> » </li> <li><a href="index.html" >The Python Standard Library</a> »</li> <li><a href="numeric.html" >9. 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