<!DOCTYPE html> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8" /> <title>fractions — Rational numbers — Python 3.7.4 documentation</title> <link rel="stylesheet" href="../_static/pydoctheme.css" type="text/css" /> <link rel="stylesheet" href="../_static/pygments.css" type="text/css" /> <script type="text/javascript" id="documentation_options" data-url_root="../" src="../_static/documentation_options.js"></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/language_data.js"></script> <script type="text/javascript" src="../_static/sidebar.js"></script> <link rel="search" type="application/opensearchdescription+xml" title="Search within Python 3.7.4 documentation" href="../_static/opensearch.xml"/> <link rel="author" title="About these documents" href="../about.html" /> <link rel="index" title="Index" href="../genindex.html" /> <link rel="search" title="Search" href="../search.html" /> <link rel="copyright" title="Copyright" href="../copyright.html" /> <link rel="next" title="random — Generate pseudo-random numbers" href="random.html" /> <link rel="prev" title="decimal — Decimal fixed point and floating point arithmetic" href="decimal.html" /> <link rel="shortcut icon" type="image/png" href="../_static/py.png" /> <link rel="canonical" href="https://docs.python.org/3/library/fractions.html" /> <script type="text/javascript" src="../_static/copybutton.js"></script> <script type="text/javascript" src="../_static/switchers.js"></script> <style> @media only screen { table.full-width-table { width: 100%; 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If <em>denominator</em> is <code class="xref py py-const docutils literal notranslate"><span class="pre">0</span></code>, it raises a <a class="reference internal" href="exceptions.html#ZeroDivisionError" title="ZeroDivisionError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ZeroDivisionError</span></code></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"><code class="xref py py-class docutils literal notranslate"><span class="pre">numbers.Rational</span></code></a> and returns a <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> instance with the same value. The next two versions accept either a <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a> or a <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><code class="xref py py-class docutils literal notranslate"><span class="pre">decimal.Decimal</span></code></a> instance, and return a <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></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"><span class="std std-ref">Floating Point Arithmetic: Issues and Limitations</span></a>), the argument to <code class="docutils literal notranslate"><span class="pre">Fraction(1.1)</span></code> is not exactly equal to 11/10, and so <code class="docutils literal notranslate"><span class="pre">Fraction(1.1)</span></code> does <em>not</em> return <code class="docutils literal notranslate"><span class="pre">Fraction(11,</span> <span class="pre">10)</span></code> as one might expect. (But see the documentation for the <a class="reference internal" href="#fractions.Fraction.limit_denominator" title="fractions.Fraction.limit_denominator"><code class="xref py py-meth docutils literal notranslate"><span class="pre">limit_denominator()</span></code></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-python3 notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="n">sign</span><span class="p">]</span> <span class="n">numerator</span> <span class="p">[</span><span class="s1">'/'</span> <span class="n">denominator</span><span class="p">]</span> </pre></div> </div> <p>where the optional <code class="docutils literal notranslate"><span class="pre">sign</span></code> may be either ‘+’ or ‘-‘ and <code class="docutils literal notranslate"><span class="pre">numerator</span></code> and <code class="docutils literal notranslate"><span class="pre">denominator</span></code> (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"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a> constructor is also accepted by the <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> constructor. In either form the input string may also have leading and/or trailing whitespace. Here are some examples:</p> <div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="k">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="s1">'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="s1">' -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="s1">'1.414213 </span><span class="se">\t\n</span><span class="s1">'</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="s1">'-.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="s1">'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="k">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="s1">'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"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> class inherits from the abstract base class <a class="reference internal" href="numbers.html#numbers.Rational" title="numbers.Rational"><code class="xref py py-class docutils literal notranslate"><span class="pre">numbers.Rational</span></code></a>, and implements all of the methods and operations from that class. <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> instances are hashable, and should be treated as immutable. In addition, <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> has the following properties and methods:</p> <div class="versionchanged"> <p><span class="versionmodified changed">Changed in version 3.2: </span>The <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> constructor now accepts <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a> and <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><code class="xref py py-class docutils literal notranslate"><span class="pre">decimal.Decimal</span></code></a> instances.</p> </div> <dl class="attribute"> <dt id="fractions.Fraction.numerator"> <code class="descname">numerator</code><a class="headerlink" href="#fractions.Fraction.numerator" title="Permalink to this definition">¶</a></dt> <dd><p>Numerator of the Fraction in lowest term.</p> </dd></dl> <dl class="attribute"> <dt id="fractions.Fraction.denominator"> <code class="descname">denominator</code><a class="headerlink" href="#fractions.Fraction.denominator" title="Permalink to this definition">¶</a></dt> <dd><p>Denominator of the Fraction in lowest term.</p> </dd></dl> <dl class="method"> <dt id="fractions.Fraction.from_float"> <code class="descname">from_float</code><span class="sig-paren">(</span><em>flt</em><span class="sig-paren">)</span><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"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> representing the exact value of <em>flt</em>, which must be a <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a>. Beware that <code class="docutils literal notranslate"><span class="pre">Fraction.from_float(0.3)</span></code> is not the same value as <code class="docutils literal notranslate"><span class="pre">Fraction(3,</span> <span class="pre">10)</span></code>.</p> <div class="admonition note"> <p class="admonition-title">Note</p> <p>From Python 3.2 onwards, you can also construct a <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> instance directly from a <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a>.</p> </div> </dd></dl> <dl class="method"> <dt id="fractions.Fraction.from_decimal"> <code class="descname">from_decimal</code><span class="sig-paren">(</span><em>dec</em><span class="sig-paren">)</span><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"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></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"><code class="xref py py-class docutils literal notranslate"><span class="pre">decimal.Decimal</span></code></a> instance.</p> <div class="admonition note"> <p class="admonition-title">Note</p> <p>From Python 3.2 onwards, you can also construct a <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> instance directly from a <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><code class="xref py py-class docutils literal notranslate"><span class="pre">decimal.Decimal</span></code></a> instance.</p> </div> </dd></dl> <dl class="method"> <dt id="fractions.Fraction.limit_denominator"> <code class="descname">limit_denominator</code><span class="sig-paren">(</span><em>max_denominator=1000000</em><span class="sig-paren">)</span><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"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> to <code class="docutils literal notranslate"><span class="pre">self</span></code> 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-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="k">import</span> <span class="n">Fraction</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="s1">'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-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">math</span> <span class="k">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> <dl class="method"> <dt id="fractions.Fraction.__floor__"> <code class="descname">__floor__</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction.__floor__" title="Permalink to this definition">¶</a></dt> <dd><p>Returns the greatest <a class="reference internal" href="functions.html#int" title="int"><code class="xref py py-class docutils literal notranslate"><span class="pre">int</span></code></a> <code class="docutils literal notranslate"><span class="pre"><=</span> <span class="pre">self</span></code>. This method can also be accessed through the <a class="reference internal" href="math.html#math.floor" title="math.floor"><code class="xref py py-func docutils literal notranslate"><span class="pre">math.floor()</span></code></a> function:</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">math</span> <span class="k">import</span> <span class="n">floor</span> <span class="gp">>>> </span><span class="n">floor</span><span class="p">(</span><span class="n">Fraction</span><span class="p">(</span><span class="mi">355</span><span class="p">,</span> <span class="mi">113</span><span class="p">))</span> <span class="go">3</span> </pre></div> </div> </dd></dl> <dl class="method"> <dt id="fractions.Fraction.__ceil__"> <code class="descname">__ceil__</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction.__ceil__" title="Permalink to this definition">¶</a></dt> <dd><p>Returns the least <a class="reference internal" href="functions.html#int" title="int"><code class="xref py py-class docutils literal notranslate"><span class="pre">int</span></code></a> <code class="docutils literal notranslate"><span class="pre">>=</span> <span class="pre">self</span></code>. This method can also be accessed through the <a class="reference internal" href="math.html#math.ceil" title="math.ceil"><code class="xref py py-func docutils literal notranslate"><span class="pre">math.ceil()</span></code></a> function.</p> </dd></dl> <dl class="method"> <dt id="fractions.Fraction.__round__"> <code class="descname">__round__</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction.__round__" title="Permalink to this definition">¶</a></dt> <dt> <code class="descname">__round__</code><span class="sig-paren">(</span><em>ndigits</em><span class="sig-paren">)</span></dt> <dd><p>The first version returns the nearest <a class="reference internal" href="functions.html#int" title="int"><code class="xref py py-class docutils literal notranslate"><span class="pre">int</span></code></a> to <code class="docutils literal notranslate"><span class="pre">self</span></code>, rounding half to even. The second version rounds <code class="docutils literal notranslate"><span class="pre">self</span></code> to the nearest multiple of <code class="docutils literal notranslate"><span class="pre">Fraction(1,</span> <span class="pre">10**ndigits)</span></code> (logically, if <code class="docutils literal notranslate"><span class="pre">ndigits</span></code> is negative), again rounding half toward even. This method can also be accessed through the <a class="reference internal" href="functions.html#round" title="round"><code class="xref py py-func docutils literal notranslate"><span class="pre">round()</span></code></a> function.</p> </dd></dl> </dd></dl> <dl class="function"> <dt id="fractions.gcd"> <code class="descclassname">fractions.</code><code class="descname">gcd</code><span class="sig-paren">(</span><em>a</em>, <em>b</em><span class="sig-paren">)</span><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 <code class="docutils literal notranslate"><span class="pre">gcd(a,</span> <span class="pre">b)</span></code> is the largest integer that divides both <em>a</em> and <em>b</em>. <code class="docutils literal notranslate"><span class="pre">gcd(a,b)</span></code> has the same sign as <em>b</em> if <em>b</em> is nonzero; otherwise it takes the sign of <em>a</em>. <code class="docutils literal notranslate"><span class="pre">gcd(0,</span> <span class="pre">0)</span></code> returns <code class="docutils literal notranslate"><span class="pre">0</span></code>.</p> <div class="deprecated"> <p><span class="versionmodified deprecated">Deprecated since version 3.5: </span>Use <a class="reference internal" href="math.html#math.gcd" title="math.gcd"><code class="xref py py-func docutils literal notranslate"><span class="pre">math.gcd()</span></code></a> instead.</p> </div> </dd></dl> <div class="admonition seealso"> <p class="admonition-title">See also</p> <dl class="simple"> <dt>Module <a class="reference internal" href="numbers.html#module-numbers" title="numbers: Numeric abstract base classes (Complex, Real, Integral, etc.)."><code class="xref py py-mod docutils literal notranslate"><span class="pre">numbers</span></code></a></dt><dd><p>The abstract base classes making up the numeric tower.</p> </dd> </dl> </div> </div> </div> </div> </div> <div class="sphinxsidebar" role="navigation" aria-label="main navigation"> <div class="sphinxsidebarwrapper"> <h4>Previous topic</h4> <p class="topless"><a href="decimal.html" title="previous chapter"><code class="xref py py-mod docutils literal notranslate"><span class="pre">decimal</span></code> — Decimal fixed point and floating point arithmetic</a></p> <h4>Next topic</h4> <p class="topless"><a href="random.html" title="next chapter"><code class="xref py py-mod docutils literal notranslate"><span class="pre">random</span></code> — Generate pseudo-random numbers</a></p> <div role="note" aria-label="source link"> <h3>This Page</h3> <ul class="this-page-menu"> <li><a href="../bugs.html">Report a Bug</a></li> <li> <a href="https://github.com/python/cpython/blob/3.7/Doc/library/fractions.rst" rel="nofollow">Show Source </a> </li> </ul> </div> </div> </div> <div class="clearer"></div> </div> <div class="related" role="navigation" aria-label="related navigation"> <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="random — Generate pseudo-random numbers" >next</a> |</li> <li class="right" > <a href="decimal.html" title="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="https://www.python.org/">Python</a> »</li> <li> <span class="language_switcher_placeholder">en</span> <span class="version_switcher_placeholder">3.7.4</span> <a href="../index.html">Documentation </a> » </li> <li class="nav-item nav-item-1"><a href="index.html" >The Python Standard Library</a> »</li> <li class="nav-item nav-item-2"><a href="numeric.html" >Numeric and Mathematical Modules</a> »</li> <li class="right"> <div class="inline-search" style="display: none" role="search"> <form class="inline-search" action="../search.html" method="get"> <input placeholder="Quick search" 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> </div> <script type="text/javascript">$('.inline-search').show(0);</script> | </li> </ul> </div> <div class="footer"> © <a href="../copyright.html">Copyright</a> 2001-2019, Python Software Foundation. <br /> The Python Software Foundation is a non-profit corporation. <a href="https://www.python.org/psf/donations/">Please donate.</a> <br /> Last updated on Jul 13, 2019. <a href="../bugs.html">Found a bug</a>? <br /> Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 2.0.1. </div> </body> </html>