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<div class="section" id="module-fractions">
<h1>10.5. <tt class="xref docutils literal"><span class="pre">fractions</span></tt> — 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>The <tt class="xref docutils literal"><span class="pre">fractions</span></tt> 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>string</em><big>)</big></dt>
<dd><p>The first version requires that <em>numerator</em> and <em>denominator</em> are
instances of <a title="numbers.Integral" class="reference external" href="numbers.html#numbers.Integral"><tt class="xref docutils literal"><span class="pre">numbers.Integral</span></tt></a> and returns a new
<a title="fractions.Fraction" class="reference internal" href="#fractions.Fraction"><tt class="xref 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 docutils literal"><span class="pre">0</span></tt>, it raises a
<a title="exceptions.ZeroDivisionError" class="reference external" href="exceptions.html#exceptions.ZeroDivisionError"><tt class="xref docutils literal"><span class="pre">ZeroDivisionError</span></tt></a>. The second version requires that
<em>other_fraction</em> is an instance of <a title="numbers.Rational" class="reference external" href="numbers.html#numbers.Rational"><tt class="xref docutils literal"><span class="pre">numbers.Rational</span></tt></a> and
returns an <a title="fractions.Fraction" class="reference internal" href="#fractions.Fraction"><tt class="xref docutils literal"><span class="pre">Fraction</span></tt></a> instance with the same value. The
last version of the constructor expects a string or unicode
instance in one of two possible forms. The first form 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. The second permitted form is that of a number
containing a decimal point:</p>
<div class="highlight-python"><pre>[sign] integer '.' [fraction] | [sign] '.' fraction</pre>
</div>
<p>where <tt class="docutils literal"><span class="pre">integer</span></tt> and <tt class="docutils literal"><span class="pre">fraction</span></tt> are strings of digits. 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="go">[40794 refs]</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>
</pre></div>
</div>
<p>The <a title="fractions.Fraction" class="reference internal" href="#fractions.Fraction"><tt class="xref docutils literal"><span class="pre">Fraction</span></tt></a> class inherits from the abstract base class
<a title="numbers.Rational" class="reference external" href="numbers.html#numbers.Rational"><tt class="xref docutils literal"><span class="pre">numbers.Rational</span></tt></a>, and implements all of the methods and
operations from that class. <a title="fractions.Fraction" class="reference internal" href="#fractions.Fraction"><tt class="xref docutils literal"><span class="pre">Fraction</span></tt></a> instances are hashable,
and should be treated as immutable. In addition,
<a title="fractions.Fraction" class="reference internal" href="#fractions.Fraction"><tt class="xref docutils literal"><span class="pre">Fraction</span></tt></a> has the following methods:</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>This class method constructs a <a title="fractions.Fraction" class="reference internal" href="#fractions.Fraction"><tt class="xref docutils literal"><span class="pre">Fraction</span></tt></a> representing the exact
value of <em>flt</em>, which must be a <a title="float" class="reference external" href="functions.html#float"><tt class="xref 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></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>This class method constructs a <a title="fractions.Fraction" class="reference internal" href="#fractions.Fraction"><tt class="xref docutils literal"><span class="pre">Fraction</span></tt></a> representing the exact
value of <em>dec</em>, which must be a <a title="decimal.Decimal" class="reference external" href="decimal.html#decimal.Decimal"><tt class="xref docutils literal"><span class="pre">decimal.Decimal</span></tt></a>.</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 title="fractions.Fraction" class="reference internal" href="#fractions.Fraction"><tt class="xref 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="o">.</span><span class="n">from_float</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="o">.</span><span class="n">from_float</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>
</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>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>.</dd></dl>
<div class="admonition-see-also admonition seealso">
<p class="first admonition-title">See also</p>
<dl class="last docutils">
<dt>Module <a title="Numeric abstract base classes (Complex, Real, Integral, etc.)." class="reference external" href="numbers.html#module-numbers"><tt class="xref docutils literal"><span class="pre">numbers</span></tt></a></dt>
<dd>The abstract base classes making up the numeric tower.</dd>
</dl>
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