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<section id="module-fractions">
<span id="fractions-rational-numbers"></span><h1><a class="reference internal" href="#module-fractions" title="fractions: Rational numbers."><code class="xref py py-mod docutils literal notranslate"><span class="pre">fractions</span></code></a> — Rational numbers<a class="headerlink" href="#module-fractions" title="Permalink to this headline">¶</a></h1>
<p><strong>Source code:</strong> <a class="reference external" href="https://github.com/python/cpython/tree/3.10/Lib/fractions.py">Lib/fractions.py</a></p>
<hr class="docutils" />
<p>The <a class="reference internal" href="#module-fractions" title="fractions: Rational numbers."><code class="xref py py-mod docutils literal notranslate"><span class="pre">fractions</span></code></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="py class">
<dt class="sig sig-object py" id="fractions.Fraction">
<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">fractions.</span></span><span class="sig-name descname"><span class="pre">Fraction</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">numerator</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">denominator</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction" title="Permalink to this definition">¶</a></dt>
<dt class="sig sig-object py">
<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">fractions.</span></span><span class="sig-name descname"><span class="pre">Fraction</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">other_fraction</span></span></em><span class="sig-paren">)</span></dt>
<dt class="sig sig-object py">
<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">fractions.</span></span><span class="sig-name descname"><span class="pre">Fraction</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">float</span></span></em><span class="sig-paren">)</span></dt>
<dt class="sig sig-object py">
<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">fractions.</span></span><span class="sig-name descname"><span class="pre">Fraction</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">decimal</span></span></em><span class="sig-paren">)</span></dt>
<dt class="sig sig-object py">
<em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">fractions.</span></span><span class="sig-name descname"><span class="pre">Fraction</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">string</span></span></em><span class="sig-paren">)</span></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"><code class="xref py py-class docutils literal notranslate"><span class="pre">numbers.Rational</span></code></a> and returns a new <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 value <code class="docutils literal notranslate"><span class="pre">numerator/denominator</span></code>. 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="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="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="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="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 <a class="reference internal" href="../glossary.html#term-hashable"><span class="xref std std-term">hashable</span></a>,
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>
<div class="versionchanged">
<p><span class="versionmodified changed">Changed in version 3.9: </span>The <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> function is now used to normalize the <em>numerator</em>
and <em>denominator</em>. <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> always return a <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> type.
Previously, the GCD type depended on <em>numerator</em> and <em>denominator</em>.</p>
</div>
<dl class="py attribute">
<dt class="sig sig-object py" id="fractions.Fraction.numerator">
<span class="sig-name descname"><span class="pre">numerator</span></span><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="py attribute">
<dt class="sig sig-object py" id="fractions.Fraction.denominator">
<span class="sig-name descname"><span class="pre">denominator</span></span><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="py method">
<dt class="sig sig-object py" id="fractions.Fraction.as_integer_ratio">
<span class="sig-name descname"><span class="pre">as_integer_ratio</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction.as_integer_ratio" title="Permalink to this definition">¶</a></dt>
<dd><p>Return a tuple of two integers, whose ratio is equal
to the Fraction and with a positive denominator.</p>
<div class="versionadded">
<p><span class="versionmodified added">New in version 3.8.</span></p>
</div>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="fractions.Fraction.from_float">
<em class="property"><span class="pre">classmethod</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">from_float</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">flt</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction.from_float" title="Permalink to this definition">¶</a></dt>
<dd><p>Alternative constructor which only accepts instances of
<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 class="reference internal" href="numbers.html#numbers.Integral" title="numbers.Integral"><code class="xref py py-class docutils literal notranslate"><span class="pre">numbers.Integral</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="py method">
<dt class="sig sig-object py" id="fractions.Fraction.from_decimal">
<em class="property"><span class="pre">classmethod</span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">from_decimal</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">dec</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction.from_decimal" title="Permalink to this definition">¶</a></dt>
<dd><p>Alternative constructor which only accepts instances of
<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> or <a class="reference internal" href="numbers.html#numbers.Integral" title="numbers.Integral"><code class="xref py py-class docutils literal notranslate"><span class="pre">numbers.Integral</span></code></a>.</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="py method">
<dt class="sig sig-object py" id="fractions.Fraction.limit_denominator">
<span class="sig-name descname"><span class="pre">limit_denominator</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">max_denominator</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1000000</span></span></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="doctest 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="kn">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="doctest 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="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>
<dl class="py method">
<dt class="sig sig-object py" id="fractions.Fraction.__floor__">
<span class="sig-name descname"><span class="pre">__floor__</span></span><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="doctest 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="kn">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="py method">
<dt class="sig sig-object py" id="fractions.Fraction.__ceil__">
<span class="sig-name descname"><span class="pre">__ceil__</span></span><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="py method">
<dt class="sig sig-object py" id="fractions.Fraction.__round__">
<span class="sig-name descname"><span class="pre">__round__</span></span><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 class="sig sig-object py">
<span class="sig-name descname"><span class="pre">__round__</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">ndigits</span></span></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>
<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>
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