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<div class="section" id="module-fractions">
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<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>
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<p><strong>Source code:</strong> <a class="reference external" href="https://github.com/python/cpython/tree/3.7/Lib/fractions.py">Lib/fractions.py</a></p>
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<hr class="docutils" />
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<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>
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<p>A Fraction instance can be constructed from a pair of integers, from
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another rational number, or from a string.</p>
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<dl class="class">
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<dt id="fractions.Fraction">
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<em class="property">class </em><code class="descclassname">fractions.</code><code class="descname">Fraction</code><span class="sig-paren">(</span><em>numerator=0</em>, <em>denominator=1</em><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction" title="Permalink to this definition">¶</a></dt>
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<dt>
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<em class="property">class </em><code class="descclassname">fractions.</code><code class="descname">Fraction</code><span class="sig-paren">(</span><em>other_fraction</em><span class="sig-paren">)</span></dt>
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<dt>
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<em class="property">class </em><code class="descclassname">fractions.</code><code class="descname">Fraction</code><span class="sig-paren">(</span><em>float</em><span class="sig-paren">)</span></dt>
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<dt>
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<em class="property">class </em><code class="descclassname">fractions.</code><code class="descname">Fraction</code><span class="sig-paren">(</span><em>decimal</em><span class="sig-paren">)</span></dt>
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<dt>
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<em class="property">class </em><code class="descclassname">fractions.</code><code class="descname">Fraction</code><span class="sig-paren">(</span><em>string</em><span class="sig-paren">)</span></dt>
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<dd><p>The first version requires that <em>numerator</em> and <em>denominator</em> are instances
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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
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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
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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
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<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
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<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
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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
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<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
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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
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argument to <code class="docutils literal notranslate"><span class="pre">Fraction(1.1)</span></code> is not exactly equal to 11/10, and so
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<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.
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(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.)
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The last version of the constructor expects a string or unicode instance.
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The usual form for this instance is:</p>
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<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>
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</pre></div>
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</div>
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<p>where the optional <code class="docutils literal notranslate"><span class="pre">sign</span></code> may be either ‘+’ or ‘-‘ and
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<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
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decimal digits. In addition, any string that represents a finite
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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
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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
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input string may also have leading and/or trailing whitespace.
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Here are some examples:</p>
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<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>
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<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>
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<span class="go">Fraction(-8, 5)</span>
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<span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="mi">123</span><span class="p">)</span>
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<span class="go">Fraction(123, 1)</span>
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<span class="gp">>>> </span><span class="n">Fraction</span><span class="p">()</span>
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<span class="go">Fraction(0, 1)</span>
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<span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="s1">'3/7'</span><span class="p">)</span>
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<span class="go">Fraction(3, 7)</span>
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<span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="s1">' -3/7 '</span><span class="p">)</span>
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<span class="go">Fraction(-3, 7)</span>
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<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>
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<span class="go">Fraction(1414213, 1000000)</span>
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<span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="s1">'-.125'</span><span class="p">)</span>
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<span class="go">Fraction(-1, 8)</span>
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<span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="s1">'7e-6'</span><span class="p">)</span>
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<span class="go">Fraction(7, 1000000)</span>
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<span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="mf">2.25</span><span class="p">)</span>
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<span class="go">Fraction(9, 4)</span>
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<span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="mf">1.1</span><span class="p">)</span>
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<span class="go">Fraction(2476979795053773, 2251799813685248)</span>
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<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">decimal</span> <span class="k">import</span> <span class="n">Decimal</span>
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<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>
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<span class="go">Fraction(11, 10)</span>
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</pre></div>
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</div>
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<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
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<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
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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,
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and should be treated as immutable. In addition,
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<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>
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<div class="versionchanged">
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<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
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<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>
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</div>
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<dl class="attribute">
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<dt id="fractions.Fraction.numerator">
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<code class="descname">numerator</code><a class="headerlink" href="#fractions.Fraction.numerator" title="Permalink to this definition">¶</a></dt>
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<dd><p>Numerator of the Fraction in lowest term.</p>
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</dd></dl>
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<dl class="attribute">
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<dt id="fractions.Fraction.denominator">
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<code class="descname">denominator</code><a class="headerlink" href="#fractions.Fraction.denominator" title="Permalink to this definition">¶</a></dt>
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<dd><p>Denominator of the Fraction in lowest term.</p>
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</dd></dl>
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<dl class="method">
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<dt id="fractions.Fraction.from_float">
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<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>
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<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
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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
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<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>
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<div class="admonition note">
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<p class="admonition-title">Note</p>
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<p>From Python 3.2 onwards, you can also construct a
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<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>
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</div>
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</dd></dl>
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<dl class="method">
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<dt id="fractions.Fraction.from_decimal">
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<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>
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<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
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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>
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<div class="admonition note">
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<p class="admonition-title">Note</p>
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<p>From Python 3.2 onwards, you can also construct a
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<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>
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instance.</p>
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</div>
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</dd></dl>
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<dl class="method">
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<dt id="fractions.Fraction.limit_denominator">
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<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>
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<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
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denominator at most max_denominator. This method is useful for finding
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rational approximations to a given floating-point number:</p>
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<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>
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<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>
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<span class="go">Fraction(355, 113)</span>
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</pre></div>
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</div>
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<p>or for recovering a rational number that’s represented as a float:</p>
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<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>
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<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>
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<span class="go">Fraction(4503599627370497, 9007199254740992)</span>
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<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>
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<span class="go">Fraction(1, 2)</span>
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<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>
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<span class="go">Fraction(11, 10)</span>
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</pre></div>
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</div>
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</dd></dl>
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<dl class="method">
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<dt id="fractions.Fraction.__floor__">
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<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>
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<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
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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>
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<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>
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<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>
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<span class="go">3</span>
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</pre></div>
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</div>
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</dd></dl>
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<dl class="method">
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<dt id="fractions.Fraction.__ceil__">
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<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>
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<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
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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>
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</dd></dl>
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<dl class="method">
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<dt id="fractions.Fraction.__round__">
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<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>
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<dt>
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<code class="descname">__round__</code><span class="sig-paren">(</span><em>ndigits</em><span class="sig-paren">)</span></dt>
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<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>,
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rounding half to even. The second version rounds <code class="docutils literal notranslate"><span class="pre">self</span></code> to the
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nearest multiple of <code class="docutils literal notranslate"><span class="pre">Fraction(1,</span> <span class="pre">10**ndigits)</span></code> (logically, if
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<code class="docutils literal notranslate"><span class="pre">ndigits</span></code> is negative), again rounding half toward even. This
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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>
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</dd></dl>
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</dd></dl>
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<dl class="function">
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<dt id="fractions.gcd">
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<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>
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<dd><p>Return the greatest common divisor of the integers <em>a</em> and <em>b</em>. If either
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<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
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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
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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>
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<span class="pre">0)</span></code> returns <code class="docutils literal notranslate"><span class="pre">0</span></code>.</p>
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<div class="deprecated">
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<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>
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</div>
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</dd></dl>
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<div class="admonition seealso">
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<p class="admonition-title">See also</p>
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<dl class="simple">
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<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>
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