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<div class="section" id="module-numbers">
<h1>10.1. <tt class="xref docutils literal"><span class="pre">numbers</span></tt> — Numeric abstract base classes<a class="headerlink" href="#module-numbers" 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">numbers</span></tt> module (<span class="target" id="index-440"></span><a class="reference external" href="http://www.python.org/dev/peps/pep-3141"><strong>PEP 3141</strong></a>) defines a hierarchy of numeric abstract
base classes which progressively define more operations. None of the types
defined in this module can be instantiated.</p>
<dl class="class">
<dt id="numbers.Number">
<em class="property">class </em><tt class="descclassname">numbers.</tt><tt class="descname">Number</tt><a class="headerlink" href="#numbers.Number" title="Permalink to this definition">¶</a></dt>
<dd>The root of the numeric hierarchy. If you just want to check if an argument
<em>x</em> is a number, without caring what kind, use <tt class="docutils literal"><span class="pre">isinstance(x,</span> <span class="pre">Number)</span></tt>.</dd></dl>
<div class="section" id="the-numeric-tower">
<h2>10.1.1. The numeric tower<a class="headerlink" href="#the-numeric-tower" title="Permalink to this headline">¶</a></h2>
<dl class="class">
<dt id="numbers.Complex">
<em class="property">class </em><tt class="descclassname">numbers.</tt><tt class="descname">Complex</tt><a class="headerlink" href="#numbers.Complex" title="Permalink to this definition">¶</a></dt>
<dd><p>Subclasses of this type describe complex numbers and include the operations
that work on the built-in <a title="complex" class="reference external" href="functions.html#complex"><tt class="xref docutils literal"><span class="pre">complex</span></tt></a> type. These are: conversions to
<a title="complex" class="reference external" href="functions.html#complex"><tt class="xref docutils literal"><span class="pre">complex</span></tt></a> and <a title="bool" class="reference external" href="functions.html#bool"><tt class="xref docutils literal"><span class="pre">bool</span></tt></a>, <a title="numbers.Complex.real" class="reference internal" href="#numbers.Complex.real"><tt class="xref docutils literal"><span class="pre">real</span></tt></a>, <a title="numbers.Complex.imag" class="reference internal" href="#numbers.Complex.imag"><tt class="xref docutils literal"><span class="pre">imag</span></tt></a>, <tt class="docutils literal"><span class="pre">+</span></tt>,
<tt class="docutils literal"><span class="pre">-</span></tt>, <tt class="docutils literal"><span class="pre">*</span></tt>, <tt class="docutils literal"><span class="pre">/</span></tt>, <a title="abs" class="reference external" href="functions.html#abs"><tt class="xref docutils literal"><span class="pre">abs()</span></tt></a>, <a title="numbers.Complex.conjugate" class="reference internal" href="#numbers.Complex.conjugate"><tt class="xref docutils literal"><span class="pre">conjugate()</span></tt></a>, <tt class="docutils literal"><span class="pre">==</span></tt>, and <tt class="docutils literal"><span class="pre">!=</span></tt>. All
except <tt class="docutils literal"><span class="pre">-</span></tt> and <tt class="docutils literal"><span class="pre">!=</span></tt> are abstract.</p>
<dl class="attribute">
<dt id="numbers.Complex.real">
<tt class="descname">real</tt><a class="headerlink" href="#numbers.Complex.real" title="Permalink to this definition">¶</a></dt>
<dd>Abstract. Retrieves the <a title="numbers.Real" class="reference internal" href="#numbers.Real"><tt class="xref docutils literal"><span class="pre">Real</span></tt></a> component of this number.</dd></dl>
<dl class="attribute">
<dt id="numbers.Complex.imag">
<tt class="descname">imag</tt><a class="headerlink" href="#numbers.Complex.imag" title="Permalink to this definition">¶</a></dt>
<dd>Abstract. Retrieves the <a title="numbers.Real" class="reference internal" href="#numbers.Real"><tt class="xref docutils literal"><span class="pre">Real</span></tt></a> component of this number.</dd></dl>
<dl class="method">
<dt id="numbers.Complex.conjugate">
<tt class="descname">conjugate</tt><big>(</big><big>)</big><a class="headerlink" href="#numbers.Complex.conjugate" title="Permalink to this definition">¶</a></dt>
<dd>Abstract. Returns the complex conjugate. For example, <tt class="docutils literal"><span class="pre">(1+3j).conjugate()</span>
<span class="pre">==</span> <span class="pre">(1-3j)</span></tt>.</dd></dl>
</dd></dl>
<dl class="class">
<dt id="numbers.Real">
<em class="property">class </em><tt class="descclassname">numbers.</tt><tt class="descname">Real</tt><a class="headerlink" href="#numbers.Real" title="Permalink to this definition">¶</a></dt>
<dd><p>To <a title="numbers.Complex" class="reference internal" href="#numbers.Complex"><tt class="xref docutils literal"><span class="pre">Complex</span></tt></a>, <a title="numbers.Real" class="reference internal" href="#numbers.Real"><tt class="xref docutils literal"><span class="pre">Real</span></tt></a> adds the operations that work on real
numbers.</p>
<p>In short, those are: a conversion to <a title="float" class="reference external" href="functions.html#float"><tt class="xref docutils literal"><span class="pre">float</span></tt></a>, <tt class="xref docutils literal"><span class="pre">trunc()</span></tt>,
<a title="round" class="reference external" href="functions.html#round"><tt class="xref docutils literal"><span class="pre">round()</span></tt></a>, <a title="math.floor" class="reference external" href="math.html#math.floor"><tt class="xref docutils literal"><span class="pre">math.floor()</span></tt></a>, <a title="math.ceil" class="reference external" href="math.html#math.ceil"><tt class="xref docutils literal"><span class="pre">math.ceil()</span></tt></a>, <a title="divmod" class="reference external" href="functions.html#divmod"><tt class="xref docutils literal"><span class="pre">divmod()</span></tt></a>, <tt class="docutils literal"><span class="pre">//</span></tt>,
<tt class="docutils literal"><span class="pre">%</span></tt>, <tt class="docutils literal"><span class="pre"><</span></tt>, <tt class="docutils literal"><span class="pre"><=</span></tt>, <tt class="docutils literal"><span class="pre">></span></tt>, and <tt class="docutils literal"><span class="pre">>=</span></tt>.</p>
<p>Real also provides defaults for <a title="complex" class="reference external" href="functions.html#complex"><tt class="xref docutils literal"><span class="pre">complex()</span></tt></a>, <a title="numbers.Complex.real" class="reference internal" href="#numbers.Complex.real"><tt class="xref docutils literal"><span class="pre">real</span></tt></a>,
<a title="numbers.Complex.imag" class="reference internal" href="#numbers.Complex.imag"><tt class="xref docutils literal"><span class="pre">imag</span></tt></a>, and <a title="numbers.Complex.conjugate" class="reference internal" href="#numbers.Complex.conjugate"><tt class="xref docutils literal"><span class="pre">conjugate()</span></tt></a>.</p>
</dd></dl>
<dl class="class">
<dt id="numbers.Rational">
<em class="property">class </em><tt class="descclassname">numbers.</tt><tt class="descname">Rational</tt><a class="headerlink" href="#numbers.Rational" title="Permalink to this definition">¶</a></dt>
<dd><p>Subtypes <a title="numbers.Real" class="reference internal" href="#numbers.Real"><tt class="xref docutils literal"><span class="pre">Real</span></tt></a> and adds
<a title="numbers.Rational.numerator" class="reference internal" href="#numbers.Rational.numerator"><tt class="xref docutils literal"><span class="pre">numerator</span></tt></a> and <a title="numbers.Rational.denominator" class="reference internal" href="#numbers.Rational.denominator"><tt class="xref docutils literal"><span class="pre">denominator</span></tt></a> properties, which
should be in lowest terms. With these, it provides a default for
<a title="float" class="reference external" href="functions.html#float"><tt class="xref docutils literal"><span class="pre">float()</span></tt></a>.</p>
<dl class="attribute">
<dt id="numbers.Rational.numerator">
<tt class="descname">numerator</tt><a class="headerlink" href="#numbers.Rational.numerator" title="Permalink to this definition">¶</a></dt>
<dd>Abstract.</dd></dl>
<dl class="attribute">
<dt id="numbers.Rational.denominator">
<tt class="descname">denominator</tt><a class="headerlink" href="#numbers.Rational.denominator" title="Permalink to this definition">¶</a></dt>
<dd>Abstract.</dd></dl>
</dd></dl>
<dl class="class">
<dt id="numbers.Integral">
<em class="property">class </em><tt class="descclassname">numbers.</tt><tt class="descname">Integral</tt><a class="headerlink" href="#numbers.Integral" title="Permalink to this definition">¶</a></dt>
<dd>Subtypes <a title="numbers.Rational" class="reference internal" href="#numbers.Rational"><tt class="xref docutils literal"><span class="pre">Rational</span></tt></a> and adds a conversion to <a title="int" class="reference external" href="functions.html#int"><tt class="xref docutils literal"><span class="pre">int</span></tt></a>.
Provides defaults for <a title="float" class="reference external" href="functions.html#float"><tt class="xref docutils literal"><span class="pre">float()</span></tt></a>, <a title="numbers.Rational.numerator" class="reference internal" href="#numbers.Rational.numerator"><tt class="xref docutils literal"><span class="pre">numerator</span></tt></a>, and
<a title="numbers.Rational.denominator" class="reference internal" href="#numbers.Rational.denominator"><tt class="xref docutils literal"><span class="pre">denominator</span></tt></a>, and bit-string operations: <tt class="docutils literal"><span class="pre"><<</span></tt>,
<tt class="docutils literal"><span class="pre">>></span></tt>, <tt class="docutils literal"><span class="pre">&</span></tt>, <tt class="docutils literal"><span class="pre">^</span></tt>, <tt class="docutils literal"><span class="pre">|</span></tt>, <tt class="docutils literal"><span class="pre">~</span></tt>.</dd></dl>
</div>
<div class="section" id="notes-for-type-implementors">
<h2>10.1.2. Notes for type implementors<a class="headerlink" href="#notes-for-type-implementors" title="Permalink to this headline">¶</a></h2>
<p>Implementors should be careful to make equal numbers equal and hash
them to the same values. This may be subtle if there are two different
extensions of the real numbers. For example, <a title="fractions.Fraction" class="reference external" href="fractions.html#fractions.Fraction"><tt class="xref docutils literal"><span class="pre">fractions.Fraction</span></tt></a>
implements <a title="hash" class="reference external" href="functions.html#hash"><tt class="xref docutils literal"><span class="pre">hash()</span></tt></a> as follows:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">denominator</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
<span class="c"># Get integers right.</span>
<span class="k">return</span> <span class="nb">hash</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">numerator</span><span class="p">)</span>
<span class="c"># Expensive check, but definitely correct.</span>
<span class="k">if</span> <span class="bp">self</span> <span class="o">==</span> <span class="nb">float</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">return</span> <span class="nb">hash</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="bp">self</span><span class="p">))</span>
<span class="k">else</span><span class="p">:</span>
<span class="c"># Use tuple's hash to avoid a high collision rate on</span>
<span class="c"># simple fractions.</span>
<span class="k">return</span> <span class="nb">hash</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">numerator</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">denominator</span><span class="p">))</span>
</pre></div>
</div>
<div class="section" id="adding-more-numeric-abcs">
<h3>10.1.2.1. Adding More Numeric ABCs<a class="headerlink" href="#adding-more-numeric-abcs" title="Permalink to this headline">¶</a></h3>
<p>There are, of course, more possible ABCs for numbers, and this would
be a poor hierarchy if it precluded the possibility of adding
those. You can add <tt class="docutils literal"><span class="pre">MyFoo</span></tt> between <a title="numbers.Complex" class="reference internal" href="#numbers.Complex"><tt class="xref docutils literal"><span class="pre">Complex</span></tt></a> and
<a title="numbers.Real" class="reference internal" href="#numbers.Real"><tt class="xref docutils literal"><span class="pre">Real</span></tt></a> with:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="k">class</span> <span class="nc">MyFoo</span><span class="p">(</span><span class="n">Complex</span><span class="p">):</span> <span class="o">...</span>
<span class="n">MyFoo</span><span class="o">.</span><span class="n">register</span><span class="p">(</span><span class="n">Real</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="implementing-the-arithmetic-operations">
<h3>10.1.2.2. Implementing the arithmetic operations<a class="headerlink" href="#implementing-the-arithmetic-operations" title="Permalink to this headline">¶</a></h3>
<p>We want to implement the arithmetic operations so that mixed-mode
operations either call an implementation whose author knew about the
types of both arguments, or convert both to the nearest built in type
and do the operation there. For subtypes of <a title="numbers.Integral" class="reference internal" href="#numbers.Integral"><tt class="xref docutils literal"><span class="pre">Integral</span></tt></a>, this
means that <a title="object.__add__" class="reference external" href="../reference/datamodel.html#object.__add__"><tt class="xref docutils literal"><span class="pre">__add__()</span></tt></a> and <a title="object.__radd__" class="reference external" href="../reference/datamodel.html#object.__radd__"><tt class="xref docutils literal"><span class="pre">__radd__()</span></tt></a> should be defined as:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="k">class</span> <span class="nc">MyIntegral</span><span class="p">(</span><span class="n">Integral</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">MyIntegral</span><span class="p">):</span>
<span class="k">return</span> <span class="n">do_my_adding_stuff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">OtherTypeIKnowAbout</span><span class="p">):</span>
<span class="k">return</span> <span class="n">do_my_other_adding_stuff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="bp">NotImplemented</span>
<span class="k">def</span> <span class="nf">__radd__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">MyIntegral</span><span class="p">):</span>
<span class="k">return</span> <span class="n">do_my_adding_stuff</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">OtherTypeIKnowAbout</span><span class="p">):</span>
<span class="k">return</span> <span class="n">do_my_other_adding_stuff</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Integral</span><span class="p">):</span>
<span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">other</span><span class="p">)</span> <span class="o">+</span> <span class="nb">int</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Real</span><span class="p">):</span>
<span class="k">return</span> <span class="nb">float</span><span class="p">(</span><span class="n">other</span><span class="p">)</span> <span class="o">+</span> <span class="nb">float</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Complex</span><span class="p">):</span>
<span class="k">return</span> <span class="nb">complex</span><span class="p">(</span><span class="n">other</span><span class="p">)</span> <span class="o">+</span> <span class="nb">complex</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="bp">NotImplemented</span>
</pre></div>
</div>
<p>There are 5 different cases for a mixed-type operation on subclasses
of <a title="numbers.Complex" class="reference internal" href="#numbers.Complex"><tt class="xref docutils literal"><span class="pre">Complex</span></tt></a>. I’ll refer to all of the above code that doesn’t
refer to <tt class="docutils literal"><span class="pre">MyIntegral</span></tt> and <tt class="docutils literal"><span class="pre">OtherTypeIKnowAbout</span></tt> as
“boilerplate”. <tt class="docutils literal"><span class="pre">a</span></tt> will be an instance of <tt class="docutils literal"><span class="pre">A</span></tt>, which is a subtype
of <a title="numbers.Complex" class="reference internal" href="#numbers.Complex"><tt class="xref docutils literal"><span class="pre">Complex</span></tt></a> (<tt class="docutils literal"><span class="pre">a</span> <span class="pre">:</span> <span class="pre">A</span> <span class="pre"><:</span> <span class="pre">Complex</span></tt>), and <tt class="docutils literal"><span class="pre">b</span> <span class="pre">:</span> <span class="pre">B</span> <span class="pre"><:</span>
<span class="pre">Complex</span></tt>. I’ll consider <tt class="docutils literal"><span class="pre">a</span> <span class="pre">+</span> <span class="pre">b</span></tt>:</p>
<blockquote>
<ol class="arabic simple">
<li>If <tt class="docutils literal"><span class="pre">A</span></tt> defines an <a title="object.__add__" class="reference external" href="../reference/datamodel.html#object.__add__"><tt class="xref docutils literal"><span class="pre">__add__()</span></tt></a> which accepts <tt class="docutils literal"><span class="pre">b</span></tt>, all is
well.</li>
<li>If <tt class="docutils literal"><span class="pre">A</span></tt> falls back to the boilerplate code, and it were to
return a value from <a title="object.__add__" class="reference external" href="../reference/datamodel.html#object.__add__"><tt class="xref docutils literal"><span class="pre">__add__()</span></tt></a>, we’d miss the possibility
that <tt class="docutils literal"><span class="pre">B</span></tt> defines a more intelligent <a title="object.__radd__" class="reference external" href="../reference/datamodel.html#object.__radd__"><tt class="xref docutils literal"><span class="pre">__radd__()</span></tt></a>, so the
boilerplate should return <a title="NotImplemented" class="reference external" href="constants.html#NotImplemented"><tt class="xref docutils literal"><span class="pre">NotImplemented</span></tt></a> from
<a title="object.__add__" class="reference external" href="../reference/datamodel.html#object.__add__"><tt class="xref docutils literal"><span class="pre">__add__()</span></tt></a>. (Or <tt class="docutils literal"><span class="pre">A</span></tt> may not implement <a title="object.__add__" class="reference external" href="../reference/datamodel.html#object.__add__"><tt class="xref docutils literal"><span class="pre">__add__()</span></tt></a> at
all.)</li>
<li>Then <tt class="docutils literal"><span class="pre">B</span></tt>‘s <a title="object.__radd__" class="reference external" href="../reference/datamodel.html#object.__radd__"><tt class="xref docutils literal"><span class="pre">__radd__()</span></tt></a> gets a chance. If it accepts
<tt class="docutils literal"><span class="pre">a</span></tt>, all is well.</li>
<li>If it falls back to the boilerplate, there are no more possible
methods to try, so this is where the default implementation
should live.</li>
<li>If <tt class="docutils literal"><span class="pre">B</span> <span class="pre"><:</span> <span class="pre">A</span></tt>, Python tries <tt class="docutils literal"><span class="pre">B.__radd__</span></tt> before
<tt class="docutils literal"><span class="pre">A.__add__</span></tt>. This is ok, because it was implemented with
knowledge of <tt class="docutils literal"><span class="pre">A</span></tt>, so it can handle those instances before
delegating to <a title="numbers.Complex" class="reference internal" href="#numbers.Complex"><tt class="xref docutils literal"><span class="pre">Complex</span></tt></a>.</li>
</ol>
</blockquote>
<p>If <tt class="docutils literal"><span class="pre">A</span> <span class="pre"><:</span> <span class="pre">Complex</span></tt> and <tt class="docutils literal"><span class="pre">B</span> <span class="pre"><:</span> <span class="pre">Real</span></tt> without sharing any other knowledge,
then the appropriate shared operation is the one involving the built
in <a title="complex" class="reference external" href="functions.html#complex"><tt class="xref docutils literal"><span class="pre">complex</span></tt></a>, and both <a title="object.__radd__" class="reference external" href="../reference/datamodel.html#object.__radd__"><tt class="xref docutils literal"><span class="pre">__radd__()</span></tt></a> s land there, so <tt class="docutils literal"><span class="pre">a+b</span>
<span class="pre">==</span> <span class="pre">b+a</span></tt>.</p>
<p>Because most of the operations on any given type will be very similar,
it can be useful to define a helper function which generates the
forward and reverse instances of any given operator. For example,
<a title="fractions.Fraction" class="reference external" href="fractions.html#fractions.Fraction"><tt class="xref docutils literal"><span class="pre">fractions.Fraction</span></tt></a> uses:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="k">def</span> <span class="nf">_operator_fallbacks</span><span class="p">(</span><span class="n">monomorphic_operator</span><span class="p">,</span> <span class="n">fallback_operator</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="nb">long</span><span class="p">,</span> <span class="n">Fraction</span><span class="p">)):</span>
<span class="k">return</span> <span class="n">monomorphic_operator</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="nb">float</span><span class="p">):</span>
<span class="k">return</span> <span class="n">fallback_operator</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">b</span><span class="p">)</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="nb">complex</span><span class="p">):</span>
<span class="k">return</span> <span class="n">fallback_operator</span><span class="p">(</span><span class="nb">complex</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">b</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="bp">NotImplemented</span>
<span class="n">forward</span><span class="o">.</span><span class="n">__name__</span> <span class="o">=</span> <span class="s">'__'</span> <span class="o">+</span> <span class="n">fallback_operator</span><span class="o">.</span><span class="n">__name__</span> <span class="o">+</span> <span class="s">'__'</span>
<span class="n">forward</span><span class="o">.</span><span class="n">__doc__</span> <span class="o">=</span> <span class="n">monomorphic_operator</span><span class="o">.</span><span class="n">__doc__</span>
<span class="k">def</span> <span class="nf">reverse</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
<span class="c"># Includes ints.</span>
<span class="k">return</span> <span class="n">monomorphic_operator</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">numbers</span><span class="o">.</span><span class="n">Real</span><span class="p">):</span>
<span class="k">return</span> <span class="n">fallback_operator</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="nb">float</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
<span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">numbers</span><span class="o">.</span><span class="n">Complex</span><span class="p">):</span>
<span class="k">return</span> <span class="n">fallback_operator</span><span class="p">(</span><span class="nb">complex</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="nb">complex</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="bp">NotImplemented</span>
<span class="n">reverse</span><span class="o">.</span><span class="n">__name__</span> <span class="o">=</span> <span class="s">'__r'</span> <span class="o">+</span> <span class="n">fallback_operator</span><span class="o">.</span><span class="n">__name__</span> <span class="o">+</span> <span class="s">'__'</span>
<span class="n">reverse</span><span class="o">.</span><span class="n">__doc__</span> <span class="o">=</span> <span class="n">monomorphic_operator</span><span class="o">.</span><span class="n">__doc__</span>
<span class="k">return</span> <span class="n">forward</span><span class="p">,</span> <span class="n">reverse</span>
<span class="k">def</span> <span class="nf">_add</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
<span class="sd">"""a + b"""</span>
<span class="k">return</span> <span class="n">Fraction</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">numerator</span> <span class="o">*</span> <span class="n">b</span><span class="o">.</span><span class="n">denominator</span> <span class="o">+</span>
<span class="n">b</span><span class="o">.</span><span class="n">numerator</span> <span class="o">*</span> <span class="n">a</span><span class="o">.</span><span class="n">denominator</span><span class="p">,</span>
<span class="n">a</span><span class="o">.</span><span class="n">denominator</span> <span class="o">*</span> <span class="n">b</span><span class="o">.</span><span class="n">denominator</span><span class="p">)</span>
<span class="n">__add__</span><span class="p">,</span> <span class="n">__radd__</span> <span class="o">=</span> <span class="n">_operator_fallbacks</span><span class="p">(</span><span class="n">_add</span><span class="p">,</span> <span class="n">operator</span><span class="o">.</span><span class="n">add</span><span class="p">)</span>
<span class="c"># ...</span>
</pre></div>
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<ul>
<li><a class="reference external" href="#">10.1. <tt class="docutils literal"><span class="pre">numbers</span></tt> — Numeric abstract base classes</a><ul>
<li><a class="reference external" href="#the-numeric-tower">10.1.1. The numeric tower</a></li>
<li><a class="reference external" href="#notes-for-type-implementors">10.1.2. Notes for type implementors</a><ul>
<li><a class="reference external" href="#adding-more-numeric-abcs">10.1.2.1. Adding More Numeric ABCs</a></li>
<li><a class="reference external" href="#implementing-the-arithmetic-operations">10.1.2.2. Implementing the arithmetic operations</a></li>
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