Edit File: numbers.html
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <title>9.1. numbers — Numeric abstract base classes — Python 2.7.5 documentation</title> <link rel="stylesheet" href="../_static/default.css" type="text/css" /> <link rel="stylesheet" href="../_static/pygments.css" type="text/css" /> <script type="text/javascript"> var DOCUMENTATION_OPTIONS = { URL_ROOT: '../', VERSION: '2.7.5', COLLAPSE_INDEX: false, FILE_SUFFIX: '.html', HAS_SOURCE: true }; </script> <script type="text/javascript" src="../_static/jquery.js"></script> <script type="text/javascript" src="../_static/underscore.js"></script> <script type="text/javascript" src="../_static/doctools.js"></script> <script type="text/javascript" src="../_static/sidebar.js"></script> <link rel="search" type="application/opensearchdescription+xml" title="Search within Python 2.7.5 documentation" href="../_static/opensearch.xml"/> <link rel="author" title="About these documents" href="../about.html" /> <link rel="copyright" title="Copyright" href="../copyright.html" /> <link rel="top" title="Python 2.7.5 documentation" href="../index.html" /> <link rel="up" title="9. Numeric and Mathematical Modules" href="numeric.html" /> <link rel="next" title="9.2. math — Mathematical functions" href="math.html" /> <link rel="prev" title="9. Numeric and Mathematical Modules" href="numeric.html" /> <link rel="shortcut icon" type="image/png" href="../_static/py.png" /> <script type="text/javascript" src="../_static/copybutton.js"></script> </head> <body> <div class="related"> <h3>Navigation</h3> <ul> <li class="right" style="margin-right: 10px"> <a href="../genindex.html" title="General Index" accesskey="I">index</a></li> <li class="right" > <a href="../py-modindex.html" title="Python Module Index" >modules</a> |</li> <li class="right" > <a href="math.html" title="9.2. math — Mathematical functions" accesskey="N">next</a> |</li> <li class="right" > <a href="numeric.html" title="9. 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Numeric and Mathematical Modules</a> »</li> </ul> </div> <div class="document"> <div class="documentwrapper"> <div class="bodywrapper"> <div class="body"> <div class="section" id="module-numbers"> <span id="numbers-numeric-abstract-base-classes"></span><h1>9.1. <a class="reference internal" href="#module-numbers" title="numbers: Numeric abstract base classes (Complex, Real, Integral, etc.)."><tt class="xref py py-mod docutils literal"><span class="pre">numbers</span></tt></a> — 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 <a class="reference internal" href="#module-numbers" title="numbers: Numeric abstract base classes (Complex, Real, Integral, etc.)."><tt class="xref py py-mod docutils literal"><span class="pre">numbers</span></tt></a> module (<span class="target" id="index-0"></span><a class="pep reference external" href="http://www.python.org/dev/peps/pep-3141"><strong>PEP 3141</strong></a>) defines a hierarchy of numeric <a class="reference internal" href="../glossary.html#term-abstract-base-class"><em class="xref std std-term">abstract base classes</em></a> 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><p>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>.</p> </dd></dl> <div class="section" id="the-numeric-tower"> <h2>9.1.1. 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 class="reference internal" href="functions.html#complex" title="complex"><tt class="xref py py-class docutils literal"><span class="pre">complex</span></tt></a> type. These are: conversions to <a class="reference internal" href="functions.html#complex" title="complex"><tt class="xref py py-class docutils literal"><span class="pre">complex</span></tt></a> and <a class="reference internal" href="functions.html#bool" title="bool"><tt class="xref py py-class docutils literal"><span class="pre">bool</span></tt></a>, <a class="reference internal" href="#numbers.Complex.real" title="numbers.Complex.real"><tt class="xref py py-attr docutils literal"><span class="pre">real</span></tt></a>, <a class="reference internal" href="#numbers.Complex.imag" title="numbers.Complex.imag"><tt class="xref py py-attr 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 class="reference internal" href="functions.html#abs" title="abs"><tt class="xref py py-func docutils literal"><span class="pre">abs()</span></tt></a>, <a class="reference internal" href="#numbers.Complex.conjugate" title="numbers.Complex.conjugate"><tt class="xref py py-meth 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><p>Abstract. Retrieves the real component of this number.</p> </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><p>Abstract. Retrieves the imaginary component of this number.</p> </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><p>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>.</p> </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 class="reference internal" href="#numbers.Complex" title="numbers.Complex"><tt class="xref py py-class docutils literal"><span class="pre">Complex</span></tt></a>, <a class="reference internal" href="#numbers.Real" title="numbers.Real"><tt class="xref py py-class 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 class="reference internal" href="functions.html#float" title="float"><tt class="xref py py-class docutils literal"><span class="pre">float</span></tt></a>, <a class="reference internal" href="math.html#math.trunc" title="math.trunc"><tt class="xref py py-func docutils literal"><span class="pre">math.trunc()</span></tt></a>, <a class="reference internal" href="functions.html#round" title="round"><tt class="xref py py-func docutils literal"><span class="pre">round()</span></tt></a>, <a class="reference internal" href="math.html#math.floor" title="math.floor"><tt class="xref py py-func docutils literal"><span class="pre">math.floor()</span></tt></a>, <a class="reference internal" href="math.html#math.ceil" title="math.ceil"><tt class="xref py py-func docutils literal"><span class="pre">math.ceil()</span></tt></a>, <a class="reference internal" href="functions.html#divmod" title="divmod"><tt class="xref py py-func 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 class="reference internal" href="functions.html#complex" title="complex"><tt class="xref py py-func docutils literal"><span class="pre">complex()</span></tt></a>, <a class="reference internal" href="#numbers.Complex.real" title="numbers.Complex.real"><tt class="xref py py-attr docutils literal"><span class="pre">real</span></tt></a>, <a class="reference internal" href="#numbers.Complex.imag" title="numbers.Complex.imag"><tt class="xref py py-attr docutils literal"><span class="pre">imag</span></tt></a>, and <a class="reference internal" href="#numbers.Complex.conjugate" title="numbers.Complex.conjugate"><tt class="xref py py-meth 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 class="reference internal" href="#numbers.Real" title="numbers.Real"><tt class="xref py py-class docutils literal"><span class="pre">Real</span></tt></a> and adds <a class="reference internal" href="#numbers.Rational.numerator" title="numbers.Rational.numerator"><tt class="xref py py-attr docutils literal"><span class="pre">numerator</span></tt></a> and <a class="reference internal" href="#numbers.Rational.denominator" title="numbers.Rational.denominator"><tt class="xref py py-attr docutils literal"><span class="pre">denominator</span></tt></a> properties, which should be in lowest terms. With these, it provides a default for <a class="reference internal" href="functions.html#float" title="float"><tt class="xref py py-func 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><p>Abstract.</p> </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><p>Abstract.</p> </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><p>Subtypes <a class="reference internal" href="#numbers.Rational" title="numbers.Rational"><tt class="xref py py-class docutils literal"><span class="pre">Rational</span></tt></a> and adds a conversion to <a class="reference internal" href="functions.html#int" title="int"><tt class="xref py py-class docutils literal"><span class="pre">int</span></tt></a>. Provides defaults for <a class="reference internal" href="functions.html#float" title="float"><tt class="xref py py-func docutils literal"><span class="pre">float()</span></tt></a>, <a class="reference internal" href="#numbers.Rational.numerator" title="numbers.Rational.numerator"><tt class="xref py py-attr docutils literal"><span class="pre">numerator</span></tt></a>, and <a class="reference internal" href="#numbers.Rational.denominator" title="numbers.Rational.denominator"><tt class="xref py py-attr docutils literal"><span class="pre">denominator</span></tt></a>. Adds abstract methods for <tt class="docutils literal"><span class="pre">**</span></tt> 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>.</p> </dd></dl> </div> <div class="section" id="notes-for-type-implementors"> <h2>9.1.2. 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 class="reference internal" href="fractions.html#fractions.Fraction" title="fractions.Fraction"><tt class="xref py py-class docutils literal"><span class="pre">fractions.Fraction</span></tt></a> implements <a class="reference internal" href="functions.html#hash" title="hash"><tt class="xref py py-func 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>9.1.2.1. 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 class="reference internal" href="#numbers.Complex" title="numbers.Complex"><tt class="xref py py-class docutils literal"><span class="pre">Complex</span></tt></a> and <a class="reference internal" href="#numbers.Real" title="numbers.Real"><tt class="xref py py-class 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>9.1.2.2. 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 class="reference internal" href="#numbers.Integral" title="numbers.Integral"><tt class="xref py py-class docutils literal"><span class="pre">Integral</span></tt></a>, this means that <a class="reference internal" href="../reference/datamodel.html#object.__add__" title="object.__add__"><tt class="xref py py-meth docutils literal"><span class="pre">__add__()</span></tt></a> and <a class="reference internal" href="../reference/datamodel.html#object.__radd__" title="object.__radd__"><tt class="xref py py-meth 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 class="reference internal" href="#numbers.Complex" title="numbers.Complex"><tt class="xref py py-class 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 class="reference internal" href="#numbers.Complex" title="numbers.Complex"><tt class="xref py py-class 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> <div><ol class="arabic simple"> <li>If <tt class="docutils literal"><span class="pre">A</span></tt> defines an <a class="reference internal" href="../reference/datamodel.html#object.__add__" title="object.__add__"><tt class="xref py py-meth 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 class="reference internal" href="../reference/datamodel.html#object.__add__" title="object.__add__"><tt class="xref py py-meth 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 class="reference internal" href="../reference/datamodel.html#object.__radd__" title="object.__radd__"><tt class="xref py py-meth docutils literal"><span class="pre">__radd__()</span></tt></a>, so the boilerplate should return <a class="reference internal" href="constants.html#NotImplemented" title="NotImplemented"><tt class="xref py py-const docutils literal"><span class="pre">NotImplemented</span></tt></a> from <a class="reference internal" href="../reference/datamodel.html#object.__add__" title="object.__add__"><tt class="xref py py-meth docutils literal"><span class="pre">__add__()</span></tt></a>. (Or <tt class="docutils literal"><span class="pre">A</span></tt> may not implement <a class="reference internal" href="../reference/datamodel.html#object.__add__" title="object.__add__"><tt class="xref py py-meth 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 class="reference internal" href="../reference/datamodel.html#object.__radd__" title="object.__radd__"><tt class="xref py py-meth 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 class="reference internal" href="#numbers.Complex" title="numbers.Complex"><tt class="xref py py-class docutils literal"><span class="pre">Complex</span></tt></a>.</li> </ol> </div></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 class="reference internal" href="functions.html#complex" title="complex"><tt class="xref py py-class docutils literal"><span class="pre">complex</span></tt></a>, and both <a class="reference internal" href="../reference/datamodel.html#object.__radd__" title="object.__radd__"><tt class="xref py py-meth 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 class="reference internal" href="fractions.html#fractions.Fraction" title="fractions.Fraction"><tt class="xref py py-class 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> </div> </div> </div> </div> </div> </div> </div> <div class="sphinxsidebar"> <div class="sphinxsidebarwrapper"> <h3><a href="../contents.html">Table Of Contents</a></h3> <ul> <li><a class="reference internal" href="#">9.1. <tt class="docutils literal"><span class="pre">numbers</span></tt> — Numeric abstract base classes</a><ul> <li><a class="reference internal" href="#the-numeric-tower">9.1.1. The numeric tower</a></li> <li><a class="reference internal" href="#notes-for-type-implementors">9.1.2. Notes for type implementors</a><ul> <li><a class="reference internal" href="#adding-more-numeric-abcs">9.1.2.1. Adding More Numeric ABCs</a></li> <li><a class="reference internal" href="#implementing-the-arithmetic-operations">9.1.2.2. Implementing the arithmetic operations</a></li> </ul> </li> </ul> </li> </ul> <h4>Previous topic</h4> <p class="topless"><a href="numeric.html" title="previous chapter">9. Numeric and Mathematical Modules</a></p> <h4>Next topic</h4> <p class="topless"><a href="math.html" title="next chapter">9.2. <tt class="docutils literal"><span class="pre">math</span></tt> — Mathematical functions</a></p> <h3>This Page</h3> <ul class="this-page-menu"> <li><a href="../bugs.html">Report a Bug</a></li> <li><a href="../_sources/library/numbers.txt" rel="nofollow">Show Source</a></li> </ul> <div id="searchbox" style="display: none"> <h3>Quick search</h3> <form class="search" action="../search.html" method="get"> <input type="text" name="q" /> <input type="submit" value="Go" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form> <p class="searchtip" style="font-size: 90%"> Enter search terms or a module, class or function name. </p> </div> <script type="text/javascript">$('#searchbox').show(0);</script> </div> </div> <div class="clearer"></div> </div> <div class="related"> <h3>Navigation</h3> <ul> <li class="right" style="margin-right: 10px"> <a href="../genindex.html" title="General Index" >index</a></li> <li class="right" > <a href="../py-modindex.html" title="Python Module Index" >modules</a> |</li> <li class="right" > <a href="math.html" title="9.2. math — Mathematical functions" >next</a> |</li> <li class="right" > <a href="numeric.html" title="9. 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