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<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">numbers</span></code> --- 数字的抽象基类</a><ul>
<li><a class="reference internal" href="#the-numeric-tower">数字的层次</a></li>
<li><a class="reference internal" href="#notes-for-type-implementors">类型接口注释。</a><ul>
<li><a class="reference internal" href="#adding-more-numeric-abcs">加入更多数字的ABC</a></li>
<li><a class="reference internal" href="#implementing-the-arithmetic-operations">实现算术运算</a></li>
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<section id="module-numbers">
<span id="numbers-numeric-abstract-base-classes"></span><h1><a class="reference internal" href="#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> --- 数字的抽象基类<a class="headerlink" href="#module-numbers" title="永久链接至标题"></a></h1>
<p><strong>源代码：</strong> <a class="reference external" href="https://github.com/python/cpython/tree/3.8/Lib/numbers.py">Lib/numbers.py</a></p>
<hr class="docutils" />
<p><a class="reference internal" href="#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> 模块 (<span class="target" id="index-2"></span><a class="pep reference external" href="https://www.python.org/dev/peps/pep-3141"><strong>PEP 3141</strong></a>) 定义了数字 <a class="reference internal" href="../glossary.html#term-abstract-base-class"><span class="xref std std-term">抽象基类</span></a> 的层次结构，其中逐级定义了更多操作。 此模块中所定义的类型都不可被实例化。</p>
<dl class="class">
<dt id="numbers.Number">
<em class="property">class </em><code class="sig-prename descclassname">numbers.</code><code class="sig-name descname">Number</code><a class="headerlink" href="#numbers.Number" title="永久链接至目标"></a></dt>
<dd><p>数字的层次结构的基础。 如果你只想确认参数 <em>x</em> 是不是数字而不关心其类型，则使用 <code class="docutils literal notranslate"><span class="pre">isinstance(x,</span> <span class="pre">Number)</span></code>。</p>
</dd></dl>
<section id="the-numeric-tower">
<h2>数字的层次<a class="headerlink" href="#the-numeric-tower" title="永久链接至标题"></a></h2>
<dl class="class">
<dt id="numbers.Complex">
<em class="property">class </em><code class="sig-prename descclassname">numbers.</code><code class="sig-name descname">Complex</code><a class="headerlink" href="#numbers.Complex" title="永久链接至目标"></a></dt>
<dd><p>内置在类型 <a class="reference internal" href="functions.html#complex" title="complex"><code class="xref py py-class docutils literal notranslate"><span class="pre">complex</span></code></a> 里的子类描述了复数和它的运算操作。这些操作有：转化至 <a class="reference internal" href="functions.html#complex" title="complex"><code class="xref py py-class docutils literal notranslate"><span class="pre">complex</span></code></a> 和 <a class="reference internal" href="functions.html#bool" title="bool"><code class="xref py py-class docutils literal notranslate"><span class="pre">bool</span></code></a>， <a class="reference internal" href="#numbers.Complex.real" title="numbers.Complex.real"><code class="xref py py-attr docutils literal notranslate"><span class="pre">real</span></code></a>、 <a class="reference internal" href="#numbers.Complex.imag" title="numbers.Complex.imag"><code class="xref py py-attr docutils literal notranslate"><span class="pre">imag</span></code></a>、<code class="docutils literal notranslate"><span class="pre">+</span></code>、<code class="docutils literal notranslate"><span class="pre">-</span></code>、<code class="docutils literal notranslate"><span class="pre">*</span></code>、<code class="docutils literal notranslate"><span class="pre">/</span></code>、 <a class="reference internal" href="functions.html#abs" title="abs"><code class="xref py py-func docutils literal notranslate"><span class="pre">abs()</span></code></a>、 <a class="reference internal" href="#numbers.Complex.conjugate" title="numbers.Complex.conjugate"><code class="xref py py-meth docutils literal notranslate"><span class="pre">conjugate()</span></code></a>、 <code class="docutils literal notranslate"><span class="pre">==</span></code> 和 <code class="docutils literal notranslate"><span class="pre">!=</span></code>。 所有的异常，<code class="docutils literal notranslate"><span class="pre">-</span></code> 和 <code class="docutils literal notranslate"><span class="pre">!=</span></code> ，都是抽象的。</p>
<dl class="attribute">
<dt id="numbers.Complex.real">
<code class="sig-name descname">real</code><a class="headerlink" href="#numbers.Complex.real" title="永久链接至目标"></a></dt>
<dd><p>抽象的。得到该数字的实数部分。</p>
</dd></dl>
<dl class="attribute">
<dt id="numbers.Complex.imag">
<code class="sig-name descname">imag</code><a class="headerlink" href="#numbers.Complex.imag" title="永久链接至目标"></a></dt>
<dd><p>抽象的。得到该数字的虚数部分。</p>
</dd></dl>
<dl class="method">
<dt id="numbers.Complex.conjugate">
<em class="property">abstractmethod </em><code class="sig-name descname">conjugate</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#numbers.Complex.conjugate" title="永久链接至目标"></a></dt>
<dd><p>抽象的。返回共轭复数。例如 <code class="docutils literal notranslate"><span class="pre">(1+3j).conjugate()</span> <span class="pre">==</span> <span class="pre">(1-3j)</span></code>。</p>
</dd></dl>
</dd></dl>
<dl class="class">
<dt id="numbers.Real">
<em class="property">class </em><code class="sig-prename descclassname">numbers.</code><code class="sig-name descname">Real</code><a class="headerlink" href="#numbers.Real" title="永久链接至目标"></a></dt>
<dd><p>相对于 <a class="reference internal" href="#numbers.Complex" title="numbers.Complex"><code class="xref py py-class docutils literal notranslate"><span class="pre">Complex</span></code></a>，<a class="reference internal" href="#numbers.Real" title="numbers.Real"><code class="xref py py-class docutils literal notranslate"><span class="pre">Real</span></code></a> 加入了只有实数才能进行的操作。</p>
<p>简单的说，它们是：转化至 <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>，<a class="reference internal" href="math.html#math.trunc" title="math.trunc"><code class="xref py py-func docutils literal notranslate"><span class="pre">math.trunc()</span></code></a>、 <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>、 <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>、 <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>、 <a class="reference internal" href="functions.html#divmod" title="divmod"><code class="xref py py-func docutils literal notranslate"><span class="pre">divmod()</span></code></a>、 <code class="docutils literal notranslate"><span class="pre">//</span></code>、 <code class="docutils literal notranslate"><span class="pre">%</span></code>、 <code class="docutils literal notranslate"><span class="pre">&lt;</span></code>、 <code class="docutils literal notranslate"><span class="pre">&lt;=</span></code>、 <code class="docutils literal notranslate"><span class="pre">&gt;</span></code>、 和 <code class="docutils literal notranslate"><span class="pre">&gt;=</span></code>。</p>
<p>实数同样默认支持 <a class="reference internal" href="functions.html#complex" title="complex"><code class="xref py py-func docutils literal notranslate"><span class="pre">complex()</span></code></a>、 <a class="reference internal" href="#numbers.Complex.real" title="numbers.Complex.real"><code class="xref py py-attr docutils literal notranslate"><span class="pre">real</span></code></a>、 <a class="reference internal" href="#numbers.Complex.imag" title="numbers.Complex.imag"><code class="xref py py-attr docutils literal notranslate"><span class="pre">imag</span></code></a> 和 <a class="reference internal" href="#numbers.Complex.conjugate" title="numbers.Complex.conjugate"><code class="xref py py-meth docutils literal notranslate"><span class="pre">conjugate()</span></code></a>。</p>
</dd></dl>
<dl class="class">
<dt id="numbers.Rational">
<em class="property">class </em><code class="sig-prename descclassname">numbers.</code><code class="sig-name descname">Rational</code><a class="headerlink" href="#numbers.Rational" title="永久链接至目标"></a></dt>
<dd><p>子类型 <a class="reference internal" href="#numbers.Real" title="numbers.Real"><code class="xref py py-class docutils literal notranslate"><span class="pre">Real</span></code></a> 并加入 <a class="reference internal" href="#numbers.Rational.numerator" title="numbers.Rational.numerator"><code class="xref py py-attr docutils literal notranslate"><span class="pre">numerator</span></code></a> 和 <a class="reference internal" href="#numbers.Rational.denominator" title="numbers.Rational.denominator"><code class="xref py py-attr docutils literal notranslate"><span class="pre">denominator</span></code></a> 两种属性，这两种属性应该属于最低的级别。加入后，这默认支持 <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-func docutils literal notranslate"><span class="pre">float()</span></code></a>。</p>
<dl class="attribute">
<dt id="numbers.Rational.numerator">
<code class="sig-name descname">numerator</code><a class="headerlink" href="#numbers.Rational.numerator" title="永久链接至目标"></a></dt>
<dd><p>抽象的。</p>
</dd></dl>
<dl class="attribute">
<dt id="numbers.Rational.denominator">
<code class="sig-name descname">denominator</code><a class="headerlink" href="#numbers.Rational.denominator" title="永久链接至目标"></a></dt>
<dd><p>抽象的。</p>
</dd></dl>
</dd></dl>
<dl class="class">
<dt id="numbers.Integral">
<em class="property">class </em><code class="sig-prename descclassname">numbers.</code><code class="sig-name descname">Integral</code><a class="headerlink" href="#numbers.Integral" title="永久链接至目标"></a></dt>
<dd><p>子类型 <a class="reference internal" href="#numbers.Rational" title="numbers.Rational"><code class="xref py py-class docutils literal notranslate"><span class="pre">Rational</span></code></a> 加上转化至 <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>。 默认支持 <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-func docutils literal notranslate"><span class="pre">float()</span></code></a>、 <a class="reference internal" href="#numbers.Rational.numerator" title="numbers.Rational.numerator"><code class="xref py py-attr docutils literal notranslate"><span class="pre">numerator</span></code></a> 和 <a class="reference internal" href="#numbers.Rational.denominator" title="numbers.Rational.denominator"><code class="xref py py-attr docutils literal notranslate"><span class="pre">denominator</span></code></a>。 在 <code class="docutils literal notranslate"><span class="pre">**</span></code> 中加入抽象方法和比特字符串的操作： <code class="docutils literal notranslate"><span class="pre">&lt;&lt;</span></code>、 <code class="docutils literal notranslate"><span class="pre">&gt;&gt;</span></code>、 <code class="docutils literal notranslate"><span class="pre">&amp;</span></code>、 <code class="docutils literal notranslate"><span class="pre">^</span></code>、 <code class="docutils literal notranslate"><span class="pre">|</span></code>、 <code class="docutils literal notranslate"><span class="pre">~</span></code>。</p>
</dd></dl>
</section>
<section id="notes-for-type-implementors">
<h2>类型接口注释。<a class="headerlink" href="#notes-for-type-implementors" title="永久链接至标题"></a></h2>
<p>实现者需要注意使相等的数字相等并拥有同样的值。当这两个数使用不同的扩展模块时，这其中的差异是很微妙的。例如，用 <a class="reference internal" href="fractions.html#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">fractions.Fraction</span></code></a> 实现 <a class="reference internal" href="functions.html#hash" title="hash"><code class="xref py py-func docutils literal notranslate"><span class="pre">hash()</span></code></a> 如下:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="fm">__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="c1"># 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="c1"># 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="c1"># Use tuple&#39;s hash to avoid a high collision rate on</span>
<span class="c1"># 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>
<section id="adding-more-numeric-abcs">
<h3>加入更多数字的ABC<a class="headerlink" href="#adding-more-numeric-abcs" title="永久链接至标题"></a></h3>
<p>当然，这里有更多支持数字的ABC，如果不加入这些，就将缺少层次感。你可以用如下方法在 <a class="reference internal" href="#numbers.Complex" title="numbers.Complex"><code class="xref py py-class docutils literal notranslate"><span class="pre">Complex</span></code></a> 和 <a class="reference internal" href="#numbers.Real" title="numbers.Real"><code class="xref py py-class docutils literal notranslate"><span class="pre">Real</span></code></a> 中加入 <code class="docutils literal notranslate"><span class="pre">MyFoo</span></code>:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><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>
</section>
<section id="implementing-the-arithmetic-operations">
<span id="id1"></span><h3>实现算术运算<a class="headerlink" href="#implementing-the-arithmetic-operations" title="永久链接至标题"></a></h3>
<p>我们希望实现计算，因此，混合模式操作要么调用一个作者知道参数类型的实现，要么转变成为最接近的内置类型并对这个执行操作。对于子类 <a class="reference internal" href="#numbers.Integral" title="numbers.Integral"><code class="xref py py-class docutils literal notranslate"><span class="pre">Integral</span></code></a>，这意味着 <a class="reference internal" href="../reference/datamodel.html#object.__add__" title="object.__add__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__add__()</span></code></a> 和 <a class="reference internal" href="../reference/datamodel.html#object.__radd__" title="object.__radd__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__radd__()</span></code></a> 必须用如下方式定义：</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><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="fm">__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="fm">__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><a class="reference internal" href="#numbers.Complex" title="numbers.Complex"><code class="xref py py-class docutils literal notranslate"><span class="pre">Complex</span></code></a> 有 5 种不同的混合类型的操作。 我将上面提到的所有代码作为“模板”称作 <code class="docutils literal notranslate"><span class="pre">MyIntegral</span></code> 和 <code class="docutils literal notranslate"><span class="pre">OtherTypeIKnowAbout</span></code>。 <code class="docutils literal notranslate"><span class="pre">a</span></code> 是 <a class="reference internal" href="#numbers.Complex" title="numbers.Complex"><code class="xref py py-class docutils literal notranslate"><span class="pre">Complex</span></code></a> 的子类型 <code class="docutils literal notranslate"><span class="pre">A</span></code> 的实例 (<code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">:</span> <span class="pre">A</span> <span class="pre">&lt;:</span> <span class="pre">Complex</span></code>)，同时 <code class="docutils literal notranslate"><span class="pre">b</span> <span class="pre">:</span> <span class="pre">B</span> <span class="pre">&lt;:</span> <span class="pre">Complex</span></code>。 我将要计算 <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">+</span> <span class="pre">b</span></code>:</p>
<blockquote>
<div><ol class="arabic simple">
<li><p>如果 <code class="docutils literal notranslate"><span class="pre">A</span></code> 被定义成一个承认 <code class="docutils literal notranslate"><span class="pre">b</span></code> 的 <a class="reference internal" href="../reference/datamodel.html#object.__add__" title="object.__add__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__add__()</span></code></a>，一切都没有问题。</p></li>
<li><p>如果 <code class="docutils literal notranslate"><span class="pre">A</span></code> 转回成“模板”失败，它将返回一个属于 <a class="reference internal" href="../reference/datamodel.html#object.__add__" title="object.__add__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__add__()</span></code></a> 的值，我们需要避免 <code class="docutils literal notranslate"><span class="pre">B</span></code> 定义了一个更加智能的 <a class="reference internal" href="../reference/datamodel.html#object.__radd__" title="object.__radd__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__radd__()</span></code></a>，因此模板需要返回一个属于 <a class="reference internal" href="../reference/datamodel.html#object.__add__" title="object.__add__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__add__()</span></code></a> 的 <a class="reference internal" href="constants.html#NotImplemented" title="NotImplemented"><code class="xref py py-const docutils literal notranslate"><span class="pre">NotImplemented</span></code></a> 。（或者 <code class="docutils literal notranslate"><span class="pre">A</span></code> 可能完全不实现 <a class="reference internal" href="../reference/datamodel.html#object.__add__" title="object.__add__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__add__()</span></code></a> 。）</p></li>
<li><p>接着看 <code class="docutils literal notranslate"><span class="pre">B</span></code> 的 <a class="reference internal" href="../reference/datamodel.html#object.__radd__" title="object.__radd__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__radd__()</span></code></a> 。如果它承认 <code class="docutils literal notranslate"><span class="pre">a</span></code> ，一切都没有问题。</p></li>
<li><p>如果没有成功回退到模板，就没有更多的方法可以去尝试，因此这里将使用默认的实现。</p></li>
<li><p>如果 <code class="docutils literal notranslate"><span class="pre">B</span> <span class="pre">&lt;:</span> <span class="pre">A</span></code> ， Python 在 <code class="docutils literal notranslate"><span class="pre">A.__add__</span></code> 之前尝试 <code class="docutils literal notranslate"><span class="pre">B.__radd__</span></code> 。 这是可行的，是通过对 <code class="docutils literal notranslate"><span class="pre">A</span></code> 的认识实现的，因此这可以在交给 <a class="reference internal" href="#numbers.Complex" title="numbers.Complex"><code class="xref py py-class docutils literal notranslate"><span class="pre">Complex</span></code></a> 处理之前处理这些实例。</p></li>
</ol>
</div></blockquote>
<p>如果 <code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">&lt;:</span> <span class="pre">Complex</span></code> 和 <code class="docutils literal notranslate"><span class="pre">B</span> <span class="pre">&lt;:</span> <span class="pre">Real</span></code> 没有共享任何资源，那么适当的共享操作涉及内置的 <a class="reference internal" href="functions.html#complex" title="complex"><code class="xref py py-class docutils literal notranslate"><span class="pre">complex</span></code></a> ，并且分别获得 <a class="reference internal" href="../reference/datamodel.html#object.__radd__" title="object.__radd__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__radd__()</span></code></a> ，因此 <code class="docutils literal notranslate"><span class="pre">a+b</span> <span class="pre">==</span> <span class="pre">b+a</span></code>。</p>
<p>由于对任何一直类型的大部分操作是十分相似的，可以定义一个帮助函数，即一个生成后续或相反的实例的生成器。例如，使用 <a class="reference internal" href="fractions.html#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">fractions.Fraction</span></code></a> 如下：</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><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="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="vm">__name__</span> <span class="o">=</span> <span class="s1">&#39;__&#39;</span> <span class="o">+</span> <span class="n">fallback_operator</span><span class="o">.</span><span class="vm">__name__</span> <span class="o">+</span> <span class="s1">&#39;__&#39;</span>
<span class="n">forward</span><span class="o">.</span><span class="vm">__doc__</span> <span class="o">=</span> <span class="n">monomorphic_operator</span><span class="o">.</span><span class="vm">__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="c1"># 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="vm">__name__</span> <span class="o">=</span> <span class="s1">&#39;__r&#39;</span> <span class="o">+</span> <span class="n">fallback_operator</span><span class="o">.</span><span class="vm">__name__</span> <span class="o">+</span> <span class="s1">&#39;__&#39;</span>
<span class="n">reverse</span><span class="o">.</span><span class="vm">__doc__</span> <span class="o">=</span> <span class="n">monomorphic_operator</span><span class="o">.</span><span class="vm">__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="w"> </span><span class="sd">&quot;&quot;&quot;a + b&quot;&quot;&quot;</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="fm">__add__</span><span class="p">,</span> <span class="fm">__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="c1"># ...</span>
</pre></div>
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<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">numbers</span></code> --- 数字的抽象基类</a><ul>
<li><a class="reference internal" href="#the-numeric-tower">数字的层次</a></li>
<li><a class="reference internal" href="#notes-for-type-implementors">类型接口注释。</a><ul>
<li><a class="reference internal" href="#adding-more-numeric-abcs">加入更多数字的ABC</a></li>
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