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literal notranslate"><span class="pre">numbers</span></code> — Numeric abstract base classes</a></li>193                <li class="right">194                    195 196    <div class="inline-search" role="search">197        <form class="inline-search" action="../search.html" method="get">198          <input placeholder="Quick search" aria-label="Quick search" type="search" name="q" id="search-box">199          <input type="submit" value="Go">200        </form>201    </div>202                     |203                </li>204            <li class="right">205<label class="theme-selector-label">206    Theme207    <select class="theme-selector" oninput="activateTheme(this.value)">208        <option value="auto" selected>Auto</option>209        <option value="light">Light</option>210        <option value="dark">Dark</option>211    </select>212</label> |</li>213            214      </ul>215    </div>    216 217    <div class="document">218      <div class="documentwrapper">219        <div class="bodywrapper">220          <div class="body" role="main">221            222  <section id="module-numbers">223<span id="numbers-numeric-abstract-base-classes"></span><h1><code class="xref py py-mod docutils literal notranslate"><span class="pre">numbers</span></code> — Numeric abstract base classes<a class="headerlink" href="#module-numbers" title="Link to this heading">¶</a></h1>224<p><strong>Source code:</strong> <a class="extlink-source reference external" href="https://github.com/python/cpython/tree/main/Lib/numbers.py">Lib/numbers.py</a></p>225<hr class="docutils" />226<p>The <code class="xref py py-mod docutils literal notranslate"><span class="pre">numbers</span></code> module (<span class="target" id="index-0"></span><a class="pep reference external" href="https://peps.python.org/pep-3141/"><strong>PEP 3141</strong></a>) defines a hierarchy of numeric227<a class="reference internal" href="../glossary.html#term-abstract-base-class"><span class="xref std std-term">abstract base classes</span></a> which progressively define228more operations.  None of the types defined in this module are intended to be instantiated.</p>229<dl class="py class">230<dt class="sig sig-object py" id="numbers.Number">231<em class="property"><span class="k"><span class="pre">class</span></span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">numbers.</span></span><span class="sig-name descname"><span class="pre">Number</span></span><a class="headerlink" href="#numbers.Number" title="Link to this definition">¶</a></dt>232<dd><p>The root of the numeric hierarchy. If you just want to check if an argument233<em>x</em> is a number, without caring what kind, use <code class="docutils literal notranslate"><span class="pre">isinstance(x,</span> <span class="pre">Number)</span></code>.</p>234</dd></dl>235 236<section id="the-numeric-tower">237<h2>The numeric tower<a class="headerlink" href="#the-numeric-tower" title="Link to this heading">¶</a></h2>238<dl class="py class">239<dt class="sig sig-object py" id="numbers.Complex">240<em class="property"><span class="k"><span class="pre">class</span></span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">numbers.</span></span><span class="sig-name descname"><span class="pre">Complex</span></span><a class="headerlink" href="#numbers.Complex" title="Link to this definition">¶</a></dt>241<dd><p>Subclasses of this type describe complex numbers and include the operations242that work on the built-in <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> type. These are: conversions to243<code class="xref py py-class docutils literal notranslate"><span class="pre">complex</span></code> and <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>,244<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>, and245<code class="docutils literal notranslate"><span class="pre">!=</span></code>. All except <code class="docutils literal notranslate"><span class="pre">-</span></code> and <code class="docutils literal notranslate"><span class="pre">!=</span></code> are abstract.</p>246<dl class="py attribute">247<dt class="sig sig-object py" id="numbers.Complex.real">248<span class="sig-name descname"><span class="pre">real</span></span><a class="headerlink" href="#numbers.Complex.real" title="Link to this definition">¶</a></dt>249<dd><p>Abstract. Retrieves the real component of this number.</p>250</dd></dl>251 252<dl class="py attribute">253<dt class="sig sig-object py" id="numbers.Complex.imag">254<span class="sig-name descname"><span class="pre">imag</span></span><a class="headerlink" href="#numbers.Complex.imag" title="Link to this definition">¶</a></dt>255<dd><p>Abstract. Retrieves the imaginary component of this number.</p>256</dd></dl>257 258<dl class="py method">259<dt class="sig sig-object py" id="numbers.Complex.conjugate">260<em class="property"><span class="k"><span class="pre">abstractmethod</span></span><span class="w"> </span></em><span class="sig-name descname"><span class="pre">conjugate</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#numbers.Complex.conjugate" title="Link to this definition">¶</a></dt>261<dd><p>Abstract. Returns the complex conjugate. For example, <code class="docutils literal notranslate"><span class="pre">(1+3j).conjugate()</span>262<span class="pre">==</span> <span class="pre">(1-3j)</span></code>.</p>263</dd></dl>264 265</dd></dl>266 267<dl class="py class">268<dt class="sig sig-object py" id="numbers.Real">269<em class="property"><span class="k"><span class="pre">class</span></span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">numbers.</span></span><span class="sig-name descname"><span class="pre">Real</span></span><a class="headerlink" href="#numbers.Real" title="Link to this definition">¶</a></dt>270<dd><p>To <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>, <code class="xref py py-class docutils literal notranslate"><span class="pre">Real</span></code> adds the operations that work on real271numbers.</p>272<p>In short, those are: a conversion to <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>,273<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>,274<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>, and <code class="docutils literal notranslate"><span class="pre">&gt;=</span></code>.</p>275<p>Real also provides defaults for <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>,276<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>, and <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>277</dd></dl>278 279<dl class="py class">280<dt class="sig sig-object py" id="numbers.Rational">281<em class="property"><span class="k"><span class="pre">class</span></span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">numbers.</span></span><span class="sig-name descname"><span class="pre">Rational</span></span><a class="headerlink" href="#numbers.Rational" title="Link to this definition">¶</a></dt>282<dd><p>Subtypes <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> and adds <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> and283<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> properties. It also provides a default for284<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>285<p>The <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> and <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> values286should be instances of <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> and should be in lowest terms with287<code class="xref py py-attr docutils literal notranslate"><span class="pre">denominator</span></code> positive.</p>288<dl class="py attribute">289<dt class="sig sig-object py" id="numbers.Rational.numerator">290<span class="sig-name descname"><span class="pre">numerator</span></span><a class="headerlink" href="#numbers.Rational.numerator" title="Link to this definition">¶</a></dt>291<dd><p>Abstract.  The numerator of this rational number.</p>292</dd></dl>293 294<dl class="py attribute">295<dt class="sig sig-object py" id="numbers.Rational.denominator">296<span class="sig-name descname"><span class="pre">denominator</span></span><a class="headerlink" href="#numbers.Rational.denominator" title="Link to this definition">¶</a></dt>297<dd><p>Abstract.  The denominator of this rational number.</p>298</dd></dl>299 300</dd></dl>301 302<dl class="py class">303<dt class="sig sig-object py" id="numbers.Integral">304<em class="property"><span class="k"><span class="pre">class</span></span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">numbers.</span></span><span class="sig-name descname"><span class="pre">Integral</span></span><a class="headerlink" href="#numbers.Integral" title="Link to this definition">¶</a></dt>305<dd><p>Subtypes <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> and adds a conversion to <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>.  Provides306defaults for <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>, and307<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>.  Adds abstract methods for <a class="reference internal" href="functions.html#pow" title="pow"><code class="xref py py-func docutils literal notranslate"><span class="pre">pow()</span></code></a> with308modulus and bit-string operations: <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>,309<code class="docutils literal notranslate"><span class="pre">~</span></code>.</p>310</dd></dl>311 312</section>313<section id="notes-for-type-implementers">314<h2>Notes for type implementers<a class="headerlink" href="#notes-for-type-implementers" title="Link to this heading">¶</a></h2>315<p>Implementers should be careful to make equal numbers equal and hash316them to the same values. This may be subtle if there are two different317extensions of the real numbers. For example, <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>318implements <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> as follows:</p>319<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span><span class="w"> </span><span class="fm">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>320    <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>321        <span class="c1"># Get integers right.</span>322        <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>323    <span class="c1"># Expensive check, but definitely correct.</span>324    <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>325        <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>326    <span class="k">else</span><span class="p">:</span>327        <span class="c1"># Use tuple&#39;s hash to avoid a high collision rate on</span>328        <span class="c1"># simple fractions.</span>329        <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>330</pre></div>331</div>332<section id="adding-more-numeric-abcs">333<h3>Adding More Numeric ABCs<a class="headerlink" href="#adding-more-numeric-abcs" title="Link to this heading">¶</a></h3>334<p>There are, of course, more possible ABCs for numbers, and this would335be a poor hierarchy if it precluded the possibility of adding336those. You can add <code class="docutils literal notranslate"><span class="pre">MyFoo</span></code> between <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> and337<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> with:</p>338<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span><span class="w"> </span><span class="nc">MyFoo</span><span class="p">(</span><span class="n">Complex</span><span class="p">):</span> <span class="o">...</span>339<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>340</pre></div>341</div>342</section>343<section id="implementing-the-arithmetic-operations">344<span id="id1"></span><h3>Implementing the arithmetic operations<a class="headerlink" href="#implementing-the-arithmetic-operations" title="Link to this heading">¶</a></h3>345<p>We want to implement the arithmetic operations so that mixed-mode346operations either call an implementation whose author knew about the347types of both arguments, or convert both to the nearest built in type348and do the operation there. For subtypes of <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>, this349means that <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> and <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> should be350defined as:</p>351<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="k">class</span><span class="w"> </span><span class="nc">MyIntegral</span><span class="p">(</span><span class="n">Integral</span><span class="p">):</span>352 353    <span class="k">def</span><span class="w"> </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>354        <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>355            <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>356        <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>357            <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>358        <span class="k">else</span><span class="p">:</span>359            <span class="k">return</span> <span class="bp">NotImplemented</span>360 361    <span class="k">def</span><span class="w"> </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>362        <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>363            <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>364        <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>365            <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>366        <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>367            <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>368        <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>369            <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>370        <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>371            <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>372        <span class="k">else</span><span class="p">:</span>373            <span class="k">return</span> <span class="bp">NotImplemented</span>374</pre></div>375</div>376<p>There are 5 different cases for a mixed-type operation on subclasses377of <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>. I’ll refer to all of the above code that doesn’t378refer to <code class="docutils literal notranslate"><span class="pre">MyIntegral</span></code> and <code class="docutils literal notranslate"><span class="pre">OtherTypeIKnowAbout</span></code> as379“boilerplate”. <code class="docutils literal notranslate"><span class="pre">a</span></code> will be an instance of <code class="docutils literal notranslate"><span class="pre">A</span></code>, which is a subtype380of <code class="xref py py-class docutils literal notranslate"><span class="pre">Complex</span></code> (<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>), and <code class="docutils literal notranslate"><span class="pre">b</span> <span class="pre">:</span> <span class="pre">B</span> <span class="pre">&lt;:</span>381<span class="pre">Complex</span></code>. I’ll consider <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">+</span> <span class="pre">b</span></code>:</p>382<ol class="arabic simple">383<li><p>If <code class="docutils literal notranslate"><span class="pre">A</span></code> defines an <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> which accepts <code class="docutils literal notranslate"><span class="pre">b</span></code>, all is384well.</p></li>385<li><p>If <code class="docutils literal notranslate"><span class="pre">A</span></code> falls back to the boilerplate code, and it were to386return a value from <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>, we’d miss the possibility387that <code class="docutils literal notranslate"><span class="pre">B</span></code> defines a more intelligent <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>, so the388boilerplate should return <a class="reference internal" href="constants.html#NotImplemented" title="NotImplemented"><code class="xref py py-data docutils literal notranslate"><span class="pre">NotImplemented</span></code></a> from389<code class="xref py py-meth docutils literal notranslate"><span class="pre">__add__()</span></code>. (Or <code class="docutils literal notranslate"><span class="pre">A</span></code> may not implement <code class="xref py py-meth docutils literal notranslate"><span class="pre">__add__()</span></code> at390all.)</p></li>391<li><p>Then <code class="docutils literal notranslate"><span class="pre">B</span></code>’s <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> gets a chance. If it accepts392<code class="docutils literal notranslate"><span class="pre">a</span></code>, all is well.</p></li>393<li><p>If it falls back to the boilerplate, there are no more possible394methods to try, so this is where the default implementation395should live.</p></li>396<li><p>If <code class="docutils literal notranslate"><span class="pre">B</span> <span class="pre">&lt;:</span> <span class="pre">A</span></code>, Python tries <code class="docutils literal notranslate"><span class="pre">B.__radd__</span></code> before397<code class="docutils literal notranslate"><span class="pre">A.__add__</span></code>. This is ok, because it was implemented with398knowledge of <code class="docutils literal notranslate"><span class="pre">A</span></code>, so it can handle those instances before399delegating to <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>.</p></li>400</ol>401<p>If <code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">&lt;:</span> <span class="pre">Complex</span></code> and <code class="docutils literal notranslate"><span class="pre">B</span> <span class="pre">&lt;:</span> <span class="pre">Real</span></code> without sharing any other knowledge,402then the appropriate shared operation is the one involving the built403in <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>, and both <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> s land there, so <code class="docutils literal notranslate"><span class="pre">a+b</span>404<span class="pre">==</span> <span class="pre">b+a</span></code>.</p>405<p>Because most of the operations on any given type will be very similar,406it can be useful to define a helper function which generates the407forward and reverse instances of any given operator. For example,408<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> uses:</p>409<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span><span class="w"> </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>410    <span class="k">def</span><span class="w"> </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>411        <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>412            <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>413        <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>414            <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>415        <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>416            <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>417        <span class="k">else</span><span class="p">:</span>418            <span class="k">return</span> <span class="bp">NotImplemented</span>419    <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>420    <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>421 422    <span class="k">def</span><span class="w"> </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>423        <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>424            <span class="c1"># Includes ints.</span>425            <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>426        <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">Real</span><span class="p">):</span>427            <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>428        <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">Complex</span><span class="p">):</span>429            <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>430        <span class="k">else</span><span class="p">:</span>431            <span class="k">return</span> <span class="bp">NotImplemented</span>432    <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>433    <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>434 435    <span class="k">return</span> <span class="n">forward</span><span class="p">,</span> <span class="n">reverse</span>436 437<span class="k">def</span><span class="w"> </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>438<span class="w">    </span><span class="sd">&quot;&quot;&quot;a + b&quot;&quot;&quot;</span>439    <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>440                    <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>441                    <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>442 443<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>444 445<span class="c1"># ...</span>446</pre></div>447</div>448</section>449</section>450</section>451 452 453            <div class="clearer"></div>454          </div>455        </div>456      </div>457      <div class="sphinxsidebar" role="navigation" aria-label="Main">458        <div class="sphinxsidebarwrapper">459  <div>460    <h3><a href="../contents.html">Table of Contents</a></h3>461    <ul>462<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">numbers</span></code> — Numeric abstract base classes</a><ul>463<li><a class="reference internal" href="#the-numeric-tower">The numeric tower</a></li>464<li><a class="reference internal" href="#notes-for-type-implementers">Notes for type implementers</a><ul>465<li><a class="reference internal" href="#adding-more-numeric-abcs">Adding More Numeric ABCs</a></li>466<li><a class="reference internal" href="#implementing-the-arithmetic-operations">Implementing the arithmetic 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