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  99. <li><a class="reference internal" href="#the-numeric-tower">The numeric tower</a></li>

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  101. <li><a class="reference internal" href="#adding-more-numeric-abcs">Adding More Numeric ABCs</a></li>

  102. <li><a class="reference internal" href="#implementing-the-arithmetic-operations">Implementing the arithmetic operations</a></li>

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  197.   <section id="module-numbers">

  198. <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> — Numeric abstract base classes<a class="headerlink" href="#module-numbers" title="Link to this heading">¶</a></h1>

  199. <p><strong>Source code:</strong> <a class="reference external" href="https://github.com/python/cpython/tree/3.12/Lib/numbers.py">Lib/numbers.py</a></p>

  200. <hr class="docutils" />

  201. <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 numeric

  202. <a class="reference internal" href="../glossary.html#term-abstract-base-class"><span class="xref std std-term">abstract base classes</span></a> which progressively define

  203. more operations.  None of the types defined in this module are intended to be instantiated.</p>

  204. <dl class="py class">

  205. <dt class="sig sig-object py" id="numbers.Number">

  206. <em class="property"><span class="pre">class</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>

  207. <dd><p>The root of the numeric hierarchy. If you just want to check if an argument

  208. <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>

  209. </dd></dl>

  210. 

  211. <section id="the-numeric-tower">

  212. <h2>The numeric tower<a class="headerlink" href="#the-numeric-tower" title="Link to this heading">¶</a></h2>

  213. <dl class="py class">

  214. <dt class="sig sig-object py" id="numbers.Complex">

  215. <em class="property"><span class="pre">class</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>

  216. <dd><p>Subclasses of this type describe complex numbers and include the operations

  217. that 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 to

  218. <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 <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>,

  219. <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>, and

  220. <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>

  221. <dl class="py attribute">

  222. <dt class="sig sig-object py" id="numbers.Complex.real">

  223. <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>

  224. <dd><p>Abstract. Retrieves the real component of this number.</p>

  225. </dd></dl>

  226. 

  227. <dl class="py attribute">

  228. <dt class="sig sig-object py" id="numbers.Complex.imag">

  229. <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>

  230. <dd><p>Abstract. Retrieves the imaginary component of this number.</p>

  231. </dd></dl>

  232. 

  233. <dl class="py method">

  234. <dt class="sig sig-object py" id="numbers.Complex.conjugate">

  235. <em class="property"><span class="pre">abstractmethod</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>

  236. <dd><p>Abstract. Returns the complex conjugate. For example, <code class="docutils literal notranslate"><span class="pre">(1+3j).conjugate()</span>

  237. <span class="pre">==</span> <span class="pre">(1-3j)</span></code>.</p>

  238. </dd></dl>

  239. 

  240. </dd></dl>

  241. 

  242. <dl class="py class">

  243. <dt class="sig sig-object py" id="numbers.Real">

  244. <em class="property"><span class="pre">class</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>

  245. <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 real

  246. numbers.</p>

  247. <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>,

  248. <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>,

  249. <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>

  250. <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>,

  251. <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>

  252. </dd></dl>

  253. 

  254. <dl class="py class">

  255. <dt class="sig sig-object py" id="numbers.Rational">

  256. <em class="property"><span class="pre">class</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>

  257. <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> and

  258. <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 for

  259. <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>

  260. <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> values

  261. should 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 with

  262. <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> positive.</p>

  263. <dl class="py attribute">

  264. <dt class="sig sig-object py" id="numbers.Rational.numerator">

  265. <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>

  266. <dd><p>Abstract.</p>

  267. </dd></dl>

  268. 

  269. <dl class="py attribute">

  270. <dt class="sig sig-object py" id="numbers.Rational.denominator">

  271. <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>

  272. <dd><p>Abstract.</p>

  273. </dd></dl>

  274. 

  275. </dd></dl>

  276. 

  277. <dl class="py class">

  278. <dt class="sig sig-object py" id="numbers.Integral">

  279. <em class="property"><span class="pre">class</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>

  280. <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>.  Provides

  281. defaults 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>, and

  282. <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> with

  283. modulus 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>,

  284. <code class="docutils literal notranslate"><span class="pre">~</span></code>.</p>

  285. </dd></dl>

  286. 

  287. </section>

  288. <section id="notes-for-type-implementors">

  289. <h2>Notes for type implementors<a class="headerlink" href="#notes-for-type-implementors" title="Link to this heading">¶</a></h2>

  290. <p>Implementors should be careful to make equal numbers equal and hash

  291. them to the same values. This may be subtle if there are two different

  292. extensions 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>

  293. implements <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>

  294. <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>

  295.     <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>

  296.         <span class="c1"># Get integers right.</span>

  297.         <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>

  298.     <span class="c1"># Expensive check, but definitely correct.</span>

  299.     <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>

  300.         <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>

  301.     <span class="k">else</span><span class="p">:</span>

  302.         <span class="c1"># Use tuple&#39;s hash to avoid a high collision rate on</span>

  303.         <span class="c1"># simple fractions.</span>

  304.         <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>

  305. </pre></div>

  306. </div>

  307. <section id="adding-more-numeric-abcs">

  308. <h3>Adding More Numeric ABCs<a class="headerlink" href="#adding-more-numeric-abcs" title="Link to this heading">¶</a></h3>

  309. <p>There are, of course, more possible ABCs for numbers, and this would

  310. be a poor hierarchy if it precluded the possibility of adding

  311. those. 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> and

  312. <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>

  313. <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>

  314. <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>

  315. </pre></div>

  316. </div>

  317. </section>

  318. <section id="implementing-the-arithmetic-operations">

  319. <span id="id1"></span><h3>Implementing the arithmetic operations<a class="headerlink" href="#implementing-the-arithmetic-operations" title="Link to this heading">¶</a></h3>

  320. <p>We want to implement the arithmetic operations so that mixed-mode

  321. operations either call an implementation whose author knew about the

  322. types of both arguments, or convert both to the nearest built in type

  323. and 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>, this

  324. means 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 be

  325. defined as:</p>

  326. <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>

  327. 

  328.     <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>

  329.         <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>

  330.             <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>

  331.         <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>

  332.             <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>

  333.         <span class="k">else</span><span class="p">:</span>

  334.             <span class="k">return</span> <span class="bp">NotImplemented</span>

  335. 

  336.     <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>

  337.         <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>

  338.             <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>

  339.         <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>

  340.             <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>

  341.         <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>

  342.             <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>

  343.         <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>

  344.             <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>

  345.         <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>

  346.             <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>

  347.         <span class="k">else</span><span class="p">:</span>

  348.             <span class="k">return</span> <span class="bp">NotImplemented</span>

  349. </pre></div>

  350. </div>

  351. <p>There are 5 different cases for a mixed-type operation on subclasses

  352. of <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’t

  353. refer to <code class="docutils literal notranslate"><span class="pre">MyIntegral</span></code> and <code class="docutils literal notranslate"><span class="pre">OtherTypeIKnowAbout</span></code> as

  354. “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 subtype

  355. of <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="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>

  356. <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>

  357. <ol class="arabic simple">

  358. <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 is

  359. well.</p></li>

  360. <li><p>If <code class="docutils literal notranslate"><span class="pre">A</span></code> falls back to the boilerplate code, and it were to

  361. return 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 possibility

  362. that <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 the

  363. boilerplate 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> from

  364. <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> at

  365. all.)</p></li>

  366. <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 accepts

  367. <code class="docutils literal notranslate"><span class="pre">a</span></code>, all is well.</p></li>

  368. <li><p>If it falls back to the boilerplate, there are no more possible

  369. methods to try, so this is where the default implementation

  370. should live.</p></li>

  371. <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> before

  372. <code class="docutils literal notranslate"><span class="pre">A.__add__</span></code>. This is ok, because it was implemented with

  373. knowledge of <code class="docutils literal notranslate"><span class="pre">A</span></code>, so it can handle those instances before

  374. delegating 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>

  375. </ol>

  376. <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,

  377. then the appropriate shared operation is the one involving the built

  378. 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>, 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>

  379. <span class="pre">==</span> <span class="pre">b+a</span></code>.</p>

  380. <p>Because most of the operations on any given type will be very similar,

  381. it can be useful to define a helper function which generates the

  382. forward and reverse instances of any given operator. For example,

  383. <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>

  384. <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>

  385.     <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>

  386.         <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>

  387.             <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>

  388.         <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>

  389.             <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>

  390.         <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>

  391.             <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>

  392.         <span class="k">else</span><span class="p">:</span>

  393.             <span class="k">return</span> <span class="bp">NotImplemented</span>

  394.     <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>

  395.     <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>

  396. 

  397.     <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>

  398.         <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>

  399.             <span class="c1"># Includes ints.</span>

  400.             <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>

  401.         <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>

  402.             <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>

  403.         <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>

  404.             <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>

  405.         <span class="k">else</span><span class="p">:</span>

  406.             <span class="k">return</span> <span class="bp">NotImplemented</span>

  407.     <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>

  408.     <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>

  409. 

  410.     <span class="k">return</span> <span class="n">forward</span><span class="p">,</span> <span class="n">reverse</span>

  411. 

  412. <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>

  413. <span class="w">    </span><span class="sd">&quot;&quot;&quot;a + b&quot;&quot;&quot;</span>

  414.     <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>

  415.                     <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>

  416.                     <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>

  417. 

  418. <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>

  419. 

  420. <span class="c1"># ...</span>

  421. </pre></div>

  422. </div>

  423. </section>

  424. </section>

  425. </section>

  426. 

  427. 

  428.             <div class="clearer"></div>

  429.           </div>

  430.         </div>

  431.       </div>

  432.       <div class="sphinxsidebar" role="navigation" aria-label="main navigation">

  433.         <div class="sphinxsidebarwrapper">

  434.   <div>

  435.     <h3><a href="../contents.html">Table of Contents</a></h3>

  436.     <ul>

  437. <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>

  438. <li><a class="reference internal" href="#the-numeric-tower">The numeric tower</a></li>

  439. <li><a class="reference internal" href="#notes-for-type-implementors">Notes for type implementors</a><ul>

  440. <li><a class="reference internal" href="#adding-more-numeric-abcs">Adding More Numeric ABCs</a></li>

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