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Code Editor : numbers.html

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 <section id="module-numbers">
<h1>9.1. <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">numbers</code></a> — Numeric abstract base classes<a class="headerlink" href="#module-numbers" title="Permalink to this headline">¶</a></h1>
<div class="versionadded">
New in version 2.6.
</div>
The <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">numbers</code></a> module (<a class="pep reference external" href="https://www.python.org/dev/peps/pep-3141">PEP 3141</a>) defines a hierarchy of numeric
<a class="reference internal" href="../glossary.html#term-abstract-base-class">abstract base classes</a> which progressively define
more operations. None of the types defined in this module can be instantiated.
<dl class="py class">
<dt class="sig sig-object py" id="numbers.Number">
class numbers.Number<a class="headerlink" href="#numbers.Number" title="Permalink to this definition">¶</a></dt>
<dd>The root of the numeric hierarchy. If you just want to check if an argument
x is a number, without caring what kind, use <code class="docutils literal notranslate">isinstance(x, Number)</code>.
</dd></dl>

<section id="the-numeric-tower">
<h2>9.1.1. The numeric tower<a class="headerlink" href="#the-numeric-tower" title="Permalink to this headline">¶</a></h2>
<dl class="py class">
<dt class="sig sig-object py" id="numbers.Complex">
class numbers.Complex<a class="headerlink" href="#numbers.Complex" title="Permalink to this definition">¶</a></dt>
<dd>Subclasses of this type describe complex numbers and include the operations
that work on the built-in <a class="reference internal" href="functions.html#complex" title="complex"><code class="xref py py-class docutils literal notranslate">complex</code></a> type. These are: conversions to
<a class="reference internal" href="functions.html#complex" title="complex"><code class="xref py py-class docutils literal notranslate">complex</code></a> and <a class="reference internal" href="functions.html#bool" title="bool"><code class="xref py py-class docutils literal notranslate">bool</code></a>, <a class="reference internal" href="#numbers.Complex.real" title="numbers.Complex.real"><code class="xref py py-attr docutils literal notranslate">real</code></a>, <a class="reference internal" href="#numbers.Complex.imag" title="numbers.Complex.imag"><code class="xref py py-attr docutils literal notranslate">imag</code></a>, <code class="docutils literal notranslate">+</code>,
<code class="docutils literal notranslate">-</code>, <code class="docutils literal notranslate">*</code>, <code class="docutils literal notranslate">/</code>, <a class="reference internal" href="functions.html#abs" title="abs"><code class="xref py py-func docutils literal notranslate">abs()</code></a>, <a class="reference internal" href="#numbers.Complex.conjugate" title="numbers.Complex.conjugate"><code class="xref py py-meth docutils literal notranslate">conjugate()</code></a>, <code class="docutils literal notranslate">==</code>, and <code class="docutils literal notranslate">!=</code>. All
except <code class="docutils literal notranslate">-</code> and <code class="docutils literal notranslate">!=</code> are abstract.
<dl class="py attribute">
<dt class="sig sig-object py" id="numbers.Complex.real">
real<a class="headerlink" href="#numbers.Complex.real" title="Permalink to this definition">¶</a></dt>
<dd>Abstract. Retrieves the real component of this number.
</dd></dl>

<dl class="py attribute">
<dt class="sig sig-object py" id="numbers.Complex.imag">
imag<a class="headerlink" href="#numbers.Complex.imag" title="Permalink to this definition">¶</a></dt>
<dd>Abstract. Retrieves the imaginary component of this number.
</dd></dl>

<dl class="py method">
<dt class="sig sig-object py" id="numbers.Complex.conjugate">
conjugate()<a class="headerlink" href="#numbers.Complex.conjugate" title="Permalink to this definition">¶</a></dt>
<dd>Abstract. Returns the complex conjugate. For example, <code class="docutils literal notranslate">(1+3j).conjugate()
== (1-3j)</code>.
</dd></dl>

</dd></dl>

<dl class="py class">
<dt class="sig sig-object py" id="numbers.Real">
class numbers.Real<a class="headerlink" href="#numbers.Real" title="Permalink to this definition">¶</a></dt>
<dd>To <a class="reference internal" href="#numbers.Complex" title="numbers.Complex"><code class="xref py py-class docutils literal notranslate">Complex</code></a>, <a class="reference internal" href="#numbers.Real" title="numbers.Real"><code class="xref py py-class docutils literal notranslate">Real</code></a> adds the operations that work on real
numbers.
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">float</code></a>, <a class="reference internal" href="math.html#math.trunc" title="math.trunc"><code class="xref py py-func docutils literal notranslate">math.trunc()</code></a>,
<a class="reference internal" href="functions.html#round" title="round"><code class="xref py py-func docutils literal notranslate">round()</code></a>, <a class="reference internal" href="math.html#math.floor" title="math.floor"><code class="xref py py-func docutils literal notranslate">math.floor()</code></a>, <a class="reference internal" href="math.html#math.ceil" title="math.ceil"><code class="xref py py-func docutils literal notranslate">math.ceil()</code></a>, <a class="reference internal" href="functions.html#divmod" title="divmod"><code class="xref py py-func docutils literal notranslate">divmod()</code></a>, <code class="docutils literal notranslate">//</code>,
<code class="docutils literal notranslate">%</code>, <code class="docutils literal notranslate">&lt;</code>, <code class="docutils literal notranslate">&lt;=</code>, <code class="docutils literal notranslate">&gt;</code>, and <code class="docutils literal notranslate">&gt;=</code>.
Real also provides defaults for <a class="reference internal" href="functions.html#complex" title="complex"><code class="xref py py-func docutils literal notranslate">complex()</code></a>, <a class="reference internal" href="#numbers.Complex.real" title="numbers.Complex.real"><code class="xref py py-attr docutils literal notranslate">real</code></a>,
<a class="reference internal" href="#numbers.Complex.imag" title="numbers.Complex.imag"><code class="xref py py-attr docutils literal notranslate">imag</code></a>, and <a class="reference internal" href="#numbers.Complex.conjugate" title="numbers.Complex.conjugate"><code class="xref py py-meth docutils literal notranslate">conjugate()</code></a>.
</dd></dl>

<dl class="py class">
<dt class="sig sig-object py" id="numbers.Rational">
class numbers.Rational<a class="headerlink" href="#numbers.Rational" title="Permalink to this definition">¶</a></dt>
<dd>Subtypes <a class="reference internal" href="#numbers.Real" title="numbers.Real"><code class="xref py py-class docutils literal notranslate">Real</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">numerator</code></a> and <a class="reference internal" href="#numbers.Rational.denominator" title="numbers.Rational.denominator"><code class="xref py py-attr docutils literal notranslate">denominator</code></a> properties, which
should be in lowest terms. With these, it provides a default for
<a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-func docutils literal notranslate">float()</code></a>.
<dl class="py attribute">
<dt class="sig sig-object py" id="numbers.Rational.numerator">
numerator<a class="headerlink" href="#numbers.Rational.numerator" title="Permalink to this definition">¶</a></dt>
<dd>Abstract.
</dd></dl>

<dl class="py attribute">
<dt class="sig sig-object py" id="numbers.Rational.denominator">
denominator<a class="headerlink" href="#numbers.Rational.denominator" title="Permalink to this definition">¶</a></dt>
<dd>Abstract.
</dd></dl>

</dd></dl>

<dl class="py class">
<dt class="sig sig-object py" id="numbers.Integral">
class numbers.Integral<a class="headerlink" href="#numbers.Integral" title="Permalink to this definition">¶</a></dt>
<dd>Subtypes <a class="reference internal" href="#numbers.Rational" title="numbers.Rational"><code class="xref py py-class docutils literal notranslate">Rational</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">int</code></a>. Provides
defaults for <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-func docutils literal notranslate">float()</code></a>, <a class="reference internal" href="#numbers.Rational.numerator" title="numbers.Rational.numerator"><code class="xref py py-attr docutils literal notranslate">numerator</code></a>, and
<a class="reference internal" href="#numbers.Rational.denominator" title="numbers.Rational.denominator"><code class="xref py py-attr docutils literal notranslate">denominator</code></a>. Adds abstract methods for <code class="docutils literal notranslate">**</code> and
bit-string operations: <code class="docutils literal notranslate">&lt;&lt;</code>, <code class="docutils literal notranslate">&gt;&gt;</code>, <code class="docutils literal notranslate">&amp;</code>, <code class="docutils literal notranslate">^</code>, <code class="docutils literal notranslate">|</code>, <code class="docutils literal notranslate">~</code>.
</dd></dl>

</section>
<section id="notes-for-type-implementors">
<h2>9.1.2. Notes for type implementors<a class="headerlink" href="#notes-for-type-implementors" title="Permalink to this headline">¶</a></h2>
Implementors should be careful to make equal numbers equal and hash
them to the same values. This may be subtle if there are two different
extensions of the real numbers. For example, <a class="reference internal" href="fractions.html#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate">fractions.Fraction</code></a>
implements <a class="reference internal" href="functions.html#hash" title="hash"><code class="xref py py-func docutils literal notranslate">hash()</code></a> as follows:
<div class="highlight-default notranslate"><div class="highlight"><pre>def __hash__(self):
 if self.denominator == 1:
 # Get integers right.
 return hash(self.numerator)
 # Expensive check, but definitely correct.
 if self == float(self):
 return hash(float(self))
 else:
 # Use tuple&#39;s hash to avoid a high collision rate on
 # simple fractions.
 return hash((self.numerator, self.denominator))
</pre></div>
</div>
<section id="adding-more-numeric-abcs">
<h3>9.1.2.1. Adding More Numeric ABCs<a class="headerlink" href="#adding-more-numeric-abcs" title="Permalink to this headline">¶</a></h3>
There are, of course, more possible ABCs for numbers, and this would
be a poor hierarchy if it precluded the possibility of adding
those. You can add <code class="docutils literal notranslate">MyFoo</code> between <a class="reference internal" href="#numbers.Complex" title="numbers.Complex"><code class="xref py py-class docutils literal notranslate">Complex</code></a> and
<a class="reference internal" href="#numbers.Real" title="numbers.Real"><code class="xref py py-class docutils literal notranslate">Real</code></a> with:
<div class="highlight-default notranslate"><div class="highlight"><pre>class MyFoo(Complex): ...
MyFoo.register(Real)
</pre></div>
</div>
</section>
<section id="implementing-the-arithmetic-operations">
<h3>9.1.2.2. Implementing the arithmetic operations<a class="headerlink" href="#implementing-the-arithmetic-operations" title="Permalink to this headline">¶</a></h3>
We want to implement the arithmetic operations so that mixed-mode
operations either call an implementation whose author knew about the
types of both arguments, or convert both to the nearest built in type
and do the operation there. For subtypes of <a class="reference internal" href="#numbers.Integral" title="numbers.Integral"><code class="xref py py-class docutils literal notranslate">Integral</code></a>, this
means that <code class="xref py py-meth docutils literal notranslate">__add__()</code> and <code class="xref py py-meth docutils literal notranslate">__radd__()</code> should be defined as:
<div class="highlight-default notranslate"><div class="highlight"><pre>class MyIntegral(Integral):

def __add__(self, other):
 if isinstance(other, MyIntegral):
 return do_my_adding_stuff(self, other)
 elif isinstance(other, OtherTypeIKnowAbout):
 return do_my_other_adding_stuff(self, other)
 else:
 return NotImplemented

def __radd__(self, other):
 if isinstance(other, MyIntegral):
 return do_my_adding_stuff(other, self)
 elif isinstance(other, OtherTypeIKnowAbout):
 return do_my_other_adding_stuff(other, self)
 elif isinstance(other, Integral):
 return int(other) + int(self)
 elif isinstance(other, Real):
 return float(other) + float(self)
 elif isinstance(other, Complex):
 return complex(other) + complex(self)
 else:
 return NotImplemented
</pre></div>
</div>
There are 5 different cases for a mixed-type operation on subclasses
of <a class="reference internal" href="#numbers.Complex" title="numbers.Complex"><code class="xref py py-class docutils literal notranslate">Complex</code></a>. I’ll refer to all of the above code that doesn’t
refer to <code class="docutils literal notranslate">MyIntegral</code> and <code class="docutils literal notranslate">OtherTypeIKnowAbout</code> as
“boilerplate”. <code class="docutils literal notranslate">a</code> will be an instance of <code class="docutils literal notranslate">A</code>, which is a subtype
of <a class="reference internal" href="#numbers.Complex" title="numbers.Complex"><code class="xref py py-class docutils literal notranslate">Complex</code></a> (<code class="docutils literal notranslate">a : A &lt;: Complex</code>), and <code class="docutils literal notranslate">b : B &lt;:
Complex</code>. I’ll consider <code class="docutils literal notranslate">a + b</code>:
<blockquote>
<div><ol class="arabic simple">
<li>If <code class="docutils literal notranslate">A</code> defines an <code class="xref py py-meth docutils literal notranslate">__add__()</code> which accepts <code class="docutils literal notranslate">b</code>, all is
well.</li>
<li>If <code class="docutils literal notranslate">A</code> falls back to the boilerplate code, and it were to
return a value from <code class="xref py py-meth docutils literal notranslate">__add__()</code>, we’d miss the possibility
that <code class="docutils literal notranslate">B</code> defines a more intelligent <code class="xref py py-meth docutils literal notranslate">__radd__()</code>, so the
boilerplate should return <a class="reference internal" href="constants.html#NotImplemented" title="NotImplemented"><code class="xref py py-const docutils literal notranslate">NotImplemented</code></a> from
<code class="xref py py-meth docutils literal notranslate">__add__()</code>. (Or <code class="docutils literal notranslate">A</code> may not implement <code class="xref py py-meth docutils literal notranslate">__add__()</code> at
all.)</li>
<li>Then <code class="docutils literal notranslate">B</code>’s <code class="xref py py-meth docutils literal notranslate">__radd__()</code> gets a chance. If it accepts
<code class="docutils literal notranslate">a</code>, all is well.</li>
<li>If it falls back to the boilerplate, there are no more possible
methods to try, so this is where the default implementation
should live.</li>
<li>If <code class="docutils literal notranslate">B &lt;: A</code>, Python tries <code class="docutils literal notranslate">B.__radd__</code> before
<code class="docutils literal notranslate">A.__add__</code>. This is ok, because it was implemented with
knowledge of <code class="docutils literal notranslate">A</code>, so it can handle those instances before
delegating to <a class="reference internal" href="#numbers.Complex" title="numbers.Complex"><code class="xref py py-class docutils literal notranslate">Complex</code></a>.</li>
</ol>
</div></blockquote>
If <code class="docutils literal notranslate">A &lt;: Complex</code> and <code class="docutils literal notranslate">B &lt;: Real</code> without sharing any other knowledge,
then the appropriate shared operation is the one involving the built
in <a class="reference internal" href="functions.html#complex" title="complex"><code class="xref py py-class docutils literal notranslate">complex</code></a>, and both <code class="xref py py-meth docutils literal notranslate">__radd__()</code> s land there, so <code class="docutils literal notranslate">a+b
== b+a</code>.
Because most of the operations on any given type will be very similar,
it can be useful to define a helper function which generates the
forward and reverse instances of any given operator. For example,
<a class="reference internal" href="fractions.html#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate">fractions.Fraction</code></a> uses:
<div class="highlight-default notranslate"><div class="highlight"><pre>def _operator_fallbacks(monomorphic_operator, fallback_operator):
 def forward(a, b):
 if isinstance(b, (int, long, Fraction)):
 return monomorphic_operator(a, b)
 elif isinstance(b, float):
 return fallback_operator(float(a), b)
 elif isinstance(b, complex):
 return fallback_operator(complex(a), b)
 else:
 return NotImplemented
 forward.__name__ = &#39;__&#39; + fallback_operator.__name__ + &#39;__&#39;
 forward.__doc__ = monomorphic_operator.__doc__

def reverse(b, a):
 if isinstance(a, Rational):
 # Includes ints.
 return monomorphic_operator(a, b)
 elif isinstance(a, numbers.Real):
 return fallback_operator(float(a), float(b))
 elif isinstance(a, numbers.Complex):
 return fallback_operator(complex(a), complex(b))
 else:
 return NotImplemented
 reverse.__name__ = &#39;__r&#39; + fallback_operator.__name__ + &#39;__&#39;
 reverse.__doc__ = monomorphic_operator.__doc__

return forward, reverse

def _add(a, b):
 &quot;&quot;&quot;a + b&quot;&quot;&quot;
 return Fraction(a.numerator * b.denominator +
 b.numerator * a.denominator,
 a.denominator * b.denominator)

__add__, __radd__ = _operator_fallbacks(_add, operator.add)

# ...
</pre></div>
</div>
</section>
</section>
</section>

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 <ul>
<li><a class="reference internal" href="#">9.1. <code class="xref py py-mod docutils literal notranslate">numbers</code> — Numeric abstract base classes</a><ul>
<li><a class="reference internal" href="#the-numeric-tower">9.1.1. The numeric tower</a></li>
<li><a class="reference internal" href="#notes-for-type-implementors">9.1.2. Notes for type implementors</a><ul>
<li><a class="reference internal" href="#adding-more-numeric-abcs">9.1.2.1. Adding More Numeric ABCs</a></li>
<li><a class="reference internal" href="#implementing-the-arithmetic-operations">9.1.2.2. Implementing the arithmetic operations</a></li>
</ul>
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</ul>
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