"numbers" --- Numeric abstract base classes
*******************************************

**Code source :** Lib/numbers.py

======================================================================

The "numbers" module (**PEP 3141**) defines a hierarchy of numeric
*abstract base classes* which progressively define more operations.
None of the types defined in this module are intended to be
instantiated.

class numbers.Number

   La base de la hiérarchie numérique. Si vous voulez juste vérifier
   qu'un argument *x* est un nombre, peu importe le type, utilisez
   "isinstance(x, Number)".


La tour numérique
=================

class numbers.Complex

   Subclasses of this type describe complex numbers and include the
   operations that work on the built-in "complex" type. These are:
   conversions to "complex" and "bool", "real", "imag", "+", "-", "*",
   "/", "**", "abs()", "conjugate()", "==", and "!=". All except "-"
   and "!=" are abstract.

   real

      Abstrait. Récupère la partie réelle de ce nombre.

   imag

      Abstrait. Retrouve la partie imaginaire de ce nombre.

   abstractmethod conjugate()

      Abstrait. Renvoie le complexe conjugué. Par exemple,
      "(1+3j).conjugate() == (1-3j)".

class numbers.Real

   To "Complex", "Real" adds the operations that work on real numbers.

   En bref, celles-ci sont : une conversion vers "float",
   "math.trunc()", "round()", "math.floor()", "math.ceil()",
   "divmod()", "//", "%", "<", "<=", ">" et ">=".

   *Real* fournit également des valeurs par défaut pour "complex()",
   "real", "imag" et "conjugate()".

class numbers.Rational

   Subtypes "Real" and adds "numerator" and "denominator" properties.
   It also provides a default for "float()".

   The "numerator" and "denominator" values should be instances of
   "Integral" and should be in lowest terms with "denominator"
   positive.

   numerator

      Abstrait.

   denominator

      Abstrait.

class numbers.Integral

   Subtypes "Rational" and adds a conversion to "int".  Provides
   defaults for "float()", "numerator", and "denominator".  Adds
   abstract methods for "pow()" with modulus and bit-string
   operations: "<<", ">>", "&", "^", "|", "~".


Notes for type implementers
===========================

Implementers 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, "fractions.Fraction"
implements "hash()" as follows:

   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's hash to avoid a high collision rate on
           # simple fractions.
           return hash((self.numerator, self.denominator))


Ajouter plus d'ABC numériques
-----------------------------

Il est bien entendu possible de créer davantage d’ABC pour les nombres
et cette hiérarchie serait médiocre si elle excluait la possibilité
d'en ajouter. Vous pouvez ajouter "MyFoo" entre "Complex" et "Real"
ainsi :

   class MyFoo(Complex): ...
   MyFoo.register(Real)


Implémentation des opérations arithmétiques
-------------------------------------------

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 "Integral", this means
that "__add__()" and "__radd__()" should be defined as:

   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

Il existe 5 cas différents pour une opération de type mixte sur des
sous-classes de "Complex". Nous nous référerons à tout le code ci-
dessus qui ne se réfère pas à "MyIntegral" et "OtherTypeIKnowAbout"
comme "expression générique". "a" est une instance de "A", qui est un
sous-type de "Complex" ("a : A <: Complex") et "b : B <: Complex".
Considérons "a + b" :

1. If "A" defines an "__add__()" which accepts "b", all is well.

2. If "A" falls back to the boilerplate code, and it were to return a
   value from "__add__()", we'd miss the possibility that "B" defines
   a more intelligent "__radd__()", so the boilerplate should return
   "NotImplemented" from "__add__()". (Or "A" may not implement
   "__add__()" at all.)

3. Then "B"'s "__radd__()" gets a chance. If it accepts "a", all is
   well.

4. Si elle fait appel au code générique, il n'y a plus de méthode
   possible à essayer, c'est donc ici que l'implémentation par défaut
   intervient.

5. Si "B < : A", Python essaie "B.__radd__" avant "A.__add__". C'est
   valable parce qu'elle est implémentée avec la connaissance de "A",
   donc elle peut gérer ces instances avant de déléguer à "Complex".

If "A <: Complex" and "B <: Real" without sharing any other knowledge,
then the appropriate shared operation is the one involving the built
in "complex", and both "__radd__()" s land there, so "a+b == b+a".

Comme la plupart des opérations sur un type donné seront très
similaires, il peut être utile de définir une fonction accessoire qui
génère les instances résultantes et inverses d'un opérateur donné. Par
exemple, "fractions.Fraction" utilise :

   def _operator_fallbacks(monomorphic_operator, fallback_operator):
       def forward(a, b):
           if isinstance(b, (int, 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__ = '__' + fallback_operator.__name__ + '__'
       forward.__doc__ = monomorphic_operator.__doc__

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

       return forward, reverse

   def _add(a, b):
       """a + b"""
       return Fraction(a.numerator * b.denominator +
                       b.numerator * a.denominator,
                       a.denominator * b.denominator)

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

   # ...
