# `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

    Abstract. The numerator of this rational number.

  denominator

    Abstract. The denominator of this rational number.

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)

    # ...