:mod:`numbers` --- Numeric abstract base classes

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

.. module:: numbers

   :synopsis: Numeric abstract base classes (Complex, Real, Integral, etc.).

.. versionadded:: 2.6

The :mod:`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 can be instantiated.

.. class:: Number

   The root of the numeric hierarchy. If you just want to check if an argument

   *x* is a number, without caring what kind, use ``isinstance(x, Number)``.

The numeric tower

-----------------

.. class:: Complex

   Subclasses of this type describe complex numbers and include the operations

   that work on the builtin :class:`complex` type. These are: conversions to

   :class:`complex` and :class:`bool`, :attr:`.real`, :attr:`.imag`, ``+``,

   ``-``, ``*``, ``/``, :func:`abs`, :meth:`conjugate`, ``==``, and ``!=``. All

   except ``-`` and ``!=`` are abstract.

   .. attribute:: real

      Abstract. Retrieves the :class:`Real` component of this number.

   .. attribute:: imag

      Abstract. Retrieves the :class:`Real` component of this number.

   .. method:: conjugate()

      Abstract. Returns the complex conjugate. For example, ``(1+3j).conjugate()

      == (1-3j)``.

.. class:: Real

   To :class:`Complex`, :class:`Real` adds the operations that work on real

   numbers.

   In short, those are: a conversion to :class:`float`, :func:`trunc`,

   :func:`round`, :func:`math.floor`, :func:`math.ceil`, :func:`divmod`, ``//``,

   ``%``, ``<``, ``<=``, ``>``, and ``>=``.

   Real also provides defaults for :func:`complex`, :attr:`Complex.real`,

   :attr:`Complex.imag`, and :meth:`Complex.conjugate`.

.. class:: Rational

   Subtypes :class:`Real` and adds

   :attr:`Rational.numerator` and :attr:`Rational.denominator` properties, which

   should be in lowest terms. With these, it provides a default for

   :func:`float`.

   .. attribute:: numerator

      Abstract.

   .. attribute:: denominator

      Abstract.

.. class:: Integral

   Subtypes :class:`Rational` and adds a conversion to :class:`int`.

   Provides defaults for :func:`float`, :attr:`Rational.numerator`, and

   :attr:`Rational.denominator`, and bit-string operations: ``<<``,

   ``>>``, ``&``, ``^``, ``|``, ``~``.

Notes for type implementors

---------------------------

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, :class:`fractions.Fraction`

implements :func:`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))

Adding More Numeric ABCs

~~~~~~~~~~~~~~~~~~~~~~~~

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 ``MyFoo`` between :class:`Complex` and

:class:`Real` with::

    class MyFoo(Complex): ...

    MyFoo.register(Real)

Implementing the arithmetic operations

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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 :class:`Integral`, this

means that :meth:`__add__` and :meth:`__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

There are 5 different cases for a mixed-type operation on subclasses

of :class:`Complex`. I'll refer to all of the above code that doesn't

refer to ``MyIntegral`` and ``OtherTypeIKnowAbout`` as

"boilerplate". ``a`` will be an instance of ``A``, which is a subtype

of :class:`Complex` (``a : A <: Complex``), and ``b : B <:

Complex``. I'll consider ``a + b``:

    1. If ``A`` defines an :meth:`__add__` which accepts ``b``, all is

       well.

    2. If ``A`` falls back to the boilerplate code, and it were to

       return a value from :meth:`__add__`, we'd miss the possibility

       that ``B`` defines a more intelligent :meth:`__radd__`, so the

       boilerplate should return :const:`NotImplemented` from

       :meth:`__add__`. (Or ``A`` may not implement :meth:`__add__` at

       all.)

    3. Then ``B``'s :meth:`__radd__` gets a chance. If it accepts

       ``a``, all is well.

    4. If it falls back to the boilerplate, there are no more possible

       methods to try, so this is where the default implementation

       should live.

    5. If ``B <: A``, Python tries ``B.__radd__`` before

       ``A.__add__``. This is ok, because it was implemented with

       knowledge of ``A``, so it can handle those instances before

       delegating to :class:`Complex`.

If ``A<:Complex`` and ``B<:Real`` without sharing any other knowledge,

then the appropriate shared operation is the one involving the built

in :class:`complex`, and both :meth:`__radd__` s land there, so ``a+b

== b+a``.

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,

:class:`fractions.Fraction` uses::

    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__ = '__' + 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, 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__ = '__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)

    # ...