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:mod:`numbers` --- Numeric abstract base classes
================================================
.. module:: numbers
:synopsis: Numeric abstract base classes (Complex, Real, Integral, etc.).
The :mod:`numbers` module (:pep:`3141`) defines a hierarchy of numeric abstract
base classes which progressively define more operations. These concepts also
provide a way to distinguish exact from inexact types. 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)``.
Exact and inexact operations
----------------------------
.. class:: Exact
Subclasses of this type have exact operations.
As long as the result of a homogenous operation is of the same type, you can
assume that it was computed exactly, and there are no round-off errors. Laws
like commutativity and associativity hold.
.. class:: Inexact
Subclasses of this type have inexact operations.
Given X, an instance of :class:`Inexact`, it is possible that ``(X + -X) + 3
== 3``, but ``X + (-X + 3) == 0``. The exact form this error takes will vary
by type, but it's generally unsafe to compare this type for equality.
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:: Complex.real
Abstract. Retrieves the :class:`Real` component of this number.
.. attribute:: Complex.imag
Abstract. Retrieves the :class:`Real` component of this number.
.. method:: Complex.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 both :class:`Real` and :class:`Exact`, 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:: Rational.numerator
Abstract.
.. attribute:: Rational.denominator
Abstract.
.. class:: Integral
Subtypes :class:`Rational` and adds a conversion to :class:`long`, the
3-argument form of :func:`pow`, and the bit-string operations: ``<<``,
``>>``, ``&``, ``^``, ``|``, ``~``. Provides defaults for :func:`float`,
:attr:`Rational.numerator`, and :attr:`Rational.denominator`.
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