:mod:`fractions` --- Rational numbers
=====================================

.. module:: fractions
   :synopsis: Rational numbers.
.. moduleauthor:: Jeffrey Yasskin <jyasskin at gmail.com>
.. sectionauthor:: Jeffrey Yasskin <jyasskin at gmail.com>
.. versionadded:: 2.6


The :mod:`fractions` module defines an immutable, infinite-precision
Rational number class.


.. class:: Fraction(numerator=0, denominator=1)
           Fraction(other_fraction)
           Fraction(string)

   The first version requires that *numerator* and *denominator* are
   instances of :class:`numbers.Integral` and returns a new
   ``Fraction`` representing ``numerator/denominator``. If
   *denominator* is :const:`0`, raises a :exc:`ZeroDivisionError`. The
   second version requires that *other_fraction* is an instance of
   :class:`numbers.Fraction` and returns an instance of
   :class:`Rational` with the same value. The third version expects a
   string of the form ``[-+]?[0-9]+(/[0-9]+)?``, optionally surrounded
   by spaces.

   Implements all of the methods and operations from
   :class:`numbers.Rational` and is immutable and hashable.


.. method:: Fraction.from_float(flt)

   This classmethod constructs a :class:`Fraction` representing the
   exact value of *flt*, which must be a :class:`float`. Beware that
   ``Fraction.from_float(0.3)`` is not the same value as ``Rational(3,
   10)``


.. method:: Fraction.from_decimal(dec)

   This classmethod constructs a :class:`Fraction` representing the
   exact value of *dec*, which must be a
   :class:`decimal.Decimal`.


.. method:: Fraction.limit_denominator(max_denominator=1000000)

   Finds and returns the closest :class:`Fraction` to ``self`` that
   has denominator at most max_denominator.  This method is useful for
   finding rational approximations to a given floating-point number:

      >>> from fractions import Fraction
      >>> Fraction('3.1415926535897932').limit_denominator(1000)
      Fraction(355L, 113L)

   or for recovering a rational number that's represented as a float:

      >>> from math import pi, cos
      >>> Fraction.from_float(cos(pi/3))
      Fraction(4503599627370497L, 9007199254740992L)
      >>> Fraction.from_float(cos(pi/3)).limit_denominator()
      Fraction(1L, 2L)


.. method:: Fraction.__floor__()

   Returns the greatest :class:`int` ``<= self``. Will be accessible
   through :func:`math.floor` in Py3k.


.. method:: Fraction.__ceil__()

   Returns the least :class:`int` ``>= self``. Will be accessible
   through :func:`math.ceil` in Py3k.


.. method:: Fraction.__round__()
            Fraction.__round__(ndigits)

   The first version returns the nearest :class:`int` to ``self``,
   rounding half to even. The second version rounds ``self`` to the
   nearest multiple of ``Fraction(1, 10**ndigits)`` (logically, if
   ``ndigits`` is negative), again rounding half toward even. Will be
   accessible through :func:`round` in Py3k.


.. seealso::

   Module :mod:`numbers`
      The abstract base classes making up the numeric tower.