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

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


The :mod:`fractions` module provides support for rational number arithmetic.


A Fraction instance can be constructed from a pair of integers, from
another rational number, or from a string.

.. 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
   :class:`Fraction` instance with value ``numerator/denominator``. If
   *denominator* is :const:`0`, it raises a
   :exc:`ZeroDivisionError`. The second version requires that
   *other_fraction* is an instance of :class:`numbers.Rational` and
   returns an :class:`Fraction` instance with the same value.  The
   last version of the constructor expects a string instance.  The
   usual form for this string is::

      [sign] numerator ['/' denominator]

   where the optional ``sign`` may be either '+' or '-' and
   ``numerator`` and ``denominator`` (if present) are strings of
   decimal digits.  In addition, any string that represents a finite
   value and is accepted by the :class:`float` constructor is also
   accepted by the :class:`Fraction` constructor.  In either form the
   input string may also have leading and/or trailing whitespace.
   Here are some examples::

      >>> from fractions import Fraction
      >>> Fraction(16, -10)
      Fraction(-8, 5)
      >>> Fraction(123)
      Fraction(123, 1)
      >>> Fraction()
      Fraction(0, 1)
      >>> Fraction('3/7')
      Fraction(3, 7)
      [40794 refs]
      >>> Fraction(' -3/7 ')
      Fraction(-3, 7)
      >>> Fraction('1.414213 \t\n')
      Fraction(1414213, 1000000)
      >>> Fraction('-.125')
      Fraction(-1, 8)
      >>> Fraction('7e-6')
      Fraction(7, 1000000)


   The :class:`Fraction` class inherits from the abstract base class
   :class:`numbers.Rational`, and implements all of the methods and
   operations from that class.  :class:`Fraction` instances are hashable,
   and should be treated as immutable.  In addition,
   :class:`Fraction` has the following methods:


   .. method:: from_float(flt)

      This class method 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 ``Fraction(3, 10)``


   .. method:: from_decimal(dec)

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


   .. method:: 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(355, 113)

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

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


   .. method:: __floor__()

      Returns the greatest :class:`int` ``<= self``.  This method can
      also be accessed through the :func:`math.floor` function:

        >>> from math import floor
        >>> floor(Fraction(355, 113))
        3


   .. method:: __ceil__()

      Returns the least :class:`int` ``>= self``.  This method can
      also be accessed through the :func:`math.ceil` function.


   .. method:: __round__()
               __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.  This
      method can also be accessed through the :func:`round` function.


.. function:: gcd(a, b)

   Return the greatest common divisor of the integers *a* and *b*.  If either
   *a* or *b* is nonzero, then the absolute value of ``gcd(a, b)`` is the
   largest integer that divides both *a* and *b*.  ``gcd(a,b)`` has the same
   sign as *b* if *b* is nonzero; otherwise it takes the sign of *a*.  ``gcd(0,
   0)`` returns ``0``.


.. seealso::

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