
``fractions`` --- Rational numbers
**********************************

The ``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 class fractions.Fraction(numerator=0, denominator=1)
class class fractions.Fraction(other_fraction)
class class fractions.Fraction(string)

   The first version requires that *numerator* and *denominator* are
   instances of ``numbers.Rational`` and returns a new ``Fraction``
   instance with value ``numerator/denominator``. If *denominator* is
   ``0``, it raises a ``ZeroDivisionError``. The second version
   requires that *other_fraction* is an instance of
   ``numbers.Rational`` and returns an ``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 ``float`` constructor is also accepted
   by the ``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 ``Fraction`` class inherits from the abstract base class
   ``numbers.Rational``, and implements all of the methods and
   operations from that class.  ``Fraction`` instances are hashable,
   and should be treated as immutable.  In addition, ``Fraction`` has
   the following methods:

   from_float(flt)

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

   from_decimal(dec)

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

   limit_denominator(max_denominator=1000000)

      Finds and returns the closest ``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)

   __floor__()

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

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

   __ceil__()

      Returns the least ``int`` ``>= self``.  This method can also be
      accessed through the ``math.ceil()`` function.

   __round__()
   __round__(ndigits)

      The first version returns the nearest ``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 ``round()`` function.

fractions.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``.

See also:

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