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

New in version 2.6.

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

   The first version requires that *numerator* and *denominator* are
   instances of ``numbers.Integral`` 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
   or unicode instance in one of two possible forms.  The first form
   is:

      [sign] numerator ['/' denominator]

   where the optional ``sign`` may be either '+' or '-' and
   ``numerator`` and ``denominator`` (if present) are strings of
   decimal digits.  The second permitted form is that of a number
   containing a decimal point:

      [sign] integer '.' [fraction] | [sign] '.' fraction

   where ``integer`` and ``fraction`` are strings of digits.  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)

   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``.

   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)

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.
