"fractions" --- Rational numbers
********************************

**Source code:** Lib/fractions.py

======================================================================

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(float)
class fractions.Fraction(decimal)
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 a "Fraction" instance with the same value.  The next two
   versions accept either a "float" or a "decimal.Decimal" instance,
   and return a "Fraction" instance with exactly the same value.  Note
   that due to the usual issues with binary floating-point (see
   Floating Point Arithmetic:  Issues and Limitations), the argument
   to "Fraction(1.1)" is not exactly equal to 11/10, and so
   "Fraction(1.1)" does *not* return "Fraction(11, 10)" as one might
   expect. (But see the documentation for the "limit_denominator()"
   method below.) The last version of the constructor expects a string
   or unicode instance. The usual form for this instance 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)
      >>> 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)
      >>> Fraction(2.25)
      Fraction(9, 4)
      >>> Fraction(1.1)
      Fraction(2476979795053773, 2251799813685248)
      >>> from decimal import Decimal
      >>> Fraction(Decimal('1.1'))
      Fraction(11, 10)

   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 properties and methods:

   Changed in version 3.2: The "Fraction" constructor now accepts
   "float" and "decimal.Decimal" instances.

   Changed in version 3.9: The "math.gcd()" function is now used to
   normalize the *numerator* and *denominator*. "math.gcd()" always
   return a "int" type. Previously, the GCD type depended on
   *numerator* and *denominator*.

   numerator

      Numerator of the Fraction in lowest term.

   denominator

      Denominator of the Fraction in lowest term.

   as_integer_ratio()

      Return a tuple of two integers, whose ratio is equal to the
      Fraction and with a positive denominator.

      New in version 3.8.

   classmethod from_float(flt)

      Alternative constructor which only accepts instances of "float"
      or "numbers.Integral". Beware that "Fraction.from_float(0.3)" is
      not the same value as "Fraction(3, 10)".

      Note:

        From Python 3.2 onwards, you can also construct a "Fraction"
        instance directly from a "float".

   classmethod from_decimal(dec)

      Alternative constructor which only accepts instances of
      "decimal.Decimal" or "numbers.Integral".

      Note:

        From Python 3.2 onwards, you can also construct a "Fraction"
        instance directly from 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(cos(pi/3))
      Fraction(4503599627370497, 9007199254740992)
      >>> Fraction(cos(pi/3)).limit_denominator()
      Fraction(1, 2)
      >>> Fraction(1.1).limit_denominator()
      Fraction(11, 10)

   __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.

See also:

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