9.5. "fractions" — Rational numbers
***********************************

New in version 2.6.

**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 methods:

   Changed in version 2.7: The "Fraction" constructor now accepts
   "float" and "decimal.Decimal" instances.

   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)".

      Note: From Python 2.7 onwards, you can also construct a
        "Fraction" instance directly from a "float".

   from_decimal(dec)

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

      Note: From Python 2.7 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)

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.