"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 浮點數
   運算：問題與限制), 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:

   3.2 版更變: The "Fraction" constructor now accepts "float" and
   "decimal.Decimal" instances.

   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.

      3.8 版新加入.

   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)".

      備註: From Python 3.2 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" instance.

      備註: 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.

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".

   3.5 版後已棄用: Use "math.gcd()" instead.

也參考:

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