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

**原始碼：**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
   (underscores may be used to delimit digits as with integral
   literals in code).  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:

   在 3.2 版的變更: The "Fraction" constructor now accepts "float" and
   "decimal.Decimal" instances.

   在 3.9 版的變更: 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*.

   在 3.11 版的變更: Underscores are now permitted when creating a
   "Fraction" instance from a string, following **PEP 515** rules.

   在 3.11 版的變更: "Fraction" implements "__int__" now to satisfy
   "typing.SupportsInt" instance checks.

   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.

      在 3.8 版新加入.

   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)".

      備註:

        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".

      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.

也參考:

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