"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 rational
numbers, from a single 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 a value equal to "numerator/denominator". If
   *denominator* is zero, 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
   (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:

   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
   returns an "int" type. Previously, the GCD type depended on
   *numerator* and *denominator*.

   Changed in version 3.11: Underscores are now permitted when
   creating a "Fraction" instance from a string, following **PEP 515**
   rules.

   Changed in version 3.11: "Fraction" implements "__int__" now to
   satisfy "typing.SupportsInt" instance checks.

   Changed in version 3.12: Space is allowed around the slash for
   string inputs: "Fraction('2 / 3')".

   Changed in version 3.12: "Fraction" instances now support float-
   style formatting, with presentation types ""e"", ""E"", ""f"",
   ""F"", ""g"", ""G"" and ""%""".

   Changed in version 3.13: Formatting of "Fraction" instances without
   a presentation type now supports fill, alignment, sign handling,
   minimum width and grouping.

   numerator

      Numerator of the Fraction in lowest term.

   denominator

      Denominator of the Fraction in lowest terms. Guaranteed to be
      positive.

   as_integer_ratio()

      Return a tuple of two integers, whose ratio is equal to the
      original Fraction.  The ratio is in lowest terms and has a
      positive denominator.

      Added in version 3.8.

   is_integer()

      Return "True" if the Fraction is an integer.

      Added in version 3.12.

   classmethod from_float(f)

      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.

   __format__(format_spec, /)

      Provides support for formatting of "Fraction" instances via the
      "str.format()" method, the "format()" built-in function, or
      Formatted string literals.

      If the "format_spec" format specification string does not end
      with one of the presentation types "'e'", "'E'", "'f'", "'F'",
      "'g'", "'G'" or "'%'" then formatting follows the general rules
      for fill, alignment, sign handling, minimum width, and grouping
      as described in the format specification mini-language. The
      "alternate form" flag "'#'" is supported: if present, it forces
      the output string to always include an explicit denominator,
      even when the value being formatted is an exact integer. The
      zero-fill flag "'0'" is not supported.

      If the "format_spec" format specification string ends with one
      of the presentation types "'e'", "'E'", "'f'", "'F'", "'g'",
      "'G'" or "'%'" then formatting follows the rules outlined for
      the "float" type in the Format Specification Mini-Language
      section.

      Here are some examples:

         >>> from fractions import Fraction
         >>> format(Fraction(103993, 33102), '_')
         '103_993/33_102'
         >>> format(Fraction(1, 7), '.^+10')
         '...+1/7...'
         >>> format(Fraction(3, 1), '')
         '3'
         >>> format(Fraction(3, 1), '#')
         '3/1'
         >>> format(Fraction(1, 7), '.40g')
         '0.1428571428571428571428571428571428571429'
         >>> format(Fraction('1234567.855'), '_.2f')
         '1_234_567.86'
         >>> f"{Fraction(355, 113):*>20.6e}"
         '********3.141593e+00'
         >>> old_price, new_price = 499, 672
         >>> "{:.2%} price increase".format(Fraction(new_price, old_price) - 1)
         '34.67% price increase'

See also:

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