"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(number)
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 *number* is an instance of
   "numbers.Rational" or has the "as_integer_ratio()" method (this
   includes "float" and "decimal.Decimal"). It returns a "Fraction"
   instance with exactly the same value. Assumed, that the
   "as_integer_ratio()" method returns a pair of coprime integers and
   last one is positive. Note that due to the usual issues with binary
   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. 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.

   Changed in version 3.14: The "Fraction" constructor now accepts any
   objects with the "as_integer_ratio()" method.

   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.

   classmethod from_number(number)

      Alternative constructor which only accepts instances of
      "numbers.Integral", "numbers.Rational", "float" or
      "decimal.Decimal", and objects with the "as_integer_ratio()"
      method, but not strings.

      Added in version 3.14.

   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.
