# `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 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 return a `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 term.

  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.

    New in version 3.8.

  is_integer()

    Return `True` if the Fraction is an integer.

    New in version 3.12.

  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)`.

    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.