# `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.