# 9.5. `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. 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.

  `numerator`

    Numerator of the Fraction in lowest term.

  `denominator`

    Denominator of the Fraction in lowest term.

  `from_float`(flt)

    This class method constructs a `Fraction` representing the exact value of flt, which must be a `float`. 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`.

  `from_decimal`(dec)

    This class method constructs a `Fraction` representing the exact value of dec, which must be a `decimal.Decimal` instance.

    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.

`fractions.``gcd`(a, b)

  Return the greatest common divisor of the integers a and b. If either a or b is nonzero, then the absolute value of `gcd(a, b)` is the largest integer that divides both a and b. `gcd(a,b)` has the same sign as b if b is nonzero; otherwise it takes the sign of a. `gcd(0, 0)` returns `0`.

  Deprecated since version 3.5: Use `math.gcd()` instead.

See also

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