:mod:fractions --- Rational numbers

.. module:: fractions :synopsis: Rational numbers.

.. moduleauthor:: Jeffrey Yasskin <jyasskin at gmail.com> .. sectionauthor:: Jeffrey Yasskin <jyasskin at gmail.com>

Source code: :source:Lib/fractions.py

The :mod: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:: Fraction(numerator=0, denominator=1) Fraction(other_fraction) Fraction(float) Fraction(decimal) Fraction(string)

The first version requires that numerator and denominator are instances of :class:numbers.Rational and returns a new :class:Fraction instance with value numerator/denominator. If denominator is :const:0, it raises a :exc:ZeroDivisionError. The second version requires that other_fraction is an instance of :class:numbers.Rational and returns a :class:Fraction instance with the same value. The next two versions accept either a :class:float or a :class:decimal.Decimal instance, and return a :class:Fraction instance with exactly the same value. Note that due to the usual issues with binary floating-point (see :ref:tut-fp-issues), 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 :meth: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 :class:float constructor is also accepted by the :class: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 :class:Fraction class inherits from the abstract base class :class:numbers.Rational, and implements all of the methods and operations from that class. :class:Fraction instances are hashable, and should be treated as immutable. In addition, :class:Fraction has the following properties and methods:

.. versionchanged:: 3.2 The :class:Fraction constructor now accepts :class:float and :class:decimal.Decimal instances.

.. attribute:: numerator

  Numerator of the Fraction in lowest term.

.. attribute:: denominator

  Denominator of the Fraction in lowest term.

.. method:: from_float(flt)

  This class method constructs a :class:`Fraction` representing the exact
  value of *flt*, which must be a :class:`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
     :class:`Fraction` instance directly from a :class:`float`.

.. method:: from_decimal(dec)

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

  .. note::

     From Python 3.2 onwards, you can also construct a
     :class:`Fraction` instance directly from a :class:`decimal.Decimal`
     instance.

.. method:: limit_denominator(max_denominator=1000000)

  Finds and returns the closest :class:`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)

.. method:: floor()

  Returns the greatest :class:`int` ``<= self``.  This method can
  also be accessed through the :func:`math.floor` function:

    >>> from math import floor
    >>> floor(Fraction(355, 113))
    3

.. method:: ceil()

  Returns the least :class:`int` ``>= self``.  This method can
  also be accessed through the :func:`math.ceil` function.

.. method:: round() round(ndigits)

  The first version returns the nearest :class:`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 :func:`round` function.

.. function:: 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:: 3.5 Use :func:math.gcd instead.

.. seealso::

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