: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.
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 (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 :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.
.. versionchanged:: 3.9 The :func:`math.gcd` function is now used to normalize the *numerator* and *denominator*. :func:`math.gcd` always return a :class:`int` type. Previously, the GCD type depended on *numerator* and *denominator*.
.. versionchanged:: 3.11 Underscores are now permitted when creating a :class:`Fraction` instance from a string, following :PEP:`515` rules.
.. versionchanged:: 3.11 :class:`Fraction` implements ``__int__`` now to satisfy ``typing.SupportsInt`` instance checks.
.. versionchanged:: 3.12
Space is allowed around the slash for string inputs: ``Fraction('2 / 3')``.
.. attribute:: numerator Numerator of the Fraction in lowest term.
.. attribute:: denominator Denominator of the Fraction in lowest term.
.. method:: as_integer_ratio() Return a tuple of two integers, whose ratio is equal to the Fraction and with a positive denominator. .. versionadded:: 3.8
.. classmethod:: from_float(flt)
Alternative constructor which only accepts instances of
:class:`float` or :class:`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
:class:`Fraction` instance directly from a :class:`float`.
.. classmethod:: from_decimal(dec)
Alternative constructor which only accepts instances of
:class:`decimal.Decimal` or :class:`numbers.Integral`.
.. 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.
.. seealso::
Module :mod:`numbers`
The abstract base classes making up the numeric tower.