:mod:`fractions` --- Rational numbers
=====================================
.. module:: fractions
:synopsis: Rational numbers.
.. moduleauthor:: Jeffrey Yasskin
.. sectionauthor:: Jeffrey Yasskin
**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``.
.. seealso::
Module :mod:`numbers`
The abstract base classes making up the numeric tower.