:mod:`fractions` --- Rational numbers
=====================================
.. module:: fractions
:synopsis: Rational numbers.
.. moduleauthor:: Jeffrey Yasskin
.. sectionauthor:: Jeffrey Yasskin
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(string)
The first version requires that *numerator* and *denominator* are
instances of :class:`numbers.Integral` 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 an :class:`Fraction` instance with the same value. The
last version of the constructor expects a string
instance in one of two possible forms. The first form is::
[sign] numerator ['/' denominator]
where the optional ``sign`` may be either '+' or '-' and
``numerator`` and ``denominator`` (if present) are strings of
decimal digits. The second permitted form is that of a number
containing a decimal point::
[sign] integer '.' [fraction] | [sign] '.' fraction
where ``integer`` and ``fraction`` are strings of digits. 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)
[40794 refs]
>>> Fraction(' -3/7 ')
Fraction(-3, 7)
>>> Fraction('1.414213 \t\n')
Fraction(1414213, 1000000)
>>> Fraction('-.125')
Fraction(-1, 8)
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 methods:
.. 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)``
.. 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.
.. 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.from_float(cos(pi/3))
Fraction(4503599627370497, 9007199254740992)
>>> Fraction.from_float(cos(pi/3)).limit_denominator()
Fraction(1, 2)
.. 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.