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183 lines
6.2 KiB
ReStructuredText
:mod:`fractions` --- Rational numbers
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=====================================
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.. module:: fractions
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:synopsis: Rational numbers.
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.. moduleauthor:: Jeffrey Yasskin <jyasskin at gmail.com>
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.. sectionauthor:: Jeffrey Yasskin <jyasskin at gmail.com>
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**Source code:** :source:`Lib/fractions.py`
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--------------
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The :mod:`fractions` module provides support for rational number arithmetic.
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A Fraction instance can be constructed from a pair of integers, from
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another rational number, or from a string.
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.. class:: Fraction(numerator=0, denominator=1)
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Fraction(other_fraction)
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Fraction(float)
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Fraction(decimal)
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Fraction(string)
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The first version requires that *numerator* and *denominator* are instances
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of :class:`numbers.Rational` and returns a new :class:`Fraction` instance
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with value ``numerator/denominator``. If *denominator* is :const:`0`, it
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raises a :exc:`ZeroDivisionError`. The second version requires that
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*other_fraction* is an instance of :class:`numbers.Rational` and returns a
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:class:`Fraction` instance with the same value. The next two versions accept
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either a :class:`float` or a :class:`decimal.Decimal` instance, and return a
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:class:`Fraction` instance with exactly the same value. Note that due to the
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usual issues with binary floating-point (see :ref:`tut-fp-issues`), the
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argument to ``Fraction(1.1)`` is not exactly equal to 11/10, and so
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``Fraction(1.1)`` does *not* return ``Fraction(11, 10)`` as one might expect.
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(But see the documentation for the :meth:`limit_denominator` method below.)
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The last version of the constructor expects a string or unicode instance.
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The usual form for this instance is::
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[sign] numerator ['/' denominator]
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where the optional ``sign`` may be either '+' or '-' and
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``numerator`` and ``denominator`` (if present) are strings of
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decimal digits. In addition, any string that represents a finite
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value and is accepted by the :class:`float` constructor is also
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accepted by the :class:`Fraction` constructor. In either form the
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input string may also have leading and/or trailing whitespace.
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Here are some examples::
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>>> from fractions import Fraction
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>>> Fraction(16, -10)
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Fraction(-8, 5)
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>>> Fraction(123)
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Fraction(123, 1)
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>>> Fraction()
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Fraction(0, 1)
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>>> Fraction('3/7')
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Fraction(3, 7)
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>>> Fraction(' -3/7 ')
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Fraction(-3, 7)
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>>> Fraction('1.414213 \t\n')
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Fraction(1414213, 1000000)
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>>> Fraction('-.125')
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Fraction(-1, 8)
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>>> Fraction('7e-6')
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Fraction(7, 1000000)
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>>> Fraction(2.25)
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Fraction(9, 4)
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>>> Fraction(1.1)
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Fraction(2476979795053773, 2251799813685248)
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>>> from decimal import Decimal
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>>> Fraction(Decimal('1.1'))
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Fraction(11, 10)
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The :class:`Fraction` class inherits from the abstract base class
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:class:`numbers.Rational`, and implements all of the methods and
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operations from that class. :class:`Fraction` instances are hashable,
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and should be treated as immutable. In addition,
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:class:`Fraction` has the following properties and methods:
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.. versionchanged:: 3.2
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The :class:`Fraction` constructor now accepts :class:`float` and
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:class:`decimal.Decimal` instances.
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.. attribute:: numerator
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Numerator of the Fraction in lowest term.
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.. attribute:: denominator
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Denominator of the Fraction in lowest term.
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.. method:: from_float(flt)
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This class method constructs a :class:`Fraction` representing the exact
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value of *flt*, which must be a :class:`float`. Beware that
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``Fraction.from_float(0.3)`` is not the same value as ``Fraction(3, 10)``.
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.. note::
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From Python 3.2 onwards, you can also construct a
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:class:`Fraction` instance directly from a :class:`float`.
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.. method:: from_decimal(dec)
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This class method constructs a :class:`Fraction` representing the exact
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value of *dec*, which must be a :class:`decimal.Decimal` instance.
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.. note::
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From Python 3.2 onwards, you can also construct a
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:class:`Fraction` instance directly from a :class:`decimal.Decimal`
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instance.
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.. method:: limit_denominator(max_denominator=1000000)
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Finds and returns the closest :class:`Fraction` to ``self`` that has
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denominator at most max_denominator. This method is useful for finding
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rational approximations to a given floating-point number:
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>>> from fractions import Fraction
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>>> Fraction('3.1415926535897932').limit_denominator(1000)
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Fraction(355, 113)
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or for recovering a rational number that's represented as a float:
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>>> from math import pi, cos
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>>> Fraction(cos(pi/3))
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Fraction(4503599627370497, 9007199254740992)
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>>> Fraction(cos(pi/3)).limit_denominator()
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Fraction(1, 2)
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>>> Fraction(1.1).limit_denominator()
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Fraction(11, 10)
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.. method:: __floor__()
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Returns the greatest :class:`int` ``<= self``. This method can
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also be accessed through the :func:`math.floor` function:
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>>> from math import floor
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>>> floor(Fraction(355, 113))
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3
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.. method:: __ceil__()
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Returns the least :class:`int` ``>= self``. This method can
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also be accessed through the :func:`math.ceil` function.
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.. method:: __round__()
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__round__(ndigits)
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The first version returns the nearest :class:`int` to ``self``,
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rounding half to even. The second version rounds ``self`` to the
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nearest multiple of ``Fraction(1, 10**ndigits)`` (logically, if
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``ndigits`` is negative), again rounding half toward even. This
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method can also be accessed through the :func:`round` function.
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.. function:: gcd(a, b)
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Return the greatest common divisor of the integers *a* and *b*. If either
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*a* or *b* is nonzero, then the absolute value of ``gcd(a, b)`` is the
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largest integer that divides both *a* and *b*. ``gcd(a,b)`` has the same
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sign as *b* if *b* is nonzero; otherwise it takes the sign of *a*. ``gcd(0,
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0)`` returns ``0``.
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.. deprecated:: 3.5
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Use :func:`math.gcd` instead.
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.. seealso::
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Module :mod:`numbers`
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The abstract base classes making up the numeric tower.
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