:mod:`fractions` --- Rational numbers ===================================== .. module:: fractions :synopsis: Rational numbers. .. moduleauthor:: Jeffrey Yasskin .. sectionauthor:: Jeffrey Yasskin .. versionadded:: 2.6 **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 methods: .. versionchanged:: 2.7 The :class:`Fraction` constructor now accepts :class:`float` and :class:`decimal.Decimal` instances. .. 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 2.7 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`. .. note:: From Python 2.7 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) .. 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.