9.5. fractions
— Rational numbers¶
Source code: Lib/fractions.py
The 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
fractions.
Fraction
(numerator=0, denominator=1)¶ 
class
fractions.
Fraction
(other_fraction) 
class
fractions.
Fraction
(float) 
class
fractions.
Fraction
(decimal) 
class
fractions.
Fraction
(string) The first version requires that numerator and denominator are instances of
numbers.Rational
and returns a newFraction
instance with valuenumerator/denominator
. If denominator is0
, it raises aZeroDivisionError
. The second version requires that other_fraction is an instance ofnumbers.Rational
and returns aFraction
instance with the same value. The next two versions accept either afloat
or adecimal.Decimal
instance, and return aFraction
instance with exactly the same value. Note that due to the usual issues with binary floatingpoint (see Floating Point Arithmetic: Issues and Limitations), the argument toFraction(1.1)
is not exactly equal to 11/10, and soFraction(1.1)
does not returnFraction(11, 10)
as one might expect. (But see the documentation for thelimit_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 ‘‘ andnumerator
anddenominator
(if present) are strings of decimal digits. In addition, any string that represents a finite value and is accepted by thefloat
constructor is also accepted by theFraction
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('7e6') 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
Fraction
class inherits from the abstract base classnumbers.Rational
, and implements all of the methods and operations from that class.Fraction
instances are hashable, and should be treated as immutable. In addition,Fraction
has the following properties and methods:Changed in version 3.2: The
Fraction
constructor now acceptsfloat
anddecimal.Decimal
instances.
numerator
¶ Numerator of the Fraction in lowest term.

denominator
¶ Denominator of the Fraction in lowest term.

from_float
(flt)¶ This class method constructs a
Fraction
representing the exact value of flt, which must be afloat
. Beware thatFraction.from_float(0.3)
is not the same value asFraction(3, 10)
.

from_decimal
(dec)¶ This class method constructs a
Fraction
representing the exact value of dec, which must be adecimal.Decimal
instance.Note
From Python 3.2 onwards, you can also construct a
Fraction
instance directly from adecimal.Decimal
instance.

limit_denominator
(max_denominator=1000000)¶ Finds and returns the closest
Fraction
toself
that has denominator at most max_denominator. This method is useful for finding rational approximations to a given floatingpoint 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)

__floor__
()¶ Returns the greatest
int
<= self
. This method can also be accessed through themath.floor()
function:>>> from math import floor >>> floor(Fraction(355, 113)) 3

__ceil__
()¶ Returns the least
int
>= self
. This method can also be accessed through themath.ceil()
function.

__round__
()¶ 
__round__
(ndigits) The first version returns the nearest
int
toself
, rounding half to even. The second version roundsself
to the nearest multiple ofFraction(1, 10**ndigits)
(logically, ifndigits
is negative), again rounding half toward even. This method can also be accessed through theround()
function.


fractions.
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)
returns0
.Deprecated since version 3.5: Use
math.gcd()
instead.
See also
 Module
numbers
The abstract base classes making up the numeric tower.