fractions — Rational numbers (2024)

Source code: Lib/

The fractions module provides support for rational number arithmetic.

A Fraction instance can be constructed from a pair of integers, fromanother 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 instancesof numbers.Rational and returns a new Fraction instancewith value numerator/denominator. If denominator is 0, itraises a ZeroDivisionError. The second version requires thatother_fraction is an instance of numbers.Rational and returns aFraction instance with the same value. The next two versions accepteither a float or a decimal.Decimal instance, and return aFraction instance with exactly the same value. Note that due to theusual issues with binary floating-point (see Floating Point Arithmetic: Issues and Limitations), theargument to Fraction(1.1) is not exactly equal to 11/10, and soFraction(1.1) does not return Fraction(11, 10) as one might expect.(But see the documentation for the 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 ‘-’ andnumerator and denominator (if present) are strings ofdecimal digits (underscores may be used to delimit digits as withintegral literals in code). In addition, any string that represents a finitevalue and is accepted by the float constructor is alsoaccepted by the Fraction constructor. In either form theinput 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 Fraction class inherits from the abstract base classnumbers.Rational, and implements all of the methods andoperations 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 accepts float anddecimal.Decimal instances.

Changed in version 3.9: The math.gcd() function is now used to normalize the numeratorand denominator. math.gcd() always return a int type.Previously, the GCD type depended on numerator and denominator.

Changed in version 3.11: Underscores are now permitted when creating a Fraction instancefrom a string, following PEP 515 rules.

Changed in version 3.11: Fraction implements __int__ now to satisfytyping.SupportsInt instance checks.

Changed in version 3.12: Space is allowed around the slash for string inputs: Fraction('2 / 3').

Changed in version 3.12: Fraction instances now support float-style formatting, withpresentation types "e", "E", "f", "F", "g", "G"and "%"".


Numerator of the Fraction in lowest term.


Denominator of the Fraction in lowest term.


Return a tuple of two integers, whose ratio is equalto the original Fraction. The ratio is in lowest termsand has a positive denominator.

Added in version 3.8.


Return True if the Fraction is an integer.

Added in version 3.12.

classmethod from_float(flt)

Alternative constructor which only accepts instances offloat or numbers.Integral. Beware thatFraction.from_float(0.3) is not the same value as Fraction(3, 10).


From Python 3.2 onwards, you can also construct aFraction instance directly from a float.

classmethod from_decimal(dec)

Alternative constructor which only accepts instances ofdecimal.Decimal or numbers.Integral.


From Python 3.2 onwards, you can also construct aFraction instance directly from a decimal.Decimalinstance.


Finds and returns the closest Fraction to self that hasdenominator at most max_denominator. This method is useful for findingrational 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)

Returns the greatest int <= self. This method canalso be accessed through the math.floor() function:

>>> from math import floor>>> floor(Fraction(355, 113))3

Returns the least int >= self. This method canalso be accessed through the math.ceil() function.


The first version returns the nearest int to self,rounding half to even. The second version rounds self to thenearest multiple of Fraction(1, 10**ndigits) (logically, ifndigits is negative), again rounding half toward even. Thismethod can also be accessed through the round() function.

__format__(format_spec, /)

Provides support for float-style formatting of Fractioninstances via the str.format() method, the format() built-infunction, or Formatted string literals. Thepresentation types "e", "E", "f", "F", "g", "G"and "%" are supported. For these presentation types, formatting for aFraction object x follows the rules outlined forthe float type in the Format Specification Mini-Language section.

Here are some examples:

>>> from fractions import Fraction>>> format(Fraction(1, 7), '.40g')'0.1428571428571428571428571428571428571429'>>> format(Fraction('1234567.855'), '_.2f')'1_234_567.86'>>> f"{Fraction(355, 113):*>20.6e}"'********3.141593e+00'>>> old_price, new_price = 499, 672>>> "{:.2%} price increase".format(Fraction(new_price, old_price) - 1)'34.67% price increase'

See also

Module numbers

The abstract base classes making up the numeric tower.

fractions — Rational numbers (2024)


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