fractions — Rational numbers¶

Source code:Lib/fractions.py

Thefractions module provides support for rational number arithmetic.

A Fraction instance can be constructed from a pair of rational numbers, froma single number, or from a string.

classfractions.Fraction(numerator=0,denominator=1)¶

classfractions.Fraction(number)

classfractions.Fraction(string)

The first version requires thatnumerator anddenominator are instancesofnumbers.Rational and returns a newFraction instancewith a value equal tonumerator/denominator.Ifdenominator is zero, it raises aZeroDivisionError.

The second version requires thatnumber is an instance ofnumbers.Rational or has theas_integer_ratio() method(this includesfloat anddecimal.Decimal).It returns aFraction instance with exactly the same value.Assumed, that theas_integer_ratio() method returns a pairof coprime integers and last one is positive.Note that due to theusual issues with binary point (seeFloating-Point Arithmetic: Issues and Limitations), theargument toFraction(1.1) is not exactly equal to 11/10, and soFraction(1.1) doesnot 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.The usual form for this instance is:

[sign]numerator['/'denominator]

where the optionalsign may be either ‘+’ or ‘-’ andnumerator anddenominator (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 thefloat constructor is alsoaccepted by theFraction constructor. In either form theinput string may also have leading and/or trailing whitespace.Here are some examples:

>>>fromfractionsimportFraction>>>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)>>>fromdecimalimportDecimal>>>Fraction(Decimal('1.1'))Fraction(11, 10)

TheFraction class inherits from the abstract base classnumbers.Rational, and implements all of the methods andoperations from that class.Fraction instances arehashable,and should be treated as immutable. In addition,Fraction has the following properties and methods:

Changed in version 3.2:TheFraction constructor now acceptsfloat anddecimal.Decimal instances.

Changed in version 3.9:Themath.gcd() function is now used to normalize thenumeratoranddenominator.math.gcd() always returns anint type.Previously, the GCD type depended onnumerator anddenominator.

Changed in version 3.11:Underscores are now permitted when creating aFraction instancefrom a string, followingPEP 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"%"".

Changed in version 3.13:Formatting ofFraction instances without a presentation typenow supports fill, alignment, sign handling, minimum width and grouping.

Changed in version 3.14:TheFraction constructor now accepts any objects with theas_integer_ratio() method.

numerator¶

Numerator of the Fraction in lowest term.

denominator¶

Denominator of the Fraction in lowest terms.Guaranteed to be positive.

as_integer_ratio()¶

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.

is_integer()¶

ReturnTrue if the Fraction is an integer.

Added in version 3.12.

classmethodfrom_float(f)¶

Alternative constructor which only accepts instances offloat ornumbers.Integral. Beware thatFraction.from_float(0.3) is not the same value asFraction(3,10).

Note

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

classmethodfrom_decimal(dec)¶

Alternative constructor which only accepts instances ofdecimal.Decimal ornumbers.Integral.

Note

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

classmethodfrom_number(number)¶

Alternative constructor which only accepts instances ofnumbers.Integral,numbers.Rational,float ordecimal.Decimal, and objects withtheas_integer_ratio() method, but not strings.

Added in version 3.14.

limit_denominator(max_denominator=1000000)¶

Finds and returns the closestFraction toself that hasdenominator at most max_denominator. This method is useful for findingrational approximations to a given floating-point number:

>>>fromfractionsimportFraction>>>Fraction('3.1415926535897932').limit_denominator(1000)Fraction(355, 113)

or for recovering a rational number that’s represented as a float:

>>>frommathimportpi,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 greatestint<=self. This method canalso be accessed through themath.floor() function:

>>>frommathimportfloor>>>floor(Fraction(355,113))3

__ceil__()¶

Returns the leastint>=self. This method canalso be accessed through themath.ceil() function.

__round__()¶

__round__(ndigits)

The first version returns the nearestint toself,rounding half to even. The second version roundsself to thenearest multiple ofFraction(1,10**ndigits) (logically, ifndigits is negative), again rounding half toward even. Thismethod can also be accessed through theround() function.

__format__(format_spec,/)¶

Provides support for formatting ofFraction instances via thestr.format() method, theformat() built-in function, orFormatted string literals.

If theformat_spec format specification string does not end with oneof the presentation types'e','E','f','F','g','G' or'%' then formatting follows the general rules for fill,alignment, sign handling, minimum width, and grouping as described in theformat specification mini-language. The “alternateform” flag'#' is supported: if present, it forces the output stringto always include an explicit denominator, even when the value beingformatted is an exact integer. The zero-fill flag'0' is notsupported.

If theformat_spec format specification string ends with one ofthe presentation types'e','E','f','F','g','G' or'%' then formatting follows the rules outlined for thefloat type in theFormat Specification Mini-Language section.

Here are some examples:

>>>fromfractionsimportFraction>>>format(Fraction(103993,33102),'_')'103_993/33_102'>>>format(Fraction(1,7),'.^+10')'...+1/7...'>>>format(Fraction(3,1),'')'3'>>>format(Fraction(3,1),'#')'3/1'>>>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'