fractions --- 有理數


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 new Fraction instance with value numerator/denominator. If denominator is 0, it raises a ZeroDivisionError. The second version requires that other_fraction is an instance of numbers.Rational and returns a Fraction instance with the same value. The next two versions accept either a float or a decimal.Decimal instance, and return a Fraction instance with exactly the same value. Note that due to the usual issues with binary floating point (see Floating-Point Arithmetic: Issues and Limitations), 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 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 (underscores may be used to delimit digits as with integral literals in code). In addition, any string that represents a finite value and is accepted by the float constructor is also accepted by the 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 Fraction class inherits from the abstract base class numbers.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:

在 3.2 版的變更: The Fraction constructor now accepts float and decimal.Decimal instances.

在 3.9 版的變更: The math.gcd() function is now used to normalize the numerator and denominator. math.gcd() always returns an int type. Previously, the GCD type depended on numerator and denominator.

在 3.11 版的變更: Underscores are now permitted when creating a Fraction instance from a string, following PEP 515 rules.

在 3.11 版的變更: Fraction implements __int__ now to satisfy typing.SupportsInt instance checks.

在 3.12 版的變更: Space is allowed around the slash for string inputs: Fraction('2 / 3').

在 3.12 版的變更: Fraction instances now support float-style formatting, with presentation 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 equal to the original Fraction. The ratio is in lowest terms and has a positive denominator.

在 3.8 版被加入.


Return True if the Fraction is an integer.

在 3.12 版被加入.

classmethod from_float(flt)

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


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

classmethod from_decimal(dec)

Alternative constructor which only accepts instances of decimal.Decimal or numbers.Integral.


From Python 3.2 onwards, you can also construct a Fraction instance directly from a decimal.Decimal instance.


Finds and returns the closest 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)

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

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

Returns the least int >= self. This method can also 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 the nearest multiple of Fraction(1, 10**ndigits) (logically, if ndigits is negative), again rounding half toward even. This method can also be accessed through the round() function.

__format__(format_spec, /)

Provides support for float-style formatting of Fraction instances via the str.format() method, the format() built-in function, or Formatted string literals. The presentation types "e", "E", "f", "F", "g", "G" and "%" are supported. For these presentation types, formatting for a Fraction object x follows the rules outlined for the float type in the 格式規格 (Format Specification) 迷你語言 section.

Here are some examples:

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


numbers 模組

The abstract base classes making up the numeric tower.