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# 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 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. 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:

Changed in version 3.2: The Fraction constructor now accepts float and decimal.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 a float. Beware that Fraction.from_float(0.3) is not the same value as Fraction(3, 10).

Note

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

from_decimal(dec)

This class method constructs a Fraction representing the exact value of dec, which must be a decimal.Decimal instance.

Note

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

limit_denominator(max_denominator=1000000)

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)
```
__floor__()

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

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

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

__round__()
__round__(ndigits)

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.

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) returns 0.

See also

Module numbers
The abstract base classes making up the numeric tower.

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9.4. decimal — Decimal fixed point and floating point arithmetic

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9.6. random — Generate pseudo-random numbers