# 9.5. fractions — Rational numbers¶

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(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 an Fraction instance with the same value. The last version of the constructor expects a string instance. The usual form for this string 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)
[40794 refs]
>>> 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)
```

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 methods:

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)
from_decimal(dec)
This class method constructs a Fraction representing the exact value of dec, which must be 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.from_float(cos(pi/3))
Fraction(4503599627370497, 9007199254740992)
>>> Fraction.from_float(cos(pi/3)).limit_denominator()
Fraction(1, 2)
```
__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.