# `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 (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:

Changed in version 3.2: The `Fraction` constructor now accepts `float` and `decimal.Decimal` instances.

Changed in version 3.9: The `math.gcd()` function is now used to normalize the numerator and 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` instance from a string, following PEP 515 rules.

Changed in version 3.11: `Fraction` implements `__int__` now to satisfy `typing.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, with presentation types `"e"`, `"E"`, `"f"`, `"F"`, `"g"`, `"G"` and `"%""`.

numerator

Numerator of the Fraction in lowest term.

denominator

Denominator of the Fraction in lowest term.

as_integer_ratio()

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.

New in version 3.8.

is_integer()

Return `True` if the Fraction is an integer.

New in version 3.12.

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)`.

Note

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`.

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.

__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 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'
```

Module `numbers`