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# Originally contributed by Sjoerd Mullender.
# Significantly modified by Jeffrey Yasskin <jyasskin at>.
"""Fraction, infinite-precision, real numbers."""
from decimal import Decimal
import math
import numbers
import operator
import re
import sys
__all__ = ['Fraction', 'gcd']
def gcd(a, b):
"""Calculate the Greatest Common Divisor of a and b.
Unless b==0, the result will have the same sign as b (so that when
b is divided by it, the result comes out positive).
import warnings
warnings.warn('fractions.gcd() is deprecated. Use math.gcd() instead.',
DeprecationWarning, 2)
if type(a) is int is type(b):
if (b or a) < 0:
return -math.gcd(a, b)
return math.gcd(a, b)
return _gcd(a, b)
def _gcd(a, b):
# Supports non-integers for backward compatibility.
while b:
a, b = b, a%b
return a
# Constants related to the hash implementation; hash(x) is based
# on the reduction of x modulo the prime _PyHASH_MODULUS.
_PyHASH_MODULUS = sys.hash_info.modulus
# Value to be used for rationals that reduce to infinity modulo
_PyHASH_INF = sys.hash_info.inf
_RATIONAL_FORMAT = re.compile(r"""
\A\s* # optional whitespace at the start, then
(?P<sign>[-+]?) # an optional sign, then
(?=\d|\.\d) # lookahead for digit or .digit
(?P<num>\d*) # numerator (possibly empty)
(?: # followed by
(?:/(?P<denom>\d+))? # an optional denominator
| # or
(?:\.(?P<decimal>\d*))? # an optional fractional part
(?:E(?P<exp>[-+]?\d+))? # and optional exponent
\s*\Z # and optional whitespace to finish
class Fraction(numbers.Rational):
"""This class implements rational numbers.
In the two-argument form of the constructor, Fraction(8, 6) will
produce a rational number equivalent to 4/3. Both arguments must
be Rational. The numerator defaults to 0 and the denominator
defaults to 1 so that Fraction(3) == 3 and Fraction() == 0.
Fractions can also be constructed from:
- numeric strings similar to those accepted by the
float constructor (for example, '-2.3' or '1e10')
- strings of the form '123/456'
- float and Decimal instances
- other Rational instances (including integers)
__slots__ = ('_numerator', '_denominator')
# We're immutable, so use __new__ not __init__
def __new__(cls, numerator=0, denominator=None, *, _normalize=True):
"""Constructs a Rational.
Takes a string like '3/2' or '1.5', another Rational instance, a
numerator/denominator pair, or a float.
>>> Fraction(10, -8)
Fraction(-5, 4)
>>> Fraction(Fraction(1, 7), 5)
Fraction(1, 35)
>>> Fraction(Fraction(1, 7), Fraction(2, 3))
Fraction(3, 14)
>>> Fraction('314')
Fraction(314, 1)
>>> Fraction('-35/4')
Fraction(-35, 4)
>>> Fraction('3.1415') # conversion from numeric string
Fraction(6283, 2000)
>>> Fraction('-47e-2') # string may include a decimal exponent
Fraction(-47, 100)
>>> Fraction(1.47) # direct construction from float (exact conversion)
Fraction(6620291452234629, 4503599627370496)
>>> Fraction(2.25)
Fraction(9, 4)
>>> Fraction(Decimal('1.47'))
Fraction(147, 100)
self = super(Fraction, cls).__new__(cls)
if denominator is None:
if type(numerator) is int:
self._numerator = numerator
self._denominator = 1
return self
elif isinstance(numerator, numbers.Rational):
self._numerator = numerator.numerator
self._denominator = numerator.denominator
return self
elif isinstance(numerator, (float, Decimal)):
# Exact conversion
self._numerator, self._denominator = numerator.as_integer_ratio()
return self
elif isinstance(numerator, str):
# Handle construction from strings.
m = _RATIONAL_FORMAT.match(numerator)
if m is None:
raise ValueError('Invalid literal for Fraction: %r' %
numerator = int('num') or '0')
denom ='denom')
if denom:
denominator = int(denom)
denominator = 1
decimal ='decimal')
if decimal:
scale = 10**len(decimal)
numerator = numerator * scale + int(decimal)
denominator *= scale
exp ='exp')
if exp:
exp = int(exp)
if exp >= 0:
numerator *= 10**exp
denominator *= 10**-exp
if'sign') == '-':
numerator = -numerator
raise TypeError("argument should be a string "
"or a Rational instance")
elif type(numerator) is int is type(denominator):
pass # *very* normal case
elif (isinstance(numerator, numbers.Rational) and
isinstance(denominator, numbers.Rational)):
numerator, denominator = (
numerator.numerator * denominator.denominator,
denominator.numerator * numerator.denominator
raise TypeError("both arguments should be "
"Rational instances")
if denominator == 0:
raise ZeroDivisionError('Fraction(%s, 0)' % numerator)
if _normalize:
if type(numerator) is int is type(denominator):
# *very* normal case
g = math.gcd(numerator, denominator)
if denominator < 0:
g = -g
g = _gcd(numerator, denominator)
numerator //= g
denominator //= g
self._numerator = numerator
self._denominator = denominator
return self
def from_float(cls, f):
"""Converts a finite float to a rational number, exactly.
Beware that Fraction.from_float(0.3) != Fraction(3, 10).
if isinstance(f, numbers.Integral):
return cls(f)
elif not isinstance(f, float):
raise TypeError("%s.from_float() only takes floats, not %r (%s)" %
(cls.__name__, f, type(f).__name__))
return cls(*f.as_integer_ratio())
def from_decimal(cls, dec):
"""Converts a finite Decimal instance to a rational number, exactly."""
from decimal import Decimal
if isinstance(dec, numbers.Integral):
dec = Decimal(int(dec))
elif not isinstance(dec, Decimal):
raise TypeError(
"%s.from_decimal() only takes Decimals, not %r (%s)" %
(cls.__name__, dec, type(dec).__name__))
return cls(*dec.as_integer_ratio())
def limit_denominator(self, max_denominator=1000000):
"""Closest Fraction to self with denominator at most max_denominator.
>>> Fraction('3.141592653589793').limit_denominator(10)
Fraction(22, 7)
>>> Fraction('3.141592653589793').limit_denominator(100)
Fraction(311, 99)
>>> Fraction(4321, 8765).limit_denominator(10000)
Fraction(4321, 8765)
# Algorithm notes: For any real number x, define a *best upper
# approximation* to x to be a rational number p/q such that:
# (1) p/q >= x, and
# (2) if p/q > r/s >= x then s > q, for any rational r/s.
# Define *best lower approximation* similarly. Then it can be
# proved that a rational number is a best upper or lower
# approximation to x if, and only if, it is a convergent or
# semiconvergent of the (unique shortest) continued fraction
# associated to x.
# To find a best rational approximation with denominator <= M,
# we find the best upper and lower approximations with
# denominator <= M and take whichever of these is closer to x.
# In the event of a tie, the bound with smaller denominator is
# chosen. If both denominators are equal (which can happen
# only when max_denominator == 1 and self is midway between
# two integers) the lower bound---i.e., the floor of self, is
# taken.
if max_denominator < 1:
raise ValueError("max_denominator should be at least 1")
if self._denominator <= max_denominator:
return Fraction(self)
p0, q0, p1, q1 = 0, 1, 1, 0
n, d = self._numerator, self._denominator
while True:
a = n//d
q2 = q0+a*q1
if q2 > max_denominator:
p0, q0, p1, q1 = p1, q1, p0+a*p1, q2
n, d = d, n-a*d
k = (max_denominator-q0)//q1
bound1 = Fraction(p0+k*p1, q0+k*q1)
bound2 = Fraction(p1, q1)
if abs(bound2 - self) <= abs(bound1-self):
return bound2
return bound1
def numerator(a):
return a._numerator
def denominator(a):
return a._denominator
def __repr__(self):
return '%s(%s, %s)' % (self.__class__.__name__,
self._numerator, self._denominator)
def __str__(self):
if self._denominator == 1:
return str(self._numerator)
return '%s/%s' % (self._numerator, self._denominator)
def _operator_fallbacks(monomorphic_operator, fallback_operator):
"""Generates forward and reverse operators given a purely-rational
operator and a function from the operator module.
Use this like:
__op__, __rop__ = _operator_fallbacks(just_rational_op, operator.op)
In general, we want to implement the arithmetic operations so
that mixed-mode operations either call an implementation whose
author knew about the types of both arguments, or convert both
to the nearest built in type and do the operation there. In
Fraction, that means that we define __add__ and __radd__ as:
def __add__(self, other):
# Both types have numerators/denominator attributes,
# so do the operation directly
if isinstance(other, (int, Fraction)):
return Fraction(self.numerator * other.denominator +
other.numerator * self.denominator,
self.denominator * other.denominator)
# float and complex don't have those operations, but we
# know about those types, so special case them.
elif isinstance(other, float):
return float(self) + other
elif isinstance(other, complex):
return complex(self) + other
# Let the other type take over.
return NotImplemented
def __radd__(self, other):
# radd handles more types than add because there's
# nothing left to fall back to.
if isinstance(other, numbers.Rational):
return Fraction(self.numerator * other.denominator +
other.numerator * self.denominator,
self.denominator * other.denominator)
elif isinstance(other, Real):
return float(other) + float(self)
elif isinstance(other, Complex):
return complex(other) + complex(self)
return NotImplemented
There are 5 different cases for a mixed-type addition on
Fraction. I'll refer to all of the above code that doesn't
refer to Fraction, float, or complex as "boilerplate". 'r'
will be an instance of Fraction, which is a subtype of
Rational (r : Fraction <: Rational), and b : B <:
Complex. The first three involve 'r + b':
1. If B <: Fraction, int, float, or complex, we handle
that specially, and all is well.
2. If Fraction falls back to the boilerplate code, and it
were to return a value from __add__, we'd miss the
possibility that B defines a more intelligent __radd__,
so the boilerplate should return NotImplemented from
__add__. In particular, we don't handle Rational
here, even though we could get an exact answer, in case
the other type wants to do something special.
3. If B <: Fraction, Python tries B.__radd__ before
Fraction.__add__. This is ok, because it was
implemented with knowledge of Fraction, so it can
handle those instances before delegating to Real or
The next two situations describe 'b + r'. We assume that b
didn't know about Fraction in its implementation, and that it
uses similar boilerplate code:
4. If B <: Rational, then __radd_ converts both to the
builtin rational type (hey look, that's us) and
5. Otherwise, __radd__ tries to find the nearest common
base ABC, and fall back to its builtin type. Since this
class doesn't subclass a concrete type, there's no
implementation to fall back to, so we need to try as
hard as possible to return an actual value, or the user
will get a TypeError.
def forward(a, b):
if isinstance(b, (int, Fraction)):
return monomorphic_operator(a, b)
elif isinstance(b, float):
return fallback_operator(float(a), b)
elif isinstance(b, complex):
return fallback_operator(complex(a), b)
return NotImplemented
forward.__name__ = '__' + fallback_operator.__name__ + '__'
forward.__doc__ = monomorphic_operator.__doc__
def reverse(b, a):
if isinstance(a, numbers.Rational):
# Includes ints.
return monomorphic_operator(a, b)
elif isinstance(a, numbers.Real):
return fallback_operator(float(a), float(b))
elif isinstance(a, numbers.Complex):
return fallback_operator(complex(a), complex(b))
return NotImplemented
reverse.__name__ = '__r' + fallback_operator.__name__ + '__'
reverse.__doc__ = monomorphic_operator.__doc__
return forward, reverse
def _add(a, b):
"""a + b"""
da, db = a.denominator, b.denominator
return Fraction(a.numerator * db + b.numerator * da,
da * db)
__add__, __radd__ = _operator_fallbacks(_add, operator.add)
def _sub(a, b):
"""a - b"""
da, db = a.denominator, b.denominator
return Fraction(a.numerator * db - b.numerator * da,
da * db)
__sub__, __rsub__ = _operator_fallbacks(_sub, operator.sub)
def _mul(a, b):
"""a * b"""
return Fraction(a.numerator * b.numerator, a.denominator * b.denominator)
__mul__, __rmul__ = _operator_fallbacks(_mul, operator.mul)
def _div(a, b):
"""a / b"""
return Fraction(a.numerator * b.denominator,
a.denominator * b.numerator)
__truediv__, __rtruediv__ = _operator_fallbacks(_div, operator.truediv)
def __floordiv__(a, b):
"""a // b"""
return math.floor(a / b)
def __rfloordiv__(b, a):
"""a // b"""
return math.floor(a / b)
def __mod__(a, b):
"""a % b"""
div = a // b
return a - b * div
def __rmod__(b, a):
"""a % b"""
div = a // b
return a - b * div
def __pow__(a, b):
"""a ** b
If b is not an integer, the result will be a float or complex
since roots are generally irrational. If b is an integer, the
result will be rational.
if isinstance(b, numbers.Rational):
if b.denominator == 1:
power = b.numerator
if power >= 0:
return Fraction(a._numerator ** power,
a._denominator ** power,
elif a._numerator >= 0:
return Fraction(a._denominator ** -power,
a._numerator ** -power,
return Fraction((-a._denominator) ** -power,
(-a._numerator) ** -power,
# A fractional power will generally produce an
# irrational number.
return float(a) ** float(b)
return float(a) ** b
def __rpow__(b, a):
"""a ** b"""
if b._denominator == 1 and b._numerator >= 0:
# If a is an int, keep it that way if possible.
return a ** b._numerator
if isinstance(a, numbers.Rational):
return Fraction(a.numerator, a.denominator) ** b
if b._denominator == 1:
return a ** b._numerator
return a ** float(b)
def __pos__(a):
"""+a: Coerces a subclass instance to Fraction"""
return Fraction(a._numerator, a._denominator, _normalize=False)
def __neg__(a):
return Fraction(-a._numerator, a._denominator, _normalize=False)
def __abs__(a):
return Fraction(abs(a._numerator), a._denominator, _normalize=False)
def __trunc__(a):
if a._numerator < 0:
return -(-a._numerator // a._denominator)
return a._numerator // a._denominator
def __floor__(a):
"""Will be math.floor(a) in 3.0."""
return a.numerator // a.denominator
def __ceil__(a):
"""Will be math.ceil(a) in 3.0."""
# The negations cleverly convince floordiv to return the ceiling.
return -(-a.numerator // a.denominator)
def __round__(self, ndigits=None):
"""Will be round(self, ndigits) in 3.0.
Rounds half toward even.
if ndigits is None:
floor, remainder = divmod(self.numerator, self.denominator)
if remainder * 2 < self.denominator:
return floor
elif remainder * 2 > self.denominator:
return floor + 1
# Deal with the half case:
elif floor % 2 == 0:
return floor
return floor + 1
shift = 10**abs(ndigits)
# See _operator_fallbacks.forward to check that the results of
# these operations will always be Fraction and therefore have
# round().
if ndigits > 0:
return Fraction(round(self * shift), shift)
return Fraction(round(self / shift) * shift)
def __hash__(self):
# XXX since this method is expensive, consider caching the result
# In order to make sure that the hash of a Fraction agrees
# with the hash of a numerically equal integer, float or
# Decimal instance, we follow the rules for numeric hashes
# outlined in the documentation. (See library docs, 'Built-in
# Types').
# dinv is the inverse of self._denominator modulo the prime
# _PyHASH_MODULUS, or 0 if self._denominator is divisible by
dinv = pow(self._denominator, _PyHASH_MODULUS - 2, _PyHASH_MODULUS)
if not dinv:
hash_ = _PyHASH_INF
hash_ = abs(self._numerator) * dinv % _PyHASH_MODULUS
result = hash_ if self >= 0 else -hash_
return -2 if result == -1 else result
def __eq__(a, b):
"""a == b"""
if type(b) is int:
return a._numerator == b and a._denominator == 1
if isinstance(b, numbers.Rational):
return (a._numerator == b.numerator and
a._denominator == b.denominator)
if isinstance(b, numbers.Complex) and b.imag == 0:
b = b.real
if isinstance(b, float):
if math.isnan(b) or math.isinf(b):
# comparisons with an infinity or nan should behave in
# the same way for any finite a, so treat a as zero.
return 0.0 == b
return a == a.from_float(b)
# Since a doesn't know how to compare with b, let's give b
# a chance to compare itself with a.
return NotImplemented
def _richcmp(self, other, op):
"""Helper for comparison operators, for internal use only.
Implement comparison between a Rational instance `self`, and
either another Rational instance or a float `other`. If
`other` is not a Rational instance or a float, return
NotImplemented. `op` should be one of the six standard
comparison operators.
# convert other to a Rational instance where reasonable.
if isinstance(other, numbers.Rational):
return op(self._numerator * other.denominator,
self._denominator * other.numerator)
if isinstance(other, float):
if math.isnan(other) or math.isinf(other):
return op(0.0, other)
return op(self, self.from_float(other))
return NotImplemented
def __lt__(a, b):
"""a < b"""
return a._richcmp(b,
def __gt__(a, b):
"""a > b"""
return a._richcmp(b,
def __le__(a, b):
"""a <= b"""
return a._richcmp(b, operator.le)
def __ge__(a, b):
"""a >= b"""
return a._richcmp(b,
def __bool__(a):
"""a != 0"""
return a._numerator != 0
# support for pickling, copy, and deepcopy
def __reduce__(self):
return (self.__class__, (str(self),))
def __copy__(self):
if type(self) == Fraction:
return self # I'm immutable; therefore I am my own clone
return self.__class__(self._numerator, self._denominator)
def __deepcopy__(self, memo):
if type(self) == Fraction:
return self # My components are also immutable
return self.__class__(self._numerator, self._denominator)