# Originally contributed by Sjoerd Mullender.
# Significantly modified by Jeffrey Yasskin <jyasskin at gmail.com>.
"""Fraction, infinite-precision, real numbers."""
from decimal import Decimal
__all__ = ['Fraction', 'gcd']
"""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).
warnings.warn('fractions.gcd() is deprecated. Use math.gcd() instead.',
if type(a) is int is type(b):
# Supports non-integers for backward compatibility.
# 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)
(?:/(?P<denom>\d+))? # an optional denominator
(?:\.(?P<decimal>\d*))? # an optional fractional part
(?:E(?P<exp>[-+]?\d+))? # and optional exponent
\s*\Z # and optional whitespace to finish
""", re.VERBOSE | re.IGNORECASE)
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(Fraction(1, 7), 5)
>>> Fraction(Fraction(1, 7), Fraction(2, 3))
>>> Fraction('3.1415') # conversion from numeric string
>>> Fraction('-47e-2') # string may include a decimal exponent
>>> Fraction(1.47) # direct construction from float (exact conversion)
Fraction(6620291452234629, 4503599627370496)
>>> Fraction(Decimal('1.47'))
self = super(Fraction, cls).__new__(cls)
if type(numerator) is int:
self._numerator = numerator
elif isinstance(numerator, numbers.Rational):
self._numerator = numerator.numerator
self._denominator = numerator.denominator
elif isinstance(numerator, (float, Decimal)):
self._numerator, self._denominator = numerator.as_integer_ratio()
elif isinstance(numerator, str):
# Handle construction from strings.
m = _RATIONAL_FORMAT.match(numerator)
raise ValueError('Invalid literal for Fraction: %r' %
numerator = int(m.group('num') or '0')
decimal = m.group('decimal')
numerator = numerator * scale + int(decimal)
if m.group('sign') == '-':
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 "
raise ZeroDivisionError('Fraction(%s, 0)' % numerator)
if type(numerator) is int is type(denominator):
g = math.gcd(numerator, denominator)
g = _gcd(numerator, denominator)
self._numerator = numerator
self._denominator = denominator
"""Converts a finite float to a rational number, exactly.
Beware that Fraction.from_float(0.3) != Fraction(3, 10).
if isinstance(f, numbers.Integral):
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):
elif not isinstance(dec, Decimal):
"%s.from_decimal() only takes Decimals, not %r (%s)" %
(cls.__name__, dec, type(dec).__name__))
return cls(*dec.as_integer_ratio())
def as_integer_ratio(self):
"""Return the integer ratio as a tuple.
Return a tuple of two integers, whose ratio is equal to the
Fraction and with a positive denominator.
return (self._numerator, self._denominator)
def limit_denominator(self, max_denominator=1000000):
"""Closest Fraction to self with denominator at most max_denominator.
>>> Fraction('3.141592653589793').limit_denominator(10)
>>> Fraction('3.141592653589793').limit_denominator(100)
>>> Fraction(4321, 8765).limit_denominator(10000)
# Algorithm notes: For any real number x, define a *best upper
# approximation* to x to be a rational number p/q such that:
# (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
# 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
raise ValueError("max_denominator should be at least 1")
if self._denominator <= max_denominator:
p0, q0, p1, q1 = 0, 1, 1, 0
n, d = self._numerator, self._denominator
p0, q0, p1, q1 = p1, q1, p0+a*p1, q2
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 '%s(%s, %s)' % (self.__class__.__name__,
self._numerator, self._denominator)
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.
__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.
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)
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
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)
forward.__name__ = '__' + fallback_operator.__name__ + '__'
forward.__doc__ = monomorphic_operator.__doc__
if isinstance(a, numbers.Rational):
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))
reverse.__name__ = '__r' + fallback_operator.__name__ + '__'
reverse.__doc__ = monomorphic_operator.__doc__
da, db = a.denominator, b.denominator
return Fraction(a.numerator * db + b.numerator * da,
__add__, __radd__ = _operator_fallbacks(_add, operator.add)
da, db = a.denominator, b.denominator
return Fraction(a.numerator * db - b.numerator * da,
__sub__, __rsub__ = _operator_fallbacks(_sub, operator.sub)
return Fraction(a.numerator * b.numerator, a.denominator * b.denominator)
__mul__, __rmul__ = _operator_fallbacks(_mul, operator.mul)
return Fraction(a.numerator * b.denominator,
a.denominator * b.numerator)
__truediv__, __rtruediv__ = _operator_fallbacks(_div, operator.truediv)
return (a.numerator * b.denominator) // (a.denominator * b.numerator)
__floordiv__, __rfloordiv__ = _operator_fallbacks(_floordiv, operator.floordiv)
da, db = a.denominator, b.denominator
div, n_mod = divmod(a.numerator * db, da * b.numerator)
return div, Fraction(n_mod, da * db)
__divmod__, __rdivmod__ = _operator_fallbacks(_divmod, divmod)
da, db = a.denominator, b.denominator
return Fraction((a.numerator * db) % (b.numerator * da), da * db)
__mod__, __rmod__ = _operator_fallbacks(_mod, operator.mod)
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
if isinstance(b, numbers.Rational):
return Fraction(a._numerator ** power,
return Fraction(a._denominator ** -power,
return Fraction((-a._denominator) ** -power,
(-a._numerator) ** -power,
# A fractional power will generally produce an
return float(a) ** float(b)
if b._denominator == 1 and b._numerator >= 0:
# If a is an int, keep it that way if possible.
if isinstance(a, numbers.Rational):
return Fraction(a.numerator, a.denominator) ** b
It is recommended that you Edit text format, this type of Fix handles quite a lot in one request
