, and >=.
Real also provides defaults for the derived operations.
"""
__slots__ = ()
@abstractmethod
def __float__(self):
"""Any Real can be converted to a native float object.
Called for float(self)."""
raise NotImplementedError
@abstractmethod
def __trunc__(self):
"""trunc(self): Truncates self to an Integral.
Returns an Integral i such that:
* i>0 iff self>0;
* abs(i) <= abs(self);
* for any Integral j satisfying the first two conditions,
abs(i) >= abs(j) [i.e. i has "maximal" abs among those].
i.e. "truncate towards 0".
"""
raise NotImplementedError
def __divmod__(self, other):
"""divmod(self, other): The pair (self // other, self % other).
Sometimes this can be computed faster than the pair of
operations.
"""
return (self // other, self % other)
def __rdivmod__(self, other):
"""divmod(other, self): The pair (self // other, self % other).
Sometimes this can be computed faster than the pair of
operations.
"""
return (other // self, other % self)
@abstractmethod
def __floordiv__(self, other):
"""self // other: The floor() of self/other."""
raise NotImplementedError
@abstractmethod
def __rfloordiv__(self, other):
"""other // self: The floor() of other/self."""
raise NotImplementedError
@abstractmethod
def __mod__(self, other):
"""self % other"""
raise NotImplementedError
@abstractmethod
def __rmod__(self, other):
"""other % self"""
raise NotImplementedError
@abstractmethod
def __lt__(self, other):
"""self < other
< on Reals defines a total ordering, except perhaps for NaN."""
raise NotImplementedError
@abstractmethod
def __le__(self, other):
"""self <= other"""
raise NotImplementedError
# Concrete implementations of Complex abstract methods.
def __complex__(self):
"""complex(self) == complex(float(self), 0)"""
return complex(float(self))
@property
def real(self):
"""Real numbers are their real component."""
return +self
@property
def imag(self):
"""Real numbers have no imaginary component."""
return 0
def conjugate(self):
"""Conjugate is a no-op for Reals."""
return +self
Real.register(float)
class Rational(Real):
""".numerator and .denominator should be in lowest terms."""
__slots__ = ()
@abstractproperty
def numerator(self):
raise NotImplementedError
@abstractproperty
def denominator(self):
raise NotImplementedError
# Concrete implementation of Real's conversion to float.
def __float__(self):
"""float(self) = self.numerator / self.denominator
It's important that this conversion use the integer's "true"
division rather than casting one side to float before dividing
so that ratios of huge integers convert without overflowing.
"""
return self.numerator / self.denominator
class Integral(Rational):
"""Integral adds a conversion to long and the bit-string operations."""
__slots__ = ()
@abstractmethod
def __long__(self):
"""long(self)"""
raise NotImplementedError
def __index__(self):
"""Called whenever an index is needed, such as in slicing"""
return long(self)
@abstractmethod
def __pow__(self, exponent, modulus=None):
"""self ** exponent % modulus, but maybe faster.
Accept the modulus argument if you want to support the
3-argument version of pow(). Raise a TypeError if exponent < 0
or any argument isn't Integral. Otherwise, just implement the
2-argument version described in Complex.
"""
raise NotImplementedError
@abstractmethod
def __lshift__(self, other):
"""self << other"""
raise NotImplementedError
@abstractmethod
def __rlshift__(self, other):
"""other << self"""
raise NotImplementedError
@abstractmethod
def __rshift__(self, other):
"""self >> other"""
raise NotImplementedError
@abstractmethod
def __rrshift__(self, other):
"""other >> self"""
raise NotImplementedError
@abstractmethod
def __and__(self, other):
"""self & other"""
raise NotImplementedError
@abstractmethod
def __rand__(self, other):
"""other & self"""
raise NotImplementedError
@abstractmethod
def __xor__(self, other):
"""self ^ other"""
raise NotImplementedError
@abstractmethod
def __rxor__(self, other):
"""other ^ self"""
raise NotImplementedError
@abstractmethod
def __or__(self, other):
"""self | other"""
raise NotImplementedError
@abstractmethod
def __ror__(self, other):
"""other | self"""
raise NotImplementedError
@abstractmethod
def __invert__(self):
"""~self"""
raise NotImplementedError
# Concrete implementations of Rational and Real abstract methods.
def __float__(self):
"""float(self) == float(long(self))"""
return float(long(self))
@property
def numerator(self):
"""Integers are their own numerators."""
return +self
@property
def denominator(self):
"""Integers have a denominator of 1."""
return 1
Integral.register(int)
Integral.register(long)