class Number(metaclass=ABCMeta): pass

class Complex(Number):
    """Complex defines the operations that work on the builtin complex type.

    In short, those are: conversion to complex, bool(), .real, .imag,
    +, -, *, /, **, abs(), .conjugate(), ==, and !=.

    If it is given heterogeneous arguments, and doesn't have special
    knowledge about them, it should fall back to the builtin complex
    type as described below.
    """

    @abstractmethod
    def __complex__(self):
        """Return a builtin complex instance."""

    def __bool__(self):
        """True if self != 0."""
        return self != 0

    @abstractproperty
    def real(self):
        """Retrieve the real component of this number.

        This should subclass Real.
        """
        raise NotImplementedError

    @abstractproperty
    def imag(self):
        """Retrieve the imaginary component of this number.

        This should subclass Real.
        """
        raise NotImplementedError

    @abstractmethod
    def __add__(self, other):
        raise NotImplementedError

    @abstractmethod
    def __radd__(self, other):
        raise NotImplementedError

    @abstractmethod
    def __neg__(self):
        raise NotImplementedError

    def __pos__(self):
        """Coerces self to whatever class defines the method."""
        raise NotImplementedError

    def __sub__(self, other):
        return self + -other

    def __rsub__(self, other):
        return -self + other

    @abstractmethod
    def __mul__(self, other):
        raise NotImplementedError

    @abstractmethod
    def __rmul__(self, other):
        raise NotImplementedError

    @abstractmethod
    def __div__(self, other):
        """a/b; should promote to float or complex when necessary."""
        raise NotImplementedError

    @abstractmethod
    def __rdiv__(self, other):
        raise NotImplementedError

    @abstractmethod
    def __pow__(self, exponent):
        """a**b; should promote to float or complex when necessary."""
        raise NotImplementedError

    @abstractmethod
    def __rpow__(self, base):
        raise NotImplementedError

    @abstractmethod
    def __abs__(self):
        """Returns the Real distance from 0."""
        raise NotImplementedError

    @abstractmethod
    def conjugate(self):
        """(x+y*i).conjugate() returns (x-y*i)."""
        raise NotImplementedError

    @abstractmethod
    def __eq__(self, other):
        raise NotImplementedError

    # __ne__ is inherited from object and negates whatever __eq__ does.

class Real(Complex):
    """To Complex, Real adds the operations that work on real numbers.

    In short, those are: conversion to float, trunc(), math.floor(),
    math.ceil(), round(), divmod(), //, %, <, <=, >, and >=.

    Real also provides defaults for some of the derived operations.
    """

    # XXX What to do about the __int__ implementation that's
    # currently present on float?  Get rid of it?

    @abstractmethod
    def __float__(self):
        """Any Real can be converted to a native float object."""
        raise NotImplementedError

    @abstractmethod
    def __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

    @abstractmethod
    def __floor__(self):
        """Finds the greatest Integral <= self."""
        raise NotImplementedError

    @abstractmethod
    def __ceil__(self):
        """Finds the least Integral >= self."""
        raise NotImplementedError

    @abstractmethod
    def __round__(self, ndigits:Integral=None):
        """Rounds self to ndigits decimal places, defaulting to 0.

        If ndigits is omitted or None, returns an Integral,
        otherwise returns a Real, preferably of the same type as
        self. Types may choose which direction to round half. For
        example, float rounds half toward even.

        """
        raise NotImplementedError

    def __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):
        """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):
        """The floor() of self/other. Integral."""
        raise NotImplementedError

    @abstractmethod
    def __rfloordiv__(self, other):
        """The floor() of other/self."""
        raise NotImplementedError

    @abstractmethod
    def __mod__(self, other):
        """self % other

        See
        https://mail.python.org/pipermail/python-3000/2006-May/001735.html
        and consider using "self/other - trunc(self/other)"
        instead if you're worried about round-off errors.
        """
        raise NotImplementedError

    @abstractmethod
    def __rmod__(self, other):
        """other % self"""
        raise NotImplementedError

    @abstractmethod
    def __lt__(self, other):
        """< on Reals defines a total ordering, except perhaps for NaN."""
        raise NotImplementedError

    @abstractmethod
    def __le__(self, other):
        raise NotImplementedError

    # __gt__ and __ge__ are automatically done by reversing the arguments.
    # (But __le__ is not computed as the opposite of __gt__!)

    # Concrete implementations of Complex abstract methods.
    # Subclasses may override these, but don't have to.

    def __complex__(self):
        return complex(float(self))

    @property
    def real(self):
        return +self

    @property
    def imag(self):
        return 0

    def conjugate(self):
        """Conjugate is a no-op for Reals."""
        return +self

class Rational(Real, Exact):
    """.numerator and .denominator should be in lowest terms."""

    @abstractproperty
    def numerator(self):
        raise NotImplementedError

    @abstractproperty
    def denominator(self):
        raise NotImplementedError

    # Concrete implementation of Real's conversion to float.
    # (This invokes Integer.__div__().)

    def __float__(self):
        return self.numerator / self.denominator

class Integral(Rational):
    """Integral adds a conversion to int and the bit-string operations."""

    @abstractmethod
    def __int__(self):
        raise NotImplementedError

    def __index__(self):
        """__index__() exists because float has __int__()."""
        return int(self)

    def __lshift__(self, other):
        return int(self) << int(other)

    def __rlshift__(self, other):
        return int(other) << int(self)

    def __rshift__(self, other):
        return int(self) >> int(other)

    def __rrshift__(self, other):
        return int(other) >> int(self)

    def __and__(self, other):
        return int(self) & int(other)

    def __rand__(self, other):
        return int(other) &